Download Diffusion Theory of Ion Permeation Through Protein Channels of

Diffusion Theory of Ion Permeation
Through Protein Channels of
Biological Membranes
Thesis submitted for the degree “Doctor of Philosophy”
by
Amit Singer
under the supervision of
Professor Zeev Schuss
Submitted to the senate of Tel Aviv University
June 2005
ii
This work was carried out under the supervision of
Professor Zeev Schuss.
Contents
Preface
Abstract
x
xii
I Diffusion of Independent Particles between Fixed
Concentrations: Analysis and Computation of the
Simulation Problem
1
1 Brownian Simulations and Unidirectional Flux in Diffusion
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Derivation of the Fokker-Planck Equation from a Path Integral
1.3 The Unidirectional Flux of the Langevin Equation . . . . . . .
1.4 The Smoluchowski Approximation to the Unidirectional Current
1.5 The Unidirectional Current in the Smoluchowski Equation . .
1.6 Brownian Simulations . . . . . . . . . . . . . . . . . . . . . . .
1.7 Summary and Discussion . . . . . . . . . . . . . . . . . . . . .
8
9
14
16
17
18
20
24
2 Memoryless Control of Boundary Concentrations of Diffusing Particles
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Equations of Motion . . . . . . . . . . . . . . . . . . .
2.3 Renewal Controls . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Probabilistic Control . . . . . . . . . . . . . . . . . . .
2.3.2 Rate Control . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 The Renewal Control . . . . . . . . . . . . . . . . . . .
2.3.4 Calculation of PL and PR : the Albedo Problem . . . .
26
27
29
31
31
32
36
39
39
iv
CONTENTS
2.4
2.3.5 Concentration Profile and Net Flux . . . . . . . . . . . 40
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 Langevin Trajectories between Fixed Concentrations
44
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2 Trajectories, Fluxes, and Boundary Concentrations . . . . . . 46
3.3 Application to Simulation . . . . . . . . . . . . . . . . . . . . 48
4 Recurrence Time of the Brownian Motion
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Short Time Asymptotics of the Fokker-Planck Equation . . . .
4.2.1 Initial layer? . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 The probability distribution of the RT . . . . . . . . .
4.2.3 Extrapolation length . . . . . . . . . . . . . . . . . . .
4.3 Integral Equation for the pdf of the RT: Short Time Asymptotics
4.3.1 Integral equation for the pdf of the RT . . . . . . . . .
4.3.2 The Laplace method for v > 0 . . . . . . . . . . . . . .
4.3.3 Laplace method for v < 0 . . . . . . . . . . . . . . . .
4.4 Mean Number of Returns . . . . . . . . . . . . . . . . . . . .
4.5 Long Time Asymptotics . . . . . . . . . . . . . . . . . . . . .
4.6 Small v asymptotics . . . . . . . . . . . . . . . . . . . . . . .
5 Narrow Escape in Three Dimensions
5.1 Introduction . . . . . . . . . . . . . . . . .
5.2 General 3D Bounded Domain . . . . . . .
5.2.1 The Neumann function and integral
5.2.2 Elliptic hole . . . . . . . . . . . . .
5.3 Explicit Computations for the Sphere . . .
5.3.1 Collins’ method . . . . . . . . . . .
5.3.2 The asymptotic expansion . . . . .
5.3.3 The MFPT . . . . . . . . . . . . .
5.4 Summary and Applications . . . . . . . . .
. . . . . .
. . . . . .
equations
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
6 Narrow Escape: The Circular Disk
6.1 Introduction . . . . . . . . . . . . . . . . . . .
6.2 Solution of a Mixed Boundary Value Problem
6.2.1 Small ε asymptotics . . . . . . . . . .
6.2.2 Expected lifetime . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
52
53
56
59
60
61
64
65
66
70
74
78
81
84
84
87
88
91
93
95
96
104
105
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
108
. 109
. 111
. 115
. 116
CONTENTS
6.2.3
6.2.4
v
Boundary layers . . . . . . . . . . . . . . . . . . . . . . 117
Flux profile . . . . . . . . . . . . . . . . . . . . . . . . 119
7 Narrow Escape: Riemann Surfaces and Non-Smooth Domains
122
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.2 Asymptotic Approximation to the MFPT on a Riemannian
Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.2.1 Expression of the MFPT using the Neumann function . 127
7.2.2 Leading order asymptotics . . . . . . . . . . . . . . . . 129
7.3 The Annulus Problem . . . . . . . . . . . . . . . . . . . . . . 132
7.4 Domains with Corners . . . . . . . . . . . . . . . . . . . . . . 139
7.5 Domains with Cusps . . . . . . . . . . . . . . . . . . . . . . . 143
7.6 Diffusion on a 2-Sphere . . . . . . . . . . . . . . . . . . . . . . 145
7.6.1 Small absorbing cap . . . . . . . . . . . . . . . . . . . 145
7.6.2 Mapping of the Riemann sphere . . . . . . . . . . . . . 147
7.6.3 Small cap with an absorbing arc . . . . . . . . . . . . . 148
8 Narrow Escape at Short Times
152
8.1 Absorbing Disk . . . . . . . . . . . . . . . . . . . . . . . . . . 152
8.2 Narrow Escape from a Disk . . . . . . . . . . . . . . . . . . . 155
8.3 Narrow Escape in an Annulus . . . . . . . . . . . . . . . . . . 156
II Non-Equilibrium Statistical Mechanics of Interacting Particles
163
9 From Langevin Equations to Partial Differential Equations 166
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
9.2 Standard Continuum Treatments . . . . . . . . . . . . . . . . 170
9.3 Equilibrium Statistical Mechanics of Simple Fluids . . . . . . 172
9.3.1 Equilibrium vs. Non-equilibrium Statistical Mechanics 174
9.4 Non-Equilibrium Statistical Mechanics . . . . . . . . . . . . . 175
9.4.1 A Trajectory Based Approach . . . . . . . . . . . . . . 175
9.4.2 The Fokker-Planck Equation . . . . . . . . . . . . . . . 176
9.5 The C-PNP System . . . . . . . . . . . . . . . . . . . . . . . . 177
9.5.1 The C-PNP Hierarchy . . . . . . . . . . . . . . . . . . 180
9.5.2 PNP revisited . . . . . . . . . . . . . . . . . . . . . . . 182
vi
CONTENTS
9.6
Summary and Discussion . . . . . . . . . . . . . . . . . . . . . 182
10 Maximum Entropy Formulation of the Kirkwood Superposition Approximation
186
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
10.2 Maximum Entropy . . . . . . . . . . . . . . . . . . . . . . . . 189
10.3 Two Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 192
10.3.1 Non-interacting particles in an external field . . . . . . 193
10.3.2 The Thermodynamic Limit . . . . . . . . . . . . . . . 193
10.4 Minimum Helmholtz Free Energy . . . . . . . . . . . . . . . . 195
10.5 Probabilistic Interpretation of the Kirkwood Closure . . . . . 197
10.6 High Level Entropy Closure . . . . . . . . . . . . . . . . . . . 201
10.6.1 The thermodynamic limit . . . . . . . . . . . . . . . . 202
10.6.2 Closure at the highest level n = N − 1 . . . . . . . . . 205
10.7 Confined Systems . . . . . . . . . . . . . . . . . . . . . . . . . 206
10.8 Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
10.9 Discussion and Summary . . . . . . . . . . . . . . . . . . . . . 210
11 Attenuation of the Electric Potential and Field in Disordered
Systems
211
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
11.2 A One-Dimensional Ionic Lattice . . . . . . . . . . . . . . . . 215
11.3 One Dimensional Random Ionic Lattice . . . . . . . . . . . . . 218
11.3.1 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . 219
11.3.2 The electrical potential as a weighted i.i.d. sum . . . . 220
11.3.3 Large and small potentials. The saddle point approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
11.3.4 Tail asymptotics . . . . . . . . . . . . . . . . . . . . . 222
11.4 Random Distances . . . . . . . . . . . . . . . . . . . . . . . . 226
11.5 Dimensions Higher than One . . . . . . . . . . . . . . . . . . . 227
11.5.1 The condition of global electroneutrality . . . . . . . . 227
11.5.2 The condition of local electroneutrality . . . . . . . . . 229
11.5.3 The liquid state . . . . . . . . . . . . . . . . . . . . . . 231
11.6 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . 232
12 Boundary Conditions and a Closure Relation for the Pair
Correlation Function in Non-Equilibrium Diffusion
233
12.1 Infinite system in Steady-State . . . . . . . . . . . . . . . . . . 233
CONTENTS
vii
12.2 Quasi Steady State: The Neumann Problem in Finite Domains 237
12.3 The Electrical Current and the Ramo-Shockley Function . . . 238
12.4 The Connection between the Diffusion Current and the Electric Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
12.4.1 The Steady State . . . . . . . . . . . . . . . . . . . . . 241
12.4.2 Quasi Steady State . . . . . . . . . . . . . . . . . . . . 242
13 Open Problems and Future Research
244
A Algebra Solution of the Albedo Problem
246
A.1 The Stationary Albedo Problem . . . . . . . . . . . . . . . . . 246
A.2 The Algebra Solution . . . . . . . . . . . . . . . . . . . . . . . 247
B Appendix of Chapter 5
B.1 Estimate of kKk2 . . . . . . .
B.1.1 Estimate of the kernel
B.2 Elliptic Hole . . . . . . . . . .
B.3 A Pathological Example . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
250
. 250
. 250
. 251
. 253
C Appendix of Chapter 6
254
C.1 Maximal Exit Time for the Circular Disk . . . . . . . . . . . . 254
C.2 Exit Times along the Ray . . . . . . . . . . . . . . . . . . . . 257
C.3 Flux Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
D Appendix of Chapter 7
267
D.1 Laplace Beltrami Operator on 2-Sphere . . . . . . . . . . . . . 267
List of Figures
1.1
1.2
1.3
1.4
Concentration profile of BD trajectories injected exactly at
the boundary . . . . . . . . . . . . . . . . . . . . . . . . . . .
Concentration profile of BD trajectories injected with the residual distribution . . . . . . . . . . . . . . . . . . . . . . . . . .
Concentration profile of BD trajectories, constant injection
rate, different time steps . . . . . . . . . . . . . . . . . . . . .
Concentration profile BD trajectories, inverse square-root injection rate law, different time steps . . . . . . . . . . . . . . .
20
22
23
24
2.1
The concentration cell of experimental electrochemistry and
molecular biophysics . . . . . . . . . . . . . . . . . . . . . . . 30
3.1
Concentration Profile of LD trajectories: Residual vs. Maxwellian
injection schemes . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1
4.2
Short time asymptotics of the RT PDF . . . . . . . . . . . . . 62
Short time asymptotics of the recurrence time pdf . . . . . . . 75
6.1
6.2
A circular disk with small absorbing boundary . . . . . . . . . 112
The boundary layer near the absorbing boundary . . . . . . . 121
7.1 Annulus with small absorbing boundary . .
7.2 Rectangle with small absorbing boundary . .
7.3 A small opening near a corner of angle α. . .
7.4 Absorbing boundary near a cusp . . . . . . .
7.5 The decapitated sphere . . . . . . . . . . . .
√
8.1 Ray of length 2 2R2 in an annulus . . . . .
8.2 The geometry of a billiard ball in an annulus
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
133
140
142
144
149
. . . . . . . . . . 159
. . . . . . . . . . 160
LIST OF FIGURES
8.3
8.4
The maximal hitting angle constraint in an annulus . . . . . . 161
The shortest ray in annulus that connects the center of the
hole and its antipodal point on the inner circle . . . . . . . . . 162
9.1
Time scales of ionic permeation processes . . . . . . . . . . . . 183
10.1 Configurations of three particles (Kirkwood) . . . . . . . . . . 198
11.1 1D semi infinite lattice of alternating charges . . . . . . . . . . 216
11.2 Integration Contour . . . . . . . . . . . . . . . . . . . . . . . . 223
11.3 2D lattice of dipoles . . . . . . . . . . . . . . . . . . . . . . . 230
12.1 Infinite domain separated by an infinite membrane with a hole 234
ix
Preface
It is a pleasure to thank the following people who made my graduate studies
into an enjoyable and unique experience.
First, I would like to thank my adviser Professor Zeev Schuss. I cannot
imagine a better adviser than Zeev, who not only taught me the fundamentals
of stochastic processes and the way of putting together good science and
mathematics, but also shared with me his vast experience in a way that
helped me in making career and life decisions. I cherish the time we spent
together.
I am mostly grateful to Professor Bob Eisenberg for his vision that the
channel problem must be treated with mathematical rigor and for his belief
in my capabilities as a mathematician. I had wonderful time working in
Bob’s department for over six months during the period of my studies. I was
lucky to be in an environment that encourages scientific independence, and
at the same time I got motivated by Bob’s enthusiasm. I would like to thank
Ardyth and Bob for making me feel at home in Chicago.
Special thanks are due to Dr. Boaz Nadler, whose Ph.D. dissertation ends
where mine starts. Our discussions were most helpful and constructive. I owe
Boaz my connection to Yale University, where my academic path continues.
This is the opportunity to thank Professor Amir Averbuch, Professor Philip
Rosenau, and Professor Koby Rubinstein for their guidance and assistance
along the way.
I want to thank Dr. David Holcman for coming up with the Narrow
Escape problem. I wish that someday I will also be able to make connections
between biology and mathematics the way he does.
I would like to express my appreciation to: Dr. Manor Mendel for calculating pair correlation functions of different closures; Dr. Duan P. Chen
for running trajectory based simulations of the recurrence time problem; Dr.
Shela Aboud for pointing out the depletion phenomenon in Brownian dy-
xi
namics simulations; Professor Douglas Henderson, Professor Roland Roth,
and Professor Stuart A. Rice for discussing with me the statistical mechanics
of liquids; and the anonymous referee of [175], for suggesting the running a
Brownian dynamics simulation.
Finally, I would like to thank my family for their never-ending support and
encouragement. This achievement is due to their investment in my education
throughout the years. Last but definitely not least, I would like to thank Or
for accompanying me during the past few years here and abroad. She is the
proof that life is best with the woman you love.
A.S.
Tel Aviv, Israel
June 2005
Abstract
In this dissertation I answer some fundamental mathematical questions that
arise in predicting the function of protein channels of biological membranes
from their structure. The mathematical problems concern both the analytic
description of the channel function as well as computer simulations. These
problems arise from the molecular level description of the physics of ionic
permeation through the channel. The prediction of the properties of ionic
channels from their known structure is an interdisciplinary field, that involves
biology, chemistry, physics, engineering, computer science, and mathematics.
Also the inverse problem, of reverse engineering of the structure from measured channel properties, is a key problem in molecular biophysics.
None of the existing continuum descriptions of ionic permeation captures
the rich phenomenology of the patch clamp experiment of Neher and Sakmann. It is therefore necessary to resort to particle simulations of the permeation process. Predicting the function of an ionic channel from its structure
by a computer simulation raises the question of connecting a small discrete
simulation volume to a continuum bath. Computer simulations are necessarily limited to a relatively small number of particles, due to computational
difficulties. Therefore, I analyze both Brownian and Langevin dynamics simulations that describe the motion of mobile ions in solution. Both models
reduce the interaction of ions with the solute (water) molecules into friction,
a noise term, and a dielectric constant. A reasonable simulation can describe
only a small portion of the experimental setup of the patch clamp experiment, the channel and its immediate surroundings. The inclusion in the
simulation of a part of the bath and its connection to the surrounding bath
are necessary, because the conditions at the boundaries of the channel are
unknown, while they are measurable in the bath, away from the channel.
The simulation of a test volume in an ionic solution, even without a
membrane and a channel, is by itself an important field of chemical physics.
xiii
There are many algorithms, procedures and protocols for particles at the
interface between the simulation region and the continuum baths (see [139]
for a complete list of references on simulations). However, none of them takes
into account the actual physics at the interface. The failure of these attempts
calls for a theory of the interface that is compatible with the physics and for
the design of simulations based on such a theory.
In Part I of this dissertation I develop such a theory of the interface
between a small simulation volume and the surrounding continuum. In this
theory I assume that the particles in the continuum region interact only
through a mean field and are otherwise independent. This assumption is
permissible, because away from the channel the effects of particle-particle
interactions at biological concentrations are adequately captured by classical
diffusion theory, as described by classical physical chemistry. The effects of
particle-particle interactions are dominant in the confined geometry of the
channel and its immediate neighborhood. A computer simulation in this
neighborhood can include all interactions, because the number of particles
in a reasonable simulation volume is manageable.
The main results of Part I are (i) The discovery of the precise range of validity of diffusion theory (Brownian motion) as a description of ionic motion
in solution. I found the correct way to use diffusion theory in simulations.
Mathematically this is expressed in the determination of the range of validity
of the Smoluchowski approximation to the Langevin equation; (ii) The design
of Brownian and Langevin simulations that do not form spurious boundary
layers, which are ubiquitous in molecular simulations in biology, chemistry,
and physics. This design is based on two new mathematical insights into
the theory of diffusion. One is the splitting of the probability flux of the
Brownian motion into unidirectional components, which is the result of a
new mathematical definition of probability flux in terms of path integrals.
The other is the application of methods of renewal theory to the theory of
diffusion. (iii) The solution of Wang and Uhlenbeck’s recurrence problem,
open since 1945. The solution is based on the conversion of the problem
from solving a boundary value problem for a partial differential equation
to an integral equation and the development of a new ray method for the
construction of its asymptotic solution. (iv) The discovery of how biology
controls many key functions by narrow valves, which mathematically is expressed as the solution of the escape problem of Brownian motion through
a small hole. I found an explicit asymptotic expression for the mean first
passage time in different geometries. Also here all the results are new.
xiv
Abstract
In Part II I develop an analytical theory of diffusion of interacting particles, which describes the permeation process inside the channel and its
immediate neighborhood. The particle-particle interactions in this neighborhood give rise to the rich nonlinear behavior of the channel current: blocking,
saturation, selectivity, and more. The main mathematical problem here is
to determine the pair correlation function of a non-equilibrium system of
interacting particles diffusing between two concentrations. This function is
the cornerstone of the statistical mechanics of simple liquids, including electrolytic solutions. This function determines the nonlinear phenomena mentioned above. Although this function is the subject of intense research in both
equilibrium and non-equilibrium statistical mechanics, the only information
about the non-equilibrium pair correlation function is the recently derived
partial differential equation it satisfies, as a part of an infinite system of coupled partial differential equations. The main results of Part II contain, among
others, (i) The discovery of the boundary conditions of the pair correlation
function in a non-equilibrium system of diffusing interacting particles, (ii) A
variational formulation of Kirkwood’s superposition approximation (closure
relation) for short range interactions, that truncates the infinite coupled system, and (iii) The proof of attenuation (screening) of the electric field and
potential in disordered systems under certain conditions, which renders the
electric interactions short rather than long range. The proof is based on
large deviations theory. In attenuated systems the results of (ii) are applicable. These results lay a part of the foundation of the diffusion theory of
interacting particles that is presently in its infancy.
I find it a fortunate coincidence that old and new tools of applied mathematics, including stochastic processes (Brownian motion and Langevin equation), path integrals, asymptotic methods in partial differential equations,
short and long time expansions, the algebra of commutators, elliptic mixed
boundary value problems, singular perturbations and boundary layer theory,
operator theory, conformal mapping, probability theory, renewal processes,
integral equations, calculus of variations and large deviations theory, combine
to produce new insights that have not been gained by traditional methods
of biology, physics, and chemistry. Even more fortunately, I find new and
old unsolved mathematical problems in the classical theory of diffusion and
make contributions toward their solution.
Part I
Diffusion of Independent
Particles between Fixed
Concentrations: Analysis and
Computation of the Simulation
Problem
2
Ion channels are proteins with a hole down their middle, embedded in
biological cell membranes. They allow ions to move through otherwise impermeable cell membranes, thereby controlling many biological processes of
great importance in health and disease. Ion channels can conduct ions of
one type much better than ions of another type and this selectivity between
ions is one of the most important part of their function. The ionic current
in a protein channel depends on the voltage and the concentrations of all
ionic species on both sides of the membrane. These are measured in the
patch clamp experiment of Neher and Sakmann, in which two baths of salt
solutions of different concentrations are separated by a patch of membrane
containing a single protein channel that connects the two baths. The current,
voltage and concentration are measured in the surrounding bath, far away
from the channel.
Channology is a relatively young science, so the existing mathematical
theories of ionic permeation in protein channels barely scratch the surface of
the mathematical iceberg of the channel problem. Also the present theory is
far from being exhaustive. In the absence of an adequate analytic description
of channel function computer simulations are most valuable. Simulations, by
virtue of their regime of applicability, often become the link between theory
and experiment. Their applications to chemistry include the study of molten
salts, dielectric interfaces, phase changes, and ionic solutions. Simulations
are a valuable tool in solid state device engineering for the design of semiconductors, and in biology similar techniques are used for the study of large
biological molecules such as proteins, enzymes, and nucleic acids, to name
but a few.
Part I of this dissertation is devoted to the development of a theory
of connection of a small simulation volume to the surrounding continuum.
Computer simulations of ions in solution have to be confined to to a relatively small number of mobile ions, due to computational limitations. Thus a
reasonable simulation can describe only a small portion of the experimental
setup of a patch clamp experiment, the channel and its immediate surroundings. The inclusion in the simulation of a part of the bath and its connection
to the surrounding bath are necessary, because the conditions at the boundaries of the channel are unknown, while they are measurable in the bath,
away from the channel.
Thus the trajectories of the particles are described individually for each
particle inside the simulation volume, while outside the simulation volume
they can be described only by their statistical properties. It follows that on
3
one side of the interface between the simulation and the surrounding bath
the description of the particles is discrete, while a continuum description has
to be used on the other side. This poses the fundamental question of how to
describe the particle trajectories at the interface.
The interface between the simulation region and the bath is sufficiently
far from the channel so that the effects of interactions between the mobile
ions and the channel, as well as the short range ion-ion interactions are
negligible. Therefore it suffices to consider the interfacing problem only for
non-interacting particles or for particles interacting with a mean field. Thus,
we assume that random ionic trajectories near and outside the interface are
independent. Specifically, in Chapter 1 we address this problem for Brownian
dynamics (BD) simulations, connected to a practically infinite surrounding
bath by an interface that serves as both a source of particles that enter the
simulation and an absorbing boundary for particles that leave the simulation.
The same problem for Langevin dynamics (LD) simulations is addressed in
Chapter 3.
The mathematical model of the interface is expected to reproduce the
physical conditions that actually exist on the boundary of the simulated
volume. These physical conditions are not merely the average electrostatic
potential and local concentrations at the boundary of the volume, but also
their fluctuation in time. It is important to recover the correct fluctuation,
because the stochastic dynamics of ions in solution are nonlinear, due to the
coupling between the electrostatic field and the motion of the mobile charges,
so that averaged boundary conditions do not reproduce correctly averaged
nonlinear response. However, in a system of noninteracting particles incorrect
fluctuation on the boundary may still produce the correct response outside
a boundary layer in the simulation region.
The boundary fluctuation consists of arrivals of new particles from the
bath into, and of the recirculation of particles in and out of the simulation
volume. The random motion of the mobile charges brings about fluctuations
in both the concentrations and the electrostatic field. Since the simulation
is confined to the volume inside the interface, the new and the recirculated
particles have to be fed into the simulation by a source that imitates the
random influx across the interface. The interface does not represent any
physical device that feeds trajectories back into the simulation, but is rather
an imaginary wall, which the physical trajectories of the diffusing particles
cross and recross any number of times. The efflux of simulated trajectories
through the interface is observed in the simulation, however, the influx of
4
new trajectories, which is the unidirectional flux (UF) of diffusion, has to be
calculated so as to reproduce the random influx with the correct statistical
properties of this stochastic process, as mentioned above. Thus the UF is
the source strength of the influx, and also the stochastic process that counts
the number of trajectories that cross the interface in one direction per unit
time.
We find the source strength needed to maintain average boundary concentrations in a manner consistent with the physical and mathematical assumptions of Brownian particles. Our new insight into the simulation algorithm is
that the source strength has to be inversely proportional to the square root
of the time step and also proportional to the average nominal concentration
at the interface. An additional new insight is the observation that new simulated trajectories should not be started at the interface, but rather injected
into the simulation volume with the residual distribution of the Brownian
dynamics displacement, as though the simulation extends across the interface outside the simulation volume. Otherwise spurious boundary layers are
formed, as has been painfully realized by simulators so far.
The source strength of injection new Brownian trajectories into a discrete
simulation is the UF of discrete diffusion. The concept of UF in diffusion
is not well defined. Specifically, the diffusion equation is a conservation law
for the net flux, but it does not define its unidirectional components, which
have to be identified in order to run a discrete simulation. Our new insight
into this problem is the splitting of the net diffusion flux into its unidirectional components by using the Wiener path integral description of diffusion,
rather than Fick’s description in terms of a partial differential equation. The
definition of the unidirectional diffusion flux is a natural outcome of the path
integral formulation [121]. It is simply the probability density of Brownian
trajectories that cross the given surface in one direction per unit time. The
diffusion equation, whose solution is the Smoluchowski approximation to the
solution of the Fokker-Planck equation for large values of the damping parameter γ, produces infinite UFs, because the Brownian motion is nowhere
differentiable with probability 1. However, the unidirectional flux in the
Fokker-Planck equation corresponding to the Langevin dynamics of the trajectories is finite for all γ, which leads to an apparent paradox. We resolve it
by showing that the unidirectional fluxes of the discretized Langevin dynamics and Brownian dynamics coincide if the time step of a discretized Brownian
dynamics is exactly twice the relaxation time, that is, only if ∆t = 2/γ.
This observation has fundamental significance in the mathematical the-
5
ory of diffusion as a stochastic process. Namely, our analysis indicates that
the mathematical Brownian motion (MBM) is an oversimplified mathematical description of the physical Brownian motion, that is, the MBM description is invalid on a time scale shorter than the relaxation time of the more
realistic Langevin dynamics into Brownian dynamics. In 1905 Einstein [41]
and, independently, in 1906 Smoluchowski [181] offered an explanation of the
Brownian motion based on kinetic theory and demonstrated, theoretically,
that the phenomenon of diffusion is the result of Brownian motion. Einstein’s
theory was later verified experimentally by Perrin [143] and Svedberg [185].
That of Smoluchowski was verified by Smoluchowski [182], Svedberg [186],
and Westgren [196]. To connect his mathematical theory with the “irregular
movement which arises from thermal molecular movement”, Einstein made
the following assumptions [41]: (1) the motion of each particle is independent of the others and (2) “the movement of one and the same particle after
different intervals of time [are] mutually independent processes, so long as
we think of these intervals of time as being chosen not too small.” Thus
Einstein’s theory is based on the assumption that the diffusing particles are
observed intermittently at time intervals that are short, but not too short.
Smoluchowski’s theory was based on Langevin’s more refined description of
the Brownian motion [111]. The study of the MBM on short time scales
pushes diffusion theory beyond its range of applicability. Thus much of the
mathematical phenomenology of the MBM, such as nowhere differentiability, local time, and so on, occurs on time scales that do not correspond to
physical diffusion.
In Chapter 2 we show that the popular memoryless (renewal) injecting
schemes of Langevin dynamics [139, and references therein] lead to the creation of spurious boundary layers, that spread out into the entire simulation
volume due to the long range electric field. These boundary layers are inherited from the solution of the steady state Fokker-Planck equation with
absorbing boundaries, also known as the steady-state albedo problem [20].
Thus, in designing a Langevin simulation, an algorithm has to be developed
to eliminate these boundary layers, because they are spurious at an interface,
as discussed above. This design problem is solved in Chapter 3, where the
correct steady state influx and efflux of Langevin trajectories at the interface is replicated by injecting new Langevin trajectories into the simulation
volume according to a new probability distribution, which is the phase space
(two-dimensional) generalization of the apparently disconnected concept of
residual time distribution of queueing (renewal) theory [97]. Our simulation
6
algorithm maintains average fixed concentrations without creating spurious
boundary layers, consistently with the assumed Langevin dynamics.
The analysis of Langevin trajectories near an absorbing boundary leads to
the following theoretical question. What is the probability density function
(pdf) of the recurrence time (RT) for the free particle motion? This problem
was raised by Wang and Uhlenbeck (1945) [193], who considered a free particle that is initiated at the origin with a positive velocity, and asked when
is the particle due to return to the origin for the first time. Marshall and
Watson (1985) [123] and Hagan, Doering and Levermore (1989) [64] found
the mean RT independently of each other. The original problem posed by
Wang and Uhlenbeck, about the pdf of the recurrence time and the return
velocity remained open and is solved in Chapter 4. Specifically, we construct
a short time asymptotic expansion of the pdf of the RT with ”beyond all orders” accuracy by the ray method [25]. We generalize the method of images
(reflection), that is well known for the one-dimensional diffusion equation,
to the Fokker-Planck equation. This is done by deriving a new renewal-type
integral equation for the pdf and its short time asymptotic solution. We also
find the long time asymptotic expansion of the solution, and its small velocity asymptotic expansion. In Appendix A we present a simplified solution of
the steady state albedo problem by using an algebra of commutators for the
Fokker-Planck operator. The solution of the recurrence problem is the pdf
of the residence time of free particles in the positive axis. This can represent
the sojourn time of a free particle inside a long one-dimensional channel.
Additional key problems in the theory of diffusion and its simulation are
how frequently do Brownian particles, confined to a given volume, escape
through a small hole in the boundary and how are the entrance points distributed on the surface of the hole. The hole may represent the mouth of an
ionic channel embedded in an impermeable membrane. To answer these questions, we consider in Chapter 5 the mean exit time from a three-dimensional
bounded domain whose boundary is reflecting, but for a small absorbing hole
(window). The calculation of the mean exit time is formulated as a mixed
Neumann-Dirichlet boundary value problem in potential theory. We find
an asymptotic expansion for the mean exit time in terms of the geometry
and the singular flux profile through the opening. This answers the above
raised questions. In Chapter 6 we consider the escape problem from a twodimensional simply connected bounded domain, and in Chapter 7 we discuss
the escape problem from a bounded domain with corners and cusps at the
boundary on a general two-dimensional Riemannian manifold as a model of
7
receptor diffusion on the surface of nerve cells. In Chapter 8 we show that
the point from which it is the hardest to exit at short times may be different
from the point where the MFPT is maximal.
Chapter 1
Brownian Simulations and
Unidirectional Flux in Diffusion
The contents of this chapter were published in [175]
The prediction of ionic currents in protein channels of biological membranes is one of the central problems of computational molecular biophysics.
Existing continuum descriptions of ionic permeation fail to capture the rich
phenomenology of the permeation process, so it is therefore necessary to
resort to particle simulations. Brownian dynamics simulations require the
connection of a small discrete simulation volume to large baths that are
maintained at fixed concentrations and voltages. The continuum baths are
connected to the simulation through interfaces, located in the baths sufficiently far from the channel. Average boundary concentrations have to be
maintained at their values in the baths by injecting and removing particles
at the interfaces. The particles injected into the simulation volume represent a unidirectional diffusion flux, while the outgoing particles represent the
unidirectional flux in the opposite direction. The classical diffusion equation
defines net diffusion flux, but not unidirectional fluxes. The stochastic formulation of classical diffusion in terms of the Wiener process leads to a Wiener
path integral, which can split the net flux into unidirectional fluxes. These
unidirectional fluxes are infinite, though the net flux is finite and agrees with
classical theory. We find that the infinite unidirectional flux is an artifact
caused by replacing the Langevin dynamics with its Smoluchowski approximation, which is classical diffusion. The Smoluchowski approximation fails
on time scales shorter than the relaxation time 1/γ of the Langevin equation.
1.1 Introduction
We find that the probability of Brownian trajectories that cross an interface
in one direction in unit time ∆t equals that of the probability of the corresponding Langevin trajectories if γ∆t = 2. That is, we find the unidirectional
flux (source strength) needed to maintain average boundary concentrations
in a manner consistent with the physics of Brownian particles. This unidirectional
flux is proportional to the concentration and inversely proportional to
√
∆t to leading order. We develop a BD simulation that maintains fixed average boundary concentrations in a manner consistent with the actual physics
of the interface and without creating spurious boundary layers.
1.1
Introduction
The prediction of ionic currents in protein channels of biological membranes
is one of the central problems of computational molecular biophysics. None
of the existing continuum descriptions of ionic permeation captures the rich
phenomenology of the patch clamp experiments [73]. It is therefore necessary
to resort to particle simulations of the permeation process [83, 84, 165, 29,
200, 189]. Computer simulations are necessarily limited to a relatively small
number of mobile ions, due to computational difficulties. Thus a reasonable
simulation can describe only a small portion of the experimental setup of a
patch clamp experiment, the channel and its immediate surroundings. The
inclusion in the simulation of a part of the bath and its connection to the
surrounding bath are necessary, because the conditions at the boundaries of
the channel are unknown, while they are measurable in the bath, away from
the channel.
Thus the trajectories of the particles are described individually for each
particle inside the simulation volume, while outside the simulation volume
they can be described only by their statistical properties. It follows that on
one side of the interface between the simulation and the surrounding bath
the description of the particles is discrete, while a continuum description has
to be used on the other side. This poses the fundamental question of how to
describe the particle trajectories at the interface, which is the subject of this
chapter.
We address this problem for Brownian dynamics (BD) simulations, connected to a practically infinite surrounding bath by an interface that serves
as both a source of particles that enter the simulation and an absorbing
boundary for particles that leave the simulation. The interface is expected
9
10
Brownian Simulations and Unidirectional Flux in Diffusion
to reproduce the physical conditions that actually exist on the boundary of
the simulated volume. These physical conditions are not merely the average electrostatic potential and local concentrations at the boundary of the
volume, but also their fluctuation in time. It is important to recover the
correct fluctuation, because the stochastic dynamics of ions in solution are
nonlinear, due to the coupling between the electrostatic field and the motion
of the mobile charges, so that averaged boundary conditions do not reproduce correctly averaged nonlinear response. In a system of noninteracting
particles incorrect fluctuation on the boundary may still produce the correct
response outside a boundary layer in the simulation region [180].
The boundary fluctuation consists of arrival of new particles from the
bath into the simulation and of the recirculation of particles in and out of
the simulation. The random motion of the mobile charges brings about the
fluctuation in both the concentrations and the electrostatic field. Since the
simulation is confined to the volume inside the interface, the new and the
recirculated particles have to be fed into the simulation by a source that
imitates the influx across the interface. The interface does not represent any
physical device that feeds trajectories back into the simulation, but is rather
an imaginary wall, which the physical trajectories of the diffusing particles
cross and recross any number of times. The efflux of simulated trajectories
through the interface is seen in the simulation, however, the influx of new
trajectories, which is the unidirectional flux (UF) of diffusion, has to be
calculated so as to reproduce the physical conditions, as mentioned above.
Thus the UF is the source strength of the influx, and also the number of
trajectories that cross the interface in one direction per unit time.
The mathematical problem of the UF begins with the description of diffusion by the diffusion equation. The diffusion equation (DE) is often considered to be an approximation of the Fokker-Planck equation (FPE) in the
Smoluchowski limit of large damping. Both equations can be written as the
conservation law
∂p
= −∇ · J .
(1.1)
∂t
The flux density J in the diffusion equation is given by
1
J (x, t) = − [ε∇p(x, t) − f (x)p(x, t)] ,
γ
where γ is the friction coefficient (or dynamical viscosity), ε =
(1.2)
kB T
, kB is
m
1.1 Introduction
11
Boltzmann’s constant, T is absolute temperature, and m is the mass of the
diffusing particle. The external acceleration field is f (x) and p(x, t) is the
density (or probability density) of the particles [166]. The flux density in
the FPE is given by where the net probability flux density vector has the
components
Jx (x, v, t) = vp(x, v, t)
(1.3)
Jv (x, v, t) = − (γv − f (x)) p(x, v, t) − εγ∇v p(x, v, t).
The density p(x, t) in the diffusion equation (1.1) with (1.2) is the probability density of the trajectories of the Smoluchowski stochastic differential
equation
r
1
2ε
ẋ = f (x) +
ẇ,
(1.4)
γ
γ
where w(t) is a vector of independent standard Wiener processes (Brownian
motions).
The density p(x, v, t) is the probability density of the phase space trajectories of the Langevin equation
p
(1.5)
ẍ + γ ẋ = f (x) + 2εγ ẇ.
In practically all conservation laws of the type (1.1) J is a net flux density
vector. It is often necessary to split it into two unidirectional components
across a given surface, such that the net flux J is their difference. Such
splitting is pretty obvious in the FPE, because the velocity v at each point
x tells the two UFs apart. Thus, in one dimension,
Z ∞
Z 0
JLR (x, t) =
vp(x, v, t) dv, JRL (x, t) = −
vp(x, v, t) dv
−∞
0
(1.6)
Z
Jnet (x, t) = JLR (x, t) − JRL (x, t) =
∞
vp(x, v, t) dv.
−∞
In contrast, the net flux J(x, t) in the DE cannot be split this way, because
velocity is not a state variable. Actually, the trajectories of a diffusion process
do not have well defined velocities, because they are nowhere differentiable
12
Brownian Simulations and Unidirectional Flux in Diffusion
with probability 1 [39]. These trajectories cross and recross every point x
infinitely many times in any time interval [t, t + ∆t], giving rise to infinite
UFs. However, the net diffusion flux is finite, as indicated in eq.(1.2). This
phenomenon was discussed in detail in [121], where a path integral description
of diffusion was used to define the UF. The unidirectional diffusion flux,
however, is finite at absorbing boundaries, where the UF equals the net
flux. The UFs measured in diffusion across biological membranes by using
radioactive tracer [73] are in effect UFs at absorbing boundaries, because the
tracer is a separate ionic species [24].
An apparent paradox arises in the Smoluchowski approximation of the
FPE by the DE, namely, the UF of the DE is infinite for all γ, while the UF of
the FPE remains finite, even in the limit γ → ∞, in which the solution of the
DE is an approximation of that of the FPE [47]. The “paradox” is resolved
by a new derivation of the FPE for LD from the Wiener path integral. This
derivation is different than the derivation of the DE or the Smoluchowski
equation from the Wiener integral (see, e.g. [102, 164, 112, 54]) by the
method of M. Kac [92]. The new derivation shows that the path integral
definition of UF in diffusion, as first introduced in [121], is consistent with
that of UF in the FPE. However, the definition of flux involves the limit
∆t → 0, that is, a time scale shorter than 1/γ, on which the solution of the
DE is not a valid approximation to that of the FPE.
This discrepancy between the Einstein and the Langevin descriptions of
the random motion of diffusing particles was hinted at by both Einstein and
Smoluchowski. Einstein [41] remarked that his diffusion theory is based on
the assumption that the diffusing particles are observed intermittently at
short time intervals, but not too short, while Smoluchowski [181] showed
that the variance of the displacement of Langevin trajectories is quadratic
in t for times much shorter than the relaxation time 1/γ, but is linear in t
for times much longer that 1/γ, which is the same as in Einstein’s theory of
diffusion [22].
The infinite unidirectional diffusion flux imposes serious limitations on
BD simulations of diffusion in a finite volume embedded in a much larger
bath. Such simulations are used, for example, in simulations of ion permeation in protein channels of biological membranes [73]. If parts of the bathing
solutions on both sides of the membrane are to be included in the simulation,
the UFs of particles into the simulation have to be calculated. Simulations
with BD would lead to increasing influxes as the time step is refined.
The method of resolution of the said “paradox” is based on the defini-
1.1 Introduction
13
tion of the UF of the Langevin dynamics (LD) in terms of the Wiener path
integral, analogous to its definition for the BD in [121]. The UF JLR (x, t)
is the probability per unit time ∆t of trajectories that are on the left of x
at time t and are on the right of x at time t + ∆t. We show that the UF
of BD coincides with that of LD if the time unit ∆t in the definition of the
unidirectional diffusion flux is exactly
∆t =
2
.
γ
(1.7)
We find the strength of the source that ensures that a given concentration is
maintained on the average at the interface in a BD simulation. The strength
of the left source JLR is to leading order independent of the efflux and depends
on the concentration CL , the damping coefficient γ, the temperature ε, and
the time step ∆t, as given in eq.(1.27). To leading order it is
r
JLR =
ε
CL + O
πγ∆t
1
.
γ
(1.8)
We also show that the coordinate of a newly injected particle has the
probability distribution of the residual of the normal distribution. Our simulation results show that no spurious boundary layers are formed with this
scheme, while they are formed if new particles are injected at the boundary.
The simulations also show that if the injection rate is fixed, there is depletion
of the population as the time step is refined, but there is no depletion if the
rate is changed according to eq.(1.8).
In Section 1.2, we derive the FPE for the LD (1.5) from the Wiener
path integral. In Section 1.3, we define the unidirectional probability flux
for LD by the path integral and show that is indeed given by (1.6). In
Section 1.4, we use the results of [47] to calculate explicitly the UF in the
Smoluchowski approximation to the solution of the FPE and to recover the
flux (1.2). In Section 1.5, we use the results of [121] to evaluate the UF
of the BD trajectories (1.4) in a finite time unit ∆t. In the limit ∆t → 0
the UF diverges, but if it is chosen as in (1.7), the UFs of LD and BD
coincide. In Section 1.6 we describe the a BD simulation of diffusion between
fixed concentrations and give results of simulations. Finally, Section 1.7 is a
summary and discussion of the results.
14
Brownian Simulations and Unidirectional Flux in Diffusion
1.2
Derivation of the Fokker-Planck Equation
from a Path Integral
The LD (1.5) of a diffusing particle can be written as the phase space system
ẋ = v,
v̇ = −γv + f (x) +
p
2εγ ẇ.
(1.9)
This means that in time ∆t the dynamics progresses according to
x(t + ∆t) = x(t) + v(t)∆t + o(∆t)
v(t + ∆t) = v(t) + [−γv(t) + f (x(t))]∆t +
(1.10)
p
2εγ ∆w + o(∆t),(1.11)
where ∆w ∼ N (0, ∆t), that is, ∆w is normally distributed with mean 0
and variance ∆t. This means that the probability density function evolves
according to the propagator
Prob{x(t + ∆t) = x, v(t + ∆t) = v} = p(x, v, t + ∆t) = o(∆t) +
Z bZ ∞
1
√
p(ξ, η, t)δ(x − ξ − η∆t)
(1.12)
4εγπ∆t a −∞
)
(
[v − η − [−γη + f (ξ)]∆t]2
dξ dη.
× exp −
4εγ∆t
To understand (1.12), we note that given that the displacement and velocity of the trajectory at time t are x(t) = ξ and v(t) = η, respectively, then
according to eq.(1.10), the displacement of the particle at time t+∆t is deterministic, independent of the value of ∆w, and is x = ξ + η∆t + o(∆t). Thus
the probability density function (pdf) of the displacement is δ(x − ξ − η∆t +
o(∆t)). It follows that the displacement contributes to the joint propagator
(1.12) of x(t) and v(t) a multiplicative factor of the Dirac δ function. Similarly, eq.(1.11) means that the conditional pdf of the velocity at time t + ∆t,
given x(t) = ξ and v(t) = η, is normal with mean η + [−γη + f (ξ)]∆t + o(∆t)
and variance 2γ∆t + o(∆t), as reflected in the exponential factor of the
propagator. If trajectories are terminated at the ends of an finite or infinite
interval (a, b), the integration over the displacement variable extends only to
that interval.
1.2 Derivation of the Fokker-Planck Equation from a Path Integral
The basis for our analysis of the UF is the following new derivation of the
Fokker-Planck equation from eq.(1.12). Integration with respect to ξ gives
Z ∞
1
p(x − η∆t, η, t)
(1.13)
p(x, v, t + ∆t) = o(∆t) + √
4εγπ∆t −∞
(
)
[v − η − [−γη + f (x − η∆t)]∆t]2
× exp −
dη.
4εγ∆t
Changing variables to
v − η − [−γη + f (x − η∆t)]∆t
√
,
2εγ∆t
and expanding in powers of ∆t, the integral takes the form
Z ∞
1
2
p(x, v, t + ∆t) = √
e−u /2 du ×
2π(1 − γ∆t + o(∆t)) −∞
−u =
(1.14)
p(x − v(1 + γ∆t)∆t + o(∆t),
p
v(1 + γ∆t) + u 2εγ∆t − f (x)∆t(1 + γ∆t) + o(∆t), t)
Reexpanding in powers of ∆t, we get
p(x − v(1 + γ∆t)∆t + o(∆t),
p
v(1 + γ∆t) + u 2εγ∆t − f (x)∆t(1 + γ∆t) + o(∆t), t)
∂p(x, v, t)
+
= p(x, v, t) − v∆t
∂x
p
∂p(x, v, t) vγ∆t + u 2εγ∆t − f (x)∆t + o(∆t) +
∂v
εγu2 ∆t
∂ 2 p(x, v, t)
+ o(∆t),
∂v 2
so (1.14) gives
p(x, v, t + ∆t) −
p(x, v, t)
1
∂p(x, v, t)
= −
v∆t
+
1 − γ∆t
1 − γ∆t
∂x
∆t ∂p(x, v, t)
(vγ − f (x)) +
1 − γ∆t
∂v
εγ∆t ∂ 2 p(x, v, t)
3/2
+
O
∆t
.
1 − γ∆t
∂v 2
15
16
Brownian Simulations and Unidirectional Flux in Diffusion
Dividing by ∆t and taking the limit ∆t → 0, we obtain the Fokker-Planck
equation in the form
∂p(x, v, t)
∂p(x, v, t)
∂
∂ 2 p(x, v, t)
= −v
+
[(γv − f (x)) p(x, v, t)] + εγ
,
∂t
∂x
∂v
∂v 2
(1.15)
which is the conservation law (1.1) with the flux components (1.3). The UF
JLR (x1 , t) is usually defined as the integral of Jx (x1 , v, t) over the positive
velocities [47, and references therein], that is,
Z
JLR (x1 , t) =
∞
vp(x1 , v, t) dv.
(1.16)
0
To show that this integral actually represents the probability of the trajectories that move from left to right across x1 per unit time, we evaluate below
the probability flux from a path integral.
1.3
The Unidirectional Flux of the Langevin
Equation
The instantaneous unidirectional probability flux from left to right at a point
x1 is defined as the probability per unit time (∆t), of Langevin trajectories
that are to the left of x1 at time t (with any velocity) and propagate to the
right of x1 at time t + ∆t (with any velocity), in the limit ∆t → 0. This can
be expressed in terms of a path integral on Langevin trajectories on the real
line as
Z
x1
Z
∞
Z
∞
Z
∞
1
p(ξ, η, t) ×
4εγπ∆t
−∞
x1
−∞
−∞
(
)
[v − η − [−γη + f (ξ)]∆t]2
δ(x − ξ − η∆t) exp −
. (1.17)
4εγ∆t
1
JLR (x1 , t) = lim
∆t→0 ∆t
dξ
dx
dη
dv √
1.4 The Smoluchowski Approximation to the Unidirectional Current
Integrating with respect to v eliminates the exponential factor and integration with respect to ξ fixes ξ at x − η∆t, so
Z Z
1
p(x − η∆t, η, t) dη dx
JLR (x1 , t) = lim
∆t→0 ∆t
x−η∆t<x1
Z x1
Z ∞
1
= lim
p(u, η, t) du
dη
∆t→0 ∆t 0
x1 −η∆t
Z ∞
=
ηp(x1 , η, t) dη.
(1.18)
0
The expression (1.18) is identical to (1.16).
1.4
The Smoluchowski Approximation to the
Unidirectional Current
The following calculations were done in [47] and are shown here for completeness. In the overdamped regime, as γ → ∞, the Smoluchowski approximation to p(x, v, t) is given by
2
1
e−v /2
1 ∂p(x, t) 1
− f (x)p(x, t) v + O
p(x, v, t) ∼ √
p(x, t) −
,
γ
∂x
γ2
2π
(1.19)
where the marginal density p(x, t) satisfies the Fokker-Planck-Smoluchowski
equation
γ
∂p(x, t)
∂ 2 p(x, t)
∂
=ε
−
[f (x)p(x, t)] .
2
∂t
∂x
∂x
(1.20)
According to (1.16) and (1.19), the UF is
Z ∞
JLR (x1 , t) =
vp(x1 , v, t) dv
0
2
∞
e−v /2
1 ∂p(x, t) 1
1
=
v√
p(x, t) −
− f (x)p(x, t) v + O
dv
γ
∂x
γ2
2π
0
r
ε
1
∂p(x, t)
1
=
p(x1 , t) −
ε
− f (x)p(x, t) + O
.
(1.21)
2π
2γ
∂x
γ2
Z
17
18
Brownian Simulations and Unidirectional Flux in Diffusion
Similarly, the UF from right to left is
Z
0
JRL (x1 , t) = −
vp(x1 , v, t) dv
−∞
r
=
(1.22)
ε
1
∂p(x, t)
1
p(x1 , t) +
ε
− f (x)p(x, t) + O
.
2π
2γ
∂x
γ2
Both UFs in (1.21) and (1.22) are finite and proportional to the marginal
density at x1 . The net flux is the difference
1 ∂p(x, t)
Jnet (x1 , t) = JLR (x1 , t) − JRL (x1 , t) = − ε
− f (x)p(x, t) ,(1.23)
γ
∂x
as in classical diffusion theory [47], [56].
1.5
The Unidirectional Current in the Smoluchowski Equation
Classical diffusion theory, however, gives a different result. In the overdamped regime the Langevin equation (1.9) is reduced to the Smoluchowski
equation [166]
γ ẋ = f (x) +
p
2εγ ẇ.
(1.24)
As in Section 1.3, the unidirectional probability current (flux) density at a
point x1 can be expressed in terms of a path integral as
JLR (x1 , t) = lim JLR (x1 , t, ∆t),
∆t→0
(1.25)
where
r
∞
∞
γζ 2
dξ
dζ exp −
×
(1.26)
JLR (x1 , t, ∆t) =
4ε
0
ξ
√
ζf (x1 )
∂p(x1 , t)
∆t
p (x1 , t) − ∆t −
p (x1 , t) + (ζ − ξ)
+O
.
2ε
∂x
γ
γ
4πε∆t
Z
Z
1.5 The Unidirectional Current in the Smoluchowski Equation
It was shown in [121] that
r
1
∂p(x1 , t)
ε
JLR (x1 , t, ∆t) =
p(x1 , t) +
f (x1 )p(x1 , t) − ε
πγ∆t
2γ
∂x
√ !
∆t
+O
.
(1.27)
3/2
γ
Similarly,
JRL (x1 , t) = lim JRL (x1 , t, ∆t),
∆t→0
where
r
∞
∞
γζ 2
JRL (x1 , t, ∆t) =
dξ
dζ exp −
×
4ε
0
ξ
√
∂p(x1 , t)
ζf (x1 )
∆t
p (x1 , t) + (ζ − ξ)
+O
.
p (x1 , t) + ∆t −
2ε
∂x
γ
(1.28)
√ !
r
ε
∆t
1
∂p(x1 , t)
=
p(x1 , t) −
f (x1 )p(x1 , t) − ε
+O
.
πγ∆t
2γ
∂x
γ 3/2
γ
4πε∆t
Z
Z
If p(x1 , t) > 0, then both JLR (x1 , t) and JRL (x1 , t) are infinite, in contradiction to the results (1.21) and (1.22). However, the net flux density is
finite and is given by
Jnet (x1 , t) =
lim {JLR (x1 , t, ∆t) − JRL (x1 , t, ∆t)}
∆t→0
1
∂
= − ε p(x1 , t) − f (x1 )p(x1 , t) ,
γ ∂x
(1.29)
which is identical to (1.23).
The apparent paradox is due to the idealized properties of the Brownian motion. More specifically, the trajectories of the Brownian motion, and
therefore also the trajectories of the solution of eq.(1.24), are nowhere differentiable, so that any trajectory of the Brownian motion crosses and recrosses
the point x1 infinitely many times in any time interval [t, t + ∆t] [86]. Obviously, such a vacillation creates infinite UFs.
Not so for the trajectories of the Langevin equation (1.9). They have
finite continuous velocities, so that the number of crossing and recrossing is
finite. We note that setting γ∆t = 2 in equations (1.27) and (1.28) gives
(1.21) and (1.22).
19
20
Brownian Simulations and Unidirectional Flux in Diffusion
Figure 1.1: The concentration profile of Brownian trajectories that are initiated at x = 0 with a normal distribution, and terminated at either x = 0 or
x = 1.
1.6
Brownian Simulations
Here we design and analyze a BD simulation of particles diffusing between
fixed concentrations. Thus, we consider the free Brownian motion (i.e., f = 0
in eq. (1.4)) in the interval [0, 1]. The trajectories were produced as follows:
a) According to the dynamics
r (1.4), new trajectories that are started at
2ε
|∆w|; b) The dynamics progresses accordx(−∆t) = 0 move to x(0) =
γ
r
2ε
ing to the Euler scheme x(t + ∆t) = x(t) +
∆w; c) The trajectory is terγ
minated if x(t) > 1 or x(t) < 0. The parameters are ε = 1, γ = 1000, ∆t = 1.
We ran 10,000 random trajectories and collected their statistics with the results shown in Figure 1.1.
1.6 Brownian Simulations
21
The simulated concentration profile is linear, but for a small depletion
layer near the left boundary x = 0, where new particles are injected. This is
inconsistent with the steady state DE, which predicts a linear concentration
profile in the entire interval [0, 1]. The discrepancy stems from part (a) of the
numerical scheme, which assumes that particles enter the simulation interval
exactly at x = 0. However, x = 0 is just an imaginary interface. Had the
simulation been run on the entire line, particles would hop into the simulation
across the imaginary boundary at x = 0 from the left, rather than exactly
at the boundary. This situation is familiar from renewal theory [97]. The
probability distribution of the distance an entering particle covers, not given
its previous location, is not normal, but rather it is the residual of the normal
distribution, given by
Z 0
(x − y)2
dy,
(1.30)
f (x) = C
exp −
2σ 2
−∞
2ε∆t
where σ 2 =
and C is determined by the normalization condition
γ
Z ∞
f (x) dx = 1. This gives
0
r
f (x) =
π
erfc
2σ
x
√
2σ
.
(1.31)
Rerunning the simulation with the entrance pdf f (x), we obtained the
expected linear concentration profile, without any depletion layers (see Figure
1.2).
Injecting particles exactly at the boundary makes their first leap into the
simulation too large, thus effectively decreasing the concentration profile near
the boundary.
Next, we changed the time step ∆t of the simulation, keeping the injection
rate of new particles constant. The population of trajectories inside the
interval was depleted when the time step was refined (see Figure 1.3). A
well behaved numerical simulation is expected to converge as the time step
is refined, rather than to result in different profiles. This shortcoming of
refining the time step is remedied by replacing the constant rate sources
with time-step-dependent sources, as predicted by eqs.(1.27)-(1.28). Figure
1.4 describes the concentration profiles for √
three different values of ∆t and
source strengths that are proportional to 1/ ∆t. The concentration profiles
22
Brownian Simulations and Unidirectional Flux in Diffusion
Figure 1.2: The concentration profile of Brownian trajectories that are initiated at x = 0 with the residual of the normal distribution, and terminated
at either x = 0 or x = 1.
1.6 Brownian Simulations
Figure 1.3: The concentration profile of Brownian trajectories that are initiated at x = 0 and terminated at either x = 0 or x = 1. Three different time
steps (∆t = 4, 1, 0.25) were used, but the injection rate of new particles remained constant. Refining the time step decreases the concentration profile.
23
24
Brownian Simulations and Unidirectional Flux in Diffusion
Figure 1.4: The concentration profile of Brownian trajectories that are initiated at x = 0 and terminated at either x = 0 or x = 1. Three different
time steps (∆t = 4, 1, 0.25)√are shown, and the injection rate of new particles is proportional to 1/ ∆t. Refining the time step does not alter the
concentration profile.
now converge when ∆t → 0. The key to this remedy is the calculation of the
UF in diffusion.
1.7
Summary and Discussion
Both Einstein [41] and Smoluchowski [181] pointed out that BD is a valid
description of diffusion only at times that are not too short. More specifically,
the Brownian approximation to the Langevin equation breaks down at times
shorter than 1/γ, the relaxation time of the medium in which the particles
diffuse.
1.7 Summary and Discussion
In a BD simulation of a channel the dynamics in the channel region may
be much more complicated than the dynamics near the interface, somewhere
inside the continuum bath, sufficiently far from the channel. Thus the net
flux is unknown, while the boundary concentration is known. It follow that
the simulation should be run with source strengths (1.27), (1.28),
r
r
1
1
ε
ε
CL + Jnet , JRL ∼
CR − Jnet .
JLR ∼
πγ∆t
2
πγ∆t
2
r
ε
However, Jnet is unknown, so neglecting it relative to
CL,R will lead
πγ∆t
to steady state boundary concentrations that are close, but not necessarily
equal to CL and CR . Thus a shooting procedure has to be adopted to adjust
the boundary fluxes so that the output concentrations agree with CL and
CR , and then the net flux can be readily found.
According to (1.27) and (1.28), the efflux and influx remain finite at the
boundaries, and agree with the fluxes of LD, if the time step in the BD
2
near the boundary. If the time step
simulation is chosen to be ∆t =
γ
is chosen differently, the fluxes remain finite, but the simulation does not
recover the UF of LD. At points away from the boundary, where correct UFs
do not have to be recovered, the simulation can proceed in coarser time steps.
The above analysis can be generalized to higher dimensions. In three
dimensions the normal component of the UF vector at a point x on a given
smooth surface represents the number of trajectories that cross the surface
from one side to the other, per unit area at x in unit time. Particles cross
the interface in one direction if their velocity satisfies v · n(x) > 0, where
n(x) is the unit normal vector to the surface at x, thus defining the domain
of integration for eq.(1.6).
The time course of injection of particles into a BD simulation can be
chosen with any inter injection probability density, as long as the mean time
between injections is chosen so that the source strength is as indicated in
(1.27) and (1.28). For example, these times can be chosen independently of
each other, without creating spurious boundary layers.
25
Chapter 2
Memoryless Control of
Boundary Concentrations of
Diffusing Particles
The contents of this chapter were published in [180]
Flux between regions of different concentration occurs in nearly every
device involving diffusion, whether an electrochemical cell, a bipolar transistor, or a protein channel in a biological membrane. Diffusion theory has
calculated that flux since the time of Fick (1855) [52], and the flux has been
known to arise from the stochastic behavior of Brownian trajectories since
the time of Einstein (1905) [41], yet the mathematical description of the behavior of trajectories corresponding to different types of boundaries is not
complete. We consider the trajectories of non-interacting particles diffusing
in a finite region connecting two baths of fixed concentrations. Inside the
region, the trajectories of diffusing particles are governed by the Langevin
equation. To maintain average concentrations at the boundaries of the region
at their values in the baths, a control mechanism is needed to set the boundary dynamics of the trajectories. Different control mechanisms are used in
Langevin and Brownian simulations of such systems. We analyze models of
controllers and derive equations for the time evolution and spatial distribution of particles inside the domain. Our analysis shows a distinct difference
between the time evolution and the steady state concentrations. While the
time evolution of the density is governed by an integral operator, the spatial
distribution is governed by the familiar Fokker-Planck operator. The bound-
2.1 Introduction
ary conditions for the time dependent density depend on the model of the
controller; however, this dependence disappears in the steady state, if the
controller is of a renewal type. Renewal-type controllers, however, produce
spurious boundary layers that can be catastrophic in simulations of charged
particles, because even a tiny net charge can have global effects. The design
of a non-renewal controller that maintains concentrations of non-interacting
particles without creating spurious boundary layers at the interface requires
the solution of the time-dependent Fokker-Planck equation with absorption
of outgoing trajectories and a source of ingoing trajectories on the boundary
(the so called albedo problem).
2.1
Introduction
We consider particles that diffuse between two regions where average concentrations are maintained at constant unequal values (see fig. 2.1). Flux
between regions of different concentration occurs in nearly every device involving diffusion, whether an electrochemical cell, a bipolar transistor, or a
protein channel in a biological membrane. Continuum theories of such diffusive systems describe the concentration field by the (time independent)
Nernst-Planck equation with fixed boundary concentrations [73, 42, 12, 140,
167, 89, 170].
The microscopic theory underlying diffusion describes motion of particles
by Langevin’s equations [12, 167, 47, 109, 166] everywhere, except at the
boundaries. The behavior of the Langevin trajectories at the boundaries depends on the interaction between the particles and the boundaries. Thus, for
example, outgoing trajectories can be terminated (absorbed); reflected (or
otherwise reinjected); delayed; and so on. None of this is described by the
Langevin equations. Brownian dynamics cannot describe such boundary behavior, because Brownian particles have no definite velocity, being functions
of infinite variation. Particles with positive (e.g., incoming) velocities can
be distinguished from those with negative (e.g., outgoing) velocities, only if
their velocity is well defined [47]. The Langevin equations are often directly
integrated in simulations [3, 132, 28, 11, 16, 139, 134, 135, 82, 83, 84].
In devices, the interaction between the trajectories and the boundaries
must be specified because the inputs, outputs, and power supplies of devices are at their boundaries; in physical systems, the boundaries are where
charge, matter, and energy are injected into a device; in biological systems
27
28
Memoryless Control of Boundary Concentrations of Diffusing Particles
boundaries represent reservoirs maintained at a (nearly) fixed electrochemical
potential by active processes of the cell.
The formulation of boundary conditions for the particle concentration is
obvious in macroscopic models, but formulation of boundary conditions for
the underlying trajectories is not so clear cut, particularly because many
different physical or computational control mechanisms can maintain a constant average density at prescribed locations, usually near the boundaries
[3, 132, 28, 11, 16, 82, 165, 29, 200, 189, 190, 191, 157]. Many boundary conditions used in Brownian and Langevin simulations produce spurious boundary layers, that do not exist at those locations in the physical systems being
simulated. Spurious boundary layers are particularly damaging to simulations of charged particles. A boundary layer leads to large fluctuations in
the electrostatic field which spreads over the entire simulation region. This
was clearly demonstrated in [139] for a problem with equal boundary concentrations in a simulation with a buffer zone.
In this chapter we provide a general description of the concentration and
flux of non-interacting particles diffusing between constant average concentrations near the boundaries. We study renewal-type controllers that maintain fixed concentrations near the boundaries, determining the time course
both of concentration (in phase space) and current. This kind of controllers
are oft en used in simulations. We show that the concentration is a weighted
sum of “left” and “right” concentrations, each of which satisfies a different integro-partial-differential equation and different boundary conditions.
In the steady state the phase space concentration is the weighted sum of
the solutions of two stationary solutions of the so called albedo problem
[96, 198, 199, 171, 20, 187]. The albedo problem was first posed by Wang
and Uhlenbeck [193] in 1945 and its analytic solution was first found by
Marshall and Watson [123]. Further progress was made by Hagan, Doering,
and Levermore [64, 65], who used complex analysis to solve the half range
expansion problem. The solution employed here was found by Klosek [103].
The weights in the sum of “left” and “right” concentrations are the rates at
which the controllers inject trajectories into the system.
Different control mechanisms that maintain the same concentrations near
the boundaries produce different time operators that govern the evolution
of the “left” and “right” concentrations. Each evolution is non-Markovian.
The removal and injection—or re-injection—of particles into the system by
renewal-type boundary controllers are described by renewal-type integral operators that govern the time evolutions of these concentrations, in contrast
2.2 Formulation
to the Fokker-Planck or Nernst-Planck equations that are commonly used.
The description of this simplified model of diffusion of non-interacting
particles is apparently new: we include a detailed description of the physical
mechanism that maintains the non-equilibrium state of the system. Similar
descriptions are needed when particles interact.
2.2
Formulation
We consider a system composed of two finite macroscopic volumes containing
electrolyte solutions of different ionic species, connected by a macroscopic or
microscopic channel. A control mechanism keeps different average concentrations in the two volumes, so that a steady current flows through the system
(see fig. 2.1), thus keeping it out of equilibrium. As seen in the figure, the
control mechanism is located only on parts of the boundaries of the system,
at macroscopic distances away from the connecting channel. The control
mechanism re-injects exiting trajectories at one or the other boundaries in
a way that maintains average fixed concentrations near the boundaries at
all times. We have in mind, for example, a typical setup used to measure
the diffusion of ions through a protein channel of a biological cell membrane
that separates two solutions of different fixed concentrations [73]. Alternatively, all trajectories are reflected at the boundary so that the system
reaches equilibrium after a long time, but the long lasting transient regime is
the non-equilibrium regime in which an almost steady current flows between
the baths. This time behavior occurs when the number of particles that flow
through the channel during the period of measurement is much smaller than
the total number of ions in either bath. The problem at hand is to describe
the steady diffusion current flowing between the two baths, in terms of the
molecular properties of the diffusing ions, such as their radii and interaction
forces, as a function of the experimentally controlled variables, such as the
concentrations in the two baths and the external potential, and as a function
of the system geometry, e.g., the geometry and charge distribution of the
channel.
The particles diffuse in a domain Ω that consists of the two macroscopic
volumes and the connecting channel. We assume that there are N h ions of
species
h (h = Ca++ , Na+ , Cl− , . . . ) in Ω, which are numbered at time t = 0,
P
h
h
h
h
h
h N = N , and we follow their trajectories, xj (t) = (xj (t), yj (t), zj (t)) at
all times t > 0 (xhj (t) is the location of the j-th ion of species h at time t).
29
30
Memoryless Control of Boundary Concentrations of Diffusing Particles
Figure 2.1: The concentration cell of experimental electrochemistry and
molecular biophysics. The region Ω typically consists of small parts of two
large baths of effectively constant concentrations, separated by a permeable
membrane in experimental electrochemistry, or (in biophysics) an impermeable membrane containing one or more channels.
2.3 Renewal Controls
31
For future use, the coordinate and velocity vectors of all ions in the 3N dimensional configuration space, are denoted
by
˙ or ṽ.
x̃ = xh1 1 , . . . , xhN1h1 , xh1 2 , . . . , xhN2h2 , . . . and x̃
2.2.1
Equations of Motion
As in [167], we assume that the motion of an ion in the solution is overdamped
diffusion in a field of force. The source of the noise and friction is the thermal
motion of the solvent (e.g., water) and both are interrelated by Einstein’s
fluctuation-dissipation principle [12]. More specifically, our starting point is
a memoryless system of N coupled Langevin equations for the dynamics of
all particles of the different species h = Ca++ , Na+ , Cl− , . . . ,
ẍhj + γ h xj ẋhj =
h
f hj
(x̃)
+
Mh
s
2γ h xhj kB T h
ẇj ,
Mh
(j = 1, 2, . . . , N h ),
(2.1)
where a dot on top of a variable means differentiation with respect to time,
γ h (xh ) is the location dependent friction coefficient per unit mass and M h is
the effective mass of an ion of species h. The force f hj (x̃) on the j-th ion of
species h includes all ion-ion interactions and thus depends on the locations
of all ions. The functions ẇhj are, by assumption, independent standard
Gaussian white noises. The parameter kB is Boltzmann’s constant and T
is absolute temperature. As seen in fig. 2.1, some parts of the boundary
∂Ω are reflecting, while other parts contain the control mechanism. At the
boundary ∂Ω, the random trajectories of the Langevin equations (2.1) are
either reflected or are redirected by the external control mechanism.
2.3
Renewal Controls
The solution of (2.1) depends on the specific choice of control mechanism.
We first analyze controls for one-dimensional non-interacting systems because
the treatment of three-dimensional interacting particle systems is more complicated. In this section we show that renewal controls (to be defined in
subsection 2.3.3) reproduce correct macroscopic properties such as total net
flux and concentration profile, but also produce non-physical boundary layers
for non-interacting diffusive particle systems.
32
Memoryless Control of Boundary Concentrations of Diffusing Particles
Consider particles diffusing in the interval Ω = [0, d]. The control mechanism maintains average concentrations CL and CR at 0 xL < xR d,
respectively, away from the boundaries, where concentrations are actually
measured. Each particle satisfies a Langevin equation
p
(2.2)
ẍ + γ ẋ + U 0 (x) = 2γ ẇ.
In order to complete the description of the dynamics we have to describe
the motion of particles at the boundaries, i.e., to describe the action of the
control mechanism.
2.3.1
Probabilistic Control
A possible control mechanism operates as follows: when a particle reaches
either one of the boundaries, it tosses a Bernoulli coin with probabilities
(L, R), L + R = 1, L, R ≥ 0. The control mechanism decides to re-enter the
particle at the left boundary x = 0 with probability L, and to re-inject the
particle to the bath at the right boundary x = d with probability R. The
re-injections occur at random times; a particle that reached the boundary at
time t, is delayed in the boundary a random time T and re-injected at time
t + T . The random time T is a non-negative random variable with pdf
q(s) ds = Prob{s ≤ T < s + ds}.
(2.3)
The velocity of injection is distributed according to pre-determined distributions sL (v) and sR (v) of the left and right sources, respectively. For example,
if both sources are Maxwellian, then
sL (v) = √
2 −v2 /2
e
= sR (−v),
2π
v>0
(2.4)
As shown below, the precise velocity distribution of the sources is unimportant for measurement of concentrations away from the boundaries.
The dynamics (2.1) and the boundary behavior provide a complete description of the trajectories, and therefore determine the probability distribution of the random particle trajectories in the system at any time. Assuming,
as we may, that the precise velocity distributions is unimportant, there are
only two parameters to be determined, namely the fixed number of particles
in the system N and the re-injection probability R. These two parameters
determine uniquely the two measured concentrations CL and CR .
2.3 Renewal Controls
33
Let pi (x, v, t) be the probability of finding the i − th particle at location
x and velocity v at time t, given that it was injected to the bath at time
t = 0, from either the left or right boundary with probabilities R and L,
and the corresponding velocity distributions sL and sR . Since the particles
are independent and interchangeable, we find that p1 = p2 = . . . = pN , and
set p(x, v, t) = p1 (x, v, t). Let p(x, v) be the steady state density of a single
particle, i.e., p(x, v) = limt→∞ p(x, v, t). The steady state concentration at
location x is given by
Z ∞
C(x) = N
p(x, v) dv.
(2.5)
−∞
We use renewal theory [97] to calculate p(x, v). Suppose the device was
turned on at time t = 0. Let t0 be the first time that the particle was injected
into the system. Then the probability of finding the particle in location (x, v)
of the phase space at time t is given by
Z t
p(x, v, t) =
p(x, v, t|t0 = s)q(s) ds.
(2.6)
0
Let τ1 be the first passage time of the particle to the boundary. Conditioning
on τ1 yields
Z ∞
Z t
p(x, v, t|t0 = s, τ1 = r)p(τ1 = r|t0 = s) dr, (2.7)
q(s) ds
p(x, v, t) =
0
0
where p(τ1 = r|t0 = s) = p(τ = r −s) = 0 for r < s. We separate the integral
into two parts
Z t
Z t
q(s) ds
p(x, v, t|t0 = s, τ1 = r)p(τ = r − s) dr
p(x, v, t) =
0
0
Z t
Z ∞
+
q(s) ds
p(x, v, t|t0 = s, τ1 = r)p(τ = r − s) dr
0
t
Z t
Z t
=
q(s) ds
p(x, v, t − r)p(τ = r − s) dr + f (x, v, t), (2.8)
0
0
where
Z
f (x, v, t) =
t
Z
0
∞
p(x, v, t|t0 = s, τ1 = r)p(τ = r − s) dr.
q(s) ds
t
(2.9)
34
Memoryless Control of Boundary Concentrations of Diffusing Particles
Changing the order of integration in (2.8) we obtain
Z t
Z t
p(x, v, t − r) dr
q(s)p(τ = r − s) ds + f (x, v, t)
p(x, v, t) =
0
0
Z t
=
p(x, v, t − r)(pτ ∗ q)(r) dr + f (x, v, t)
0
= (p ∗ pτ ∗ q)(t) + f (x, v, t),
(2.10)
where ∗ denotes convolution. Taking the Laplace transform of the equation
gives
p̂(x, v, θ) =
fˆ(x, v, θ)
.
1 − p̂τ (θ)q̂(θ)
(2.11)
The steady state distribution is given by
θfˆ(x, v, θ)
.
θ→0 1 − p̂τ (θ)q̂(θ)
p(x, v) = lim p(x, v, t) = lim θp̂(x, v, θ) = lim
t→∞
θ→0
(2.12)
Both numerator and denominator of the right hand side vanish as θ tends to
0. Expanding the denominator in Taylor series, we find that
p(x, v) =
fˆ(x, v, θ = 0)
,
hτ i + hT i
(2.13)
where hτ i is the mean first passage time (MFPT), and hT i is the mean delay
time before re-injection.
To evaluate fˆ(x, v, θ = 0), we consider a Langevin particle in the interval
[0, d] which is injected at time t = 0 at x = 0 with velocity distribution
sL (v). When the particle reaches one of the boundaries, it is absorbed, and
its trajectory is terminated at once. Let p̃L (x, v, t) be the probability density
function of the particle (p should not to be confused with p̃L ; the subscript
L stands for left.) The density p̃L satisfies the Fokker-Planck equation
∂ p̃L
∂ p̃L
∂
∂ 2 p̃L
= Lx,v p̃L = −v
+
[(γv + U 0 (x)) p̃L ] + γ 2 ,
∂t
∂x
∂v
∂v
(2.14)
with the initial condition
p̃L (x, v, t = 0) = δ(x − 0+ )sL (v),
(2.15)
2.3 Renewal Controls
35
and the absorbing boundary conditions
p̃L (x = 0− , v, t) = 0, v > 0,
(2.16)
p̃L (x = d, v, t) = 0, v < 0.
(2.17)
Equations (2.14)-(2.17) define the time dependent albedo problem. In the
limit of high friction a new time scale is often used [166]
t̂ = t/γ,
(2.18)
1 ∂ p̃L
= Lx,v p̃L .
γ ∂ t̂
(2.19)
so eq.(2.14) is rewritten as
We define the function
Z
PL (x, v) =
∞
0
1
p̃L (x, v, t̂) dt̂ =
γ
Z
∞
p̃L (x, v, t) dt.
(2.20)
0
The function γPL (x, v) is the average time that a particle spends at location
(x, v) prior to its absorption, given that it was injected from the left electrode
at time t = 0. It follows from equations (2.14)-(2.17) that PL , the solution
of the steady state albedo problem, satisfies
1
Lx,v PL = − δ(x − 0+ )sL (v),
γ
(2.21)
with the absorbing boundary conditions
PL (x = 0− , v) = 0, v > 0,
(2.22)
PL (x = d, v) = 0, v < 0.
The MFPT to the boundary hτL i of a particle that was injected from the left
electrode is given by
Z
d
Z
∞
hτL i =
γPL (x, v) dx dv.
0
−∞
(2.23)
36
Memoryless Control of Boundary Concentrations of Diffusing Particles
Similarly, we define γPR as the mean time spent by a trajectory at the point
(x, v) prior to its absorbtion, given that it was injected to the bath from the
right electrode at x = d at time t = 0. The function PR satisfies similar
equations, and its integral is the MFPT hτR i.
Using the definition of f (x, v, t), equation (2.9), and changing the order
of integration, we find that
Z ∞
ˆ
f (x, v, t) dt
(2.24)
f (x, v, θ = 0) =
0
Z ∞ Z t
Z ∞
=
dt
q(s) ds
p(x, v, t|t0 = s, τ1 = r)p(τ = r − s) dr
0
0
∞
Z
t
Z
0
Z
p(τ = r − s) dr
q(s) ds
=
∞
s
r
p(x, v, t|t0 = s, τ1 = r) dt
s
We identify the inner two integrals as the mean total time that a particle
had spent
in the (x, v) location of phase space prior to its first absorbtion.
Z
∞
Since
q(s) ds = 1, we find that
0
fˆ(x, v, θ = 0) = LγPL (x, v) + RγPR (x, v),
and
p(x, v) =
LγPL (x, v) + RγPR (x, v)
,
LhτL i + RhτR i + hT i
(2.25)
(2.26)
from which the concentration in phase space is given by
C(x, v) = N p(x, v) = N
LγPL (x, v) + RγPR (x, v)
.
LhτL i + RhτR i + hT i
(2.27)
Equation (2.27) relates the probabilistic control mechanism to its resulting
phase space steady state concentration, that satisfies the steady state FokkerPlanck equation with flux boundary conditions (2.21).
2.3.2
Rate Control
Another possible renewal control consists of two sources, placed at the left
and right boundaries, which inject particles into the system. When a particle reaches the right or left boundary, its trajectory is terminated at once.
2.3 Renewal Controls
37
The sources inject particles at identical independent distributed (i.i.d) interarrival random times TL and TR , whose probability density functions are
fL (t) and fR (t), respectively. The rates of injection are defined as
λL =
1
,
hTL i
λR =
1
.
hTR i
(2.28)
Note that the number of particles in the system does not remain fixed for
this rate control mechanism. For any rectangle A ⊂ [0, d] × R, we denote by
NAL (t) the number of particles in A at time t, that were originated at the left
source. Then NAL (t) satisfies a set of renewal equations [97]
Prob{NAL (t) = 0} = Prob{(x(t), v(t)) ∈
/ A}
Z t
Z
L
fL (s)Prob{NA (t − s) = 0} ds +
·
0
∞
fL (s) ds .
t
(2.29)
Prob{NAL (t)
= 1} = Prob{(x(t), v(t)) ∈ A}
Z t
Z
L
fL (s)Prob{NA (t − s) = 0} ds +
·
0
∞
fL (s) ds
t
+Prob{(x(t), v(t)) ∈
/ A} ·
Z t
fL (s)Prob{NAL (t − s) = 1} ds.
(2.30)
0
Prob{NAL (t) = n} = Prob{(x(t), v(t)) ∈ A} ·
Z t
fL (s)Prob{NAL (t − s) = n − 1} ds
0
+Prob{(x(t), v(t)) ∈
/ A} ·
Z t
fL (s)Prob{NAL (t − s) = n} ds,
0
n > 1.
(2.31)
38
Memoryless Control of Boundary Concentrations of Diffusing Particles
Thus, the expected value of NAL (t) is given by
hNAL (t)i
=
∞
X
nProb{NAL (t) = n}
n=1
= Prob{(x(t), v(t)) ∈ A} ·
Z t
Z
L
fL (s)hNA (t − s)i ds +
∞
fL (s) ds
0
0
t
Z
fL (s)hNAL (t − s)i ds
+Prob{(x(t), v(t)) ∈
/ A} ·
0
Z
t
= Prob{(x(t), v(t)) ∈ A} +
fL (s)hNAL (t − s)i ds. (2.32)
0
Dividing by the area |A| of A and taking the limit |A| → 0, we obtain the
number of particles per unit length and per unit velocity, which we call the
phase space density C L (x, v, t). It satisfies the renewal equation
C L (x, v, t) = p̃L (x, v, t) + (fL ∗ C L )(x, v, t),
(2.33)
Taking the Laplace transform with respect to t, we find that
Ĉ L (x, v, θ) =
p̃ˆL (x, v, θ)
,
1 − fˆL (θ)
(2.34)
and the steady state density is given by
θp̃ˆL (x, v, θ)
θ→0 1 − fˆ (θ)
L
C L (x, v) = lim C L (x, v, t) = lim
t→∞
=
p̃ˆL (x, v, 0)
= λL p̃ˆL (x, v, 0).
hTL i
(2.35)
We obtain from (2.20) that
p̃ˆL (x, v, 0) =
Z
∞
p̃L (x, v, t) dt = γPL (x, v).
(2.36)
0
The linearity of the expectation implies that the steady state concentration
is
C(x, v) = C L (x, v) + C R (x, v) = γλL PL (x, v) + γλR PR (x, v).
(2.37)
2.3 Renewal Controls
2.3.3
39
The Renewal Control
Even though the two control models described above are different, and have
different time evolution (e.g., the number of particles inside the domain is
bounded by N for the former, and unbounded for the latter), they have
identical steady state phase space concentrations. Indeed, choosing
λL =
NL
,
LhτL i + RhτR i + hT i
λR =
NR
,
LhτL i + RhτR i + hT i
(2.38)
we find that eqs. (2.27) and (2.37) are identical. This is no mere coincidence:
both controls are special cases of renewal controls.
Definition 1. A source that injects particles into the domain at random
times
0 = T0 ≤ T1 ≤ T2 . . . ≤ Tn ≤ . . ., such that Yn = Tn − Tn−1 are i.i.d with
hY1 i < ∞ is called a renewal source.
Definition 2. A control made of renewal sources located at the absorbing
boundary of the domain is called a renewal control.
Theorem 1. The steady state phase space concentration of a renewal control
1
1
, λR =
are the rates of
is given by equation (2.37), where λL =
L
hY1 i
hY1R i
the left and right renewal sources, respectively.
Proof. The proof is given in the previous subsection.
2.3.4
Calculation of PL and PR : the Albedo Problem
As seen above, all renewal control mechanisms require the knowledge of PL
and PR , which are the solutions of the steady state albedo problem. It was
shown in [103] that PL is given by
PL (x, v) = √
1
2
e−v /2 e−U (x)/ Q(x, v),
2π
(2.39)
OU T
where Q = QLBL + QR
, with QL,R
BL + Q
BL the boundary layer solutions,
which decay exponentially fast away from the boundaries, and QOU T the
outer solution, given by
Z x
1 U (x)/
U (z)/
OU T
+ D + O(γ −2 ),
(2.40)
dz − ve
Q
(x, v) = C
e
γ
0
40
Memoryless Control of Boundary Concentrations of Diffusing Particles
with
eU (0)/ ζ 12 + B0L
√ Rd
+ O(γ −2 ),
C =
U
(z)/
γ e
dz
0
eU (0)/
1
L
D = − √
ζ
+ B0 + O(γ −2 ),
2
γ (2.41)
where B0L is a constant that depends on the velocity distribution
of the left
source, and ζ denotes the Riemann zeta function (ζ 21 = −1.46035 . . .).
The outer solution QOU T approximates Q at distances O(γ −1 ) away from the
boundaries. Similar expressions can be written for PR .
2.3.5
Concentration Profile and Net Flux
Equation (2.27) gives the concentration at x, which is established by the
probabilistic control mechanism, as
C(x) =
N γ (LPL (x) + RPR (x))
.
LhτL i + RhτR i + hT i
(2.42)
Therefore,
LPL (x1 ) + RPR (x1 )
CL
=
.
CR
LPL (x2 ) + RPR (x2 )
(2.43)
We now solve equation (2.43) for the yet-undetermined parameter L that
keeps constant concentrations CL and CR . Since L = 1 − R, the solution is
given by
L=
CR PR (x1 ) − CL PR (x2 )
.
CL [PL (x2 ) − PR (x2 )] − CR [PL (x1 ) − PR (x1 )]
(2.44)
Substituting in equation (2.42) we find that
Nγ
CL
=
,
hτ i + hT i
LPL (x1 ) + RPR (x1 )
(2.45)
and the two parameters of the control mechanism, N and L, are uniquely
determined. We assume that the left and right sources have the same velocity
2.4 Discussion
41
density distribution, sL (v) = sR (−v), v > 0, which guarantees B0L = B0R ≡
B0 . The resulting concentration at x away from the boundary is given by
Z x
Z x2
U (z)/
(U (x2 )−U (x))/
(U (x1 )−U (x))/
eU (z)/ dz
CL e
e
dz + CR e
x1
x
Z x2
C(x) =
,
U (z)/
e
dz
x1
(2.46)
which
is the same as given in eq.(3.5) of [47]. Note that the constant factor
ζ 12 + B0 cancels out and therefore it cannot be seen in measuring concentrations. The total net flux is given by
Z ∞
Z ∞
Nγ
vp(x, v) dv =
J(x) = N
vP (x, v) dv,
(2.47)
·
hτ i + hT i −∞
−∞
where P = LPL + RPR . The flux is constant and to leading order in γ −1 is
given by
CL eU (x1 )/ − CR eU (x2 )/
Z x2
J=
.
(2.48)
γ
U (z)/
e
dz
x1
We see that the macroscopic net flux (2.48) is O(γ −1 ), and coincides with that
given in eq.(3.7) of [47] and in [73]. Theorem 1 then implies that eqs.(2.46)
and (2.48) describe the concentration and the flux for all renewal control
mechanisms.
2.4
Discussion
The renewal controls studied here maintain systems of non interacting particles at constant average concentrations near the boundaries, and away from
the boundaries they produce the stationary Nernst-Planck equation of classical diffusion theory.
We have proven that all renewal controls produce the same steady state
concentration and flux, even though their time evolutions can differ qualitatively. However, renewal controls—that are widely used in computer
simulations—are problematic because they produce spurious boundary layers. These boundary layers are expected to appear in interacting particle
systems driven out of equilibrium by renewal controls.
42
Memoryless Control of Boundary Concentrations of Diffusing Particles
The existence of such boundary layers may be of little importance if the
particles interact only through short range forces, such as Lennard-Jones
forces, or the forces that prevent overlap of hard spheres. However, the
boundary layers can have a catastrophic effect for particles that interact
through long range forces, such as ions that interact electrostatically. The
net charge carried by only a tiny fraction of the total number of ions is,
after all, responsible for electrical signalling in the nervous system and the
electrical potentials in electrochemical cells and these potentials extend over
large distances, from micron to many meters, e.g., in the neurons of whales
[194] as well as in inorganic applications from batteries to the trans-Atlantic
cable [75, 161, 162, 163, 61].
The boundary behavior of diffusing particles has been studied for many
types of boundaries, including absorbing, reflecting, sticky boundaries, and
more [120, 97]. In [165] a sequence of Markovian jump processes is constructed such that their transition probability densities converge to the solution of the Nernst-Planck equation with given boundary conditions, including
fixed concentrations and sticky boundaries.
As mentioned above, replacing the baths with renewal sources is a mathematical idealization that can produce artificial boundary effects. The renewal
control effectively terminates trajectories at boundaries and starts new trajectories there. Most experiments do not. In real physical systems, particles
that reach the boundary usually move into a ‘guard’ region, from which they
often return to the domain (with some probability), with a given time distribution. To capture this behavior by a mathematical model, the entire pdf
of the first passage time for the albedo problem has to be found, not only
its first moment. The spurious boundary layers will be avoided if the correct
time course of recycling trajectories in and out of the domain is used. We
postpone this calculation, which we could not find in the literature, to a
future paper.
The time evolution of systems whose average concentrations near the
boundaries are maintained by renewal controls is complicated and cannot
be described, in general, by a single partial differential equation. We have
shown that the phase space concentration is a sum of two components, each of
which satisfies a different integral-partial-differential equation with different
boundary conditions. Only in the steady state does the concentration satisfy
the Fokker-Planck equation with boundary conditions identical to those of
the steady state albedo problem. Although the overdamped limit is a useful
approximation inside the domain, it cannot be used near the boundaries,
2.4 Discussion
where the full Fokker-Planck equation has to be solved. For particle systems
with only short range interactions, the outer solution—which is the solution
to the Smoluchowski equation—determines the concentration and correlation
functions away from the boundaries. One can hope that a simple boundary
condition can be found for such systems, similar to the simple boundary
condition that exists for non interacting systems.
From the theoretical point of view, the absence of a rigorous mathematical
theory of the boundary behavior of Brownian trajectories diffusing between
fixed concentrations, based on the physical theory of the Brownian motion, is
a serious gap in classical physics. This chapter is a step toward the bridging
of this gap.
43
Chapter 3
Langevin Trajectories between
Fixed Concentrations
The contents of this chapter were published in [137]
We consider the trajectories of particles diffusing between two infinite
baths of fixed concentrations connected by a channel, e.g. a protein channel
of a biological membrane. The steady state influx and efflux of Langevin
trajectories at the boundaries of a finite volume containing the channel and
parts of the two baths is replicated by termination of outgoing trajectories
and injection according to a residual phase space density. We present a simulation scheme that maintains averaged fixed concentrations without creating
spurious boundary layers, consistent with the assumed physics. This complements the results of [47] and comprises the solution of the problem of
connecting a finite Langevin dynamics simulation to a continuum bath.
3.1
Introduction
We consider particles that diffuse in a domain Ω connecting two regions,
where fixed, but possibly different concentrations are maintained by connection to practically infinite reservoirs. This is the situation in the diffusion of
ions through a protein channel of a biological membrane that separates two
salt solutions of different fixed concentrations [73].
Continuum theories of such diffusive systems describe the concentration
field by the Nernst-Planck equation (NPE) with fixed boundary concentrations [73, 12, 140, 167]. On the other hand, the underlying microscopic theory
3.1 Introduction
of diffusion describes the motion of the diffusing particles by Langevin’s equations [12], [167, 47, 166]. This means that on a microscopic scale there are
fluctuations in the concentrations at the boundaries. The question of the
boundary behavior of the Langevin trajectories (LT), corresponding to fixed
boundary concentrations, arises both in theory and in the practice of particle
simulations of diffusive motion [3, 28, 11, 16, 139, 134, 135, 82].
When the concentrations are maintained by connection to infinite reservoirs, there are no physical sources and absorbers of trajectories at any definite location in the reservoir or in Ω. The boundaries in this setup can be
chosen anywhere in the reservoirs, where the average concentrations are effectively fixed. Nothing unusual happens to the LT there. Upon reaching the
boundary they simply cross into the reservoir and may cross the boundary
back and forth any number of times. Limiting the system to a finite region
necessarily puts sources and absorbers at the interfaces with the baths, as
described in [175].
The boundary behavior of diffusing particles in a finite domain Ω has
been studied in various cases, including absorbing, reflecting, sticky boundaries, and many other modes of boundary behavior [120], [97]. In [165] a
sequence of Markovian jump processes is constructed such that their transition probability densities converge to the solution of the Nernst-Planck
equation with given boundary conditions, including fixed concentrations and
sticky boundaries. Brownian dynamics simulations with different boundary
protocols seem to indicate that density fluctuations near the channels are
independent of the boundary conditions, if the boundaries are moved sufficiently far away from the channel [29]. However, as shown in [180], many
boundary protocols for maintaining fixed concentrations lead to the formation of spurious boundary layers, which in the case of charged particles may
produce large long range fluctuations in the electric field that spread throughout the entire simulation volume Ω. The analytic structure of these boundary
layers was determined in [123, 64], following several numerical investigations
(e.g, [20]).
It seems that the boundary behavior of LT of particles diffusing between
fixed concentrations has not been described mathematically in an adequate
way. From the theoretical point of view, the absence of a rigorous mathematical theory of the boundary behavior of LT diffusing between fixed concentrations, based on the physical theory of the Brownian motion, is a serious
lacuna in classical physics.
It is the purpose of this letter to analyze the boundary behavior of LT
45
46
Langevin Trajectories between Fixed Concentrations
between fixed concentrations and to design a Langevin simulation that does
not form spurious boundary layers. We find the joint probability density
function of the velocities and locations, where new simulated LT are injected
into a given simulation volume, while maintaining the fixed concentrations.
As the time step decreases the simulated density converges to the solution
of the Fokker-Planck equation (FPE) with the imposed boundary conditions
without forming boundary layers.
3.2
Trajectories, Fluxes, and Boundary Concentrations
We assume fixed concentrations CL and CR on the left and right interfaces
between Ω and the baths B, respectively, with all other boundaries of Ω being
impermeable walls, where the normal particle flux vanishes. We assume that
the particles interact only with a mean field, whose potential is Φ(x), so the
diffusive motion of a particle in the channel and in the reservoirs is described
by the Langevin equation (LE)
p
2γ(x)ε ẇ
ẍ + γ(x)ẋ + ∇x Φ(x) =
(3.1)
x(0) = x0 , v(0) = v 0 ,
where γ(x) is the (state-dependent) friction per unit mass, ε is a thermal
factor, and ẇ is a vector of standard independent Gaussian δ-correlated
white noises [166].
The probability density function (pdf) of finding the trajectory of the
diffusing particle at location x with velocity v at time t, given its initial
position, satisfies the Fokker-Planck equation (FPE) in the bath and in the
reservoirs,
∂p
= −v · ∇x p + γ(x)ε∆v p
∂t
h
i
+∇v · γ(x)v + ∇x Φ(x) p,
(3.2)
p(x, v, 0 | x0 , v 0 ) = δ(x − x0 , v − v 0 ).
In the Smoluchowski limit of large friction the stationary solution of (3.2)
3.2 Trajectories, Fluxes, and Boundary Concentrations
47
admits the form [47]
e−|v | /2ε
p(x, v) =
(2πε)3/2
2
J (x) · v
+O
p(x) +
ε
1
γ2
(3.3)
where the flux density vector J (x) is given by
1
1
J (x) = −
ε∇p(x) + p(x) ∇Φ(x) + O
,
γ (x)
γ2
and p(x) satisfies
1
−∇ · J (x) = ∇ ·
ε∇p(x) + p(x) ∇Φ (x) = 0.
γ (x)
In one dimension, the stationary pdfs of velocities of the particles crossing
the interface into the given volume are
pL (v) ∼
2
Jv
e−v /2ε
√
1+
εCL
2πε
J
1
√
+
2 CL 2πε
for v > 0,
(3.4)
−v 2 /2ε
pR (v) ∼
e
Jv
√
1−
εCR
2πε
J
1
√
+
2 CR 2πε
for v < 0,
where J is the net probability flux through the channel. The source strengths
(unidirectional fluxes at the interfaces) are given by [47]
r
JL =
ε
J
CL −
+O
2π
2
ε
J
CR +
+O
2π
2
1
γ2
1
γ2
(3.5)
r
JR =
.
48
Langevin Trajectories between Fixed Concentrations
3.3
Application to Simulation
Langevin simulations of ion permeation in a protein channel of a biological
membrane have to include a part of the surrounding bath, because boundary conditions at the ends of the channel are unknown. The boundary of
the simulation has to be interfaced with the bath in a manner that does not
distort the physics. This means that new LT have to be injected into the simulation at the correct rate and with the correct distribution of displacement
and velocity, for otherwise, spurious boundary layers will form [180].
Consider a single simulated trajectory that jumps according to the discretized LE (3.1)
x(t + ∆t) = x(t) + v(t)∆t,
(3.6)
p
v(t + ∆t) = v(t)(1 − γ∆t) − ∇x Φ(x(t))∆t + 2εγ ∆w(t),
where ∆w is normally distributed with zero mean and variance ∆t. The
trajectory is terminated when it exits Ω for the first time. The problem at
hand is to determine an injection scheme of new trajectories into Ω such that
the interface concentrations are maintained on the average at their nominal
values CL and CR and the simulated density profile satisfies (3.3).
To be consistent with (3.3), the injection rate has to be equal to the
unidirectional flux at the boundary (3.5). New trajectories have to be injected
with displacement and velocity as though the simulation extends outside
Ω, consistently with the scheme (3.6), because the interface is a fictitious
boundary. The scheme (3.6) can move the trajectory from the bath B into
Ω from any point ξ ∈ B and with any velocity η. The probability that a
trajectory, which is moved with time step ∆t from the bath into Ω, or from
Ω into the bath will land exactly on the boundary is zero. It follows that the
pdf of the point (x, v), where the trajectory lands in Ω in one time step, at
time t0 = t + ∆t, say, given that it started at a bath point (ξ, η) (in phase
space) is, according to (3.6),
Pr{x(t0 ) = x, v(t0 ) = v | x(t) = ξ, v(t) = η}
(3.7)
(
=
|v − η + (γη + ∇Φ(ξ))∆t|2
δ(x − ξ − η∆t)
exp
−
(4πεγ∆t)3/2
4εγ∆t
)
+ o(∆t).
3.3 Application to Simulation
49
The stationary pdf p(ξ, η) of such a bath point is given in (3.3). The conditional probability of such a landing is
Pr{x, v | x ∈ Ω, ξ ∈ B} =
Z
Z
dη
dξ Pr{v(t0 ) = v, x(t0 ) = x | ξ, η}p(ξ, η)
3
R
B
,
Pr{x ∈ Ω, ξ ∈ B}
(3.8)
where the denominator is a normalization constant such that
Z
Z
dv dx Pr{x, v | x ∈ Ω, ξ ∈ B} = 1.
R3
Ω
Thus the first point of a new trajectory is chosen according to the pdf (3.8)
and the subsequent points are generated according (3.6), that is, with the
transition pdf (3.7), until the trajectory leaves Ω. By construction, this
scheme recovers the joint pdf (3.3) in Ω, so no spurious boundary layer is
formed.
As an example, we consider a one-dimensional Langevin dynamics simulation of diffusion of free particles between fixed concentrations on a given
interval. Assuming that in a channel of length L
√
(CL − CR ) ε
CL ,
γL
which means that γ is sufficiently large, the flux term in eq.(3.3) is negligible
relative to the concentration term. The concentration term is linear with
slope J and thus can be approximated by a constant, so that p(ξ) = p(0) +
O (γ −1 ) in the left bath. Actually, the value of p(0) 6= 0 is unimportant,
because it cancels out in the normalized pdf (3.8), which comes out to be
v2
2ε[1 + (γ∆t)2 ]
p
2ε∆t 1 + (γ∆t)2
exp −
Pr{x, v | x > 0, ξ < 0} =
(3.9)
s
×erfc 
1 + (γ∆t)2
4εγ∆t

x
1 − γ∆t 
−v
.
∆t
1 + (γ∆t)2
50
Langevin Trajectories between Fixed Concentrations
In the limit ∆t → 0 we obtain from eq.(3.9)
Pr{x, v | x > 0, ξ < 0} →
2δ(x)H(v) −v2 /2ε
√
e
,
2πε
(3.10)
where H(v) is the Heaviside unit step function. This means that with the said
approximation, LT enter at x = 0 with a Maxwellian distribution of positive
velocities. Without the approximation the limiting distribution of velocities
is (3.4). Note, however, that injecting trajectories by any Markovian scheme,
with the limiting distribution (3.10) and with any time step ∆t, creates a
boundary layer [180].
A LD simulation with CL 6= 0, CR = 0, and the parameters γ = 100, ε =
1, L = 1, ∆t = 10−4 with 25000 trajectories, once with a Maxwellian distribution of velocities at the boundary x = 0 (red) and once with the pdf (3.9)
(blue) shows that a boundary layer is formed in the former, but not in the
latter (see Figure 3.1).
An alternative way to interpret eq.(3.9) is to view the simulation (3.6) as
a discrete time Markovian process (x(t), v(t)) that never enters or exits Ω
exactly at the boundary. If, however, we run a simulation in which particles
are inserted at the boundary, the time of insertion has to be random, rather
than a lattice time n∆t. Thus the time of the first jump from the boundary
into the domain is the residual time ∆t0 between the moment of insertion
and the next lattice time (n + 1)∆t. The probability density of jump size in
both variables has to be randomized with ∆t0 , with the result (3.9).
3.3 Application to Simulation
Figure 3.1: Left panel: Concentration against displacement of a LD simulation with injecting particles according to the residual distribution (3.9)
(blue), and according to the Maxwellian velocity distribution (3.10) exactly
at the boundary (red). The two graphs are almost identical, except for a
small boundary layer near x = 0 in red. Right panel:
√ Zoom in of the concentration profile in the boundary layer x < 0.01 = ε/γ.
51
Chapter 4
Recurrence Time of the
Brownian Motion
Consider the motion of a diffusing particle governed by the Langevin equation. The particle is initiated at the origin x = 0 with a specific velocity
v0 > 0 and its trajectory is terminated once it returns to the origin. In 1945,
Wang and Uhlenbeck [193] posed the problem of finding the probability density function of return at time t, calling it “the recurrence time problem”.
We partially answer this question by showing that the pdf of the recurrence
time (RT) has a “beyond all orders” short time asymptotic expansion. We
find that the asymptotic pdf is in good agreement with Langevin dynamics
simulations. The Laplace transform of the pdf of the RT can be expanded
in eigenfunctions using the half range expansion technique. However, this
procedure does not reveal much about the pdf. Instead, we find an integral
equation for the RT density and extract its short time asymptotics. We find
a reflection rule for the Fokker-Planck equation for rays hitting an absorbing
boundary, in a manner that extends the reflection rule of the diffusion equation (method of images in one dimension; eikonals in higher dimensions). A
closely related problem is that of first passage times to x = 0 of particles
initiated away from the boundary at x0 > 0, often called the Milne problem.
Hagan, Doering and Levermore (1988) [65] solved this problem by constructing an initial time layer of size 2/γ, when particles almost do not see the
absorbing boundary at the origin prior to that time. However, in the RT
problem, the particle is initiated at the location of the absorbing boundary,
so arbitrarily short return times are possible. Indeed, our analysis, as well as
numerical simulations, show that a large percentage of the particles (depend-
4.1 Introduction
53
ing on the initial velocity) return at times shorter than 2/γ, even though the
mean first passage time is infinite (as in the Gambler’s Ruin paradox [51]).
We find the velocity distribution at the RT. We extend our results to the
case of a Brownian motion with an external constant (gravitational) force
field, and find that the beyond all orders short time asymptotic expansion is
altered only by a change in the pre-exponential factor. In addition, we find
the long time asymptotics of the RT pdf, and its asymptotic behavior for
small velocities. We find that the mean number of crossings of the initiation
point in the limit of small initial velocity is asymptotically logarithmic in v0 .
4.1
Introduction
The one dimensional Brownian motion is described by the Langevin equation
p
(4.1)
ẍ + γ ẋ = −g + 2γε ẇ,
where γ is the friction (damping) parameter, ε is the noise strength (temperature), g is the force field and ẇ is a δ-correlated white (Gaussian) noise.
Wang and Uhlenbeck (1945) [193] posed several problems in the theory of
the Brownian motion, some of which are yet to be solved. The RT problem
is to find the probability distribution of the first time that the Brownian
motion returns to the location it was initiated at t = 0. A different problem,
also posed in [193], is the first passage time problem, which is to find the
probability time distribution of the Brownian motion for hitting a specific
point located away from the location of initiation for the first time. Both
problems remained open for many years, resisting all solution attempts, even
for the free Brownian motion in the absence of external field, i.e., g = 0.
Let p(x, v, t| x0 , v0 ) be the probability density function (pdf) to find the
diffusive particle at time t at position x with velocity v, given it was initiated
at time t = 0 at location x0 ≥ 0 with initial velocity v0 . In the absence of an
external field, the pdf satisfies the Fokker-Planck equation
∂p
∂
∂2p
∂p
= −v
+
(γvp) + εγ 2 ,
∂t
∂x ∂v
∂v
with the initial condition
p(x, v, t = 0) = δ(x − x0 )δ(v − v0 ),
(4.2)
54
Recurrence Time of the Brownian Motion
indicating the position and velocity at t = 0, and the absorbing boundary
condition
p(x = 0, v, t) = 0, v > 0.
(4.3)
The absorbing boundary condition reflects the fact that the first passage
time of particles with and without absorption is the same. The motion after
this time is irrelevant, so we may assume that the particle is absorbed, and
there are no incoming trajectories from left to right (characterized by positive
velocity) at the origin. In the RT problem x0 = 0, that is, the particle is
initiated at the location of the absorbing boundary with a positive velocity
v0 > 0, whereas the first passage time to the origin problems are identified
with x0 > 0 and the initial velocity is not necessarily positive. In both
problems the goal is to find the pdf p(x, v, t).
What makes these problems hard is the form in which the absorbing
boundary condition (4.3) is given for half the velocity space. Other boundary
conditions, such as reflecting boundary conditions (for g = 0), or moving the
absorbing boundary to infinity, are much simpler. For example, replacing the
absorbing boundary condition (4.3) with boundary conditions at x = ±∞,
oversimplifies the problem, whose explicit solution was given in [22]. The
reflecting boundary problem is solved by the method of images, however, the
absorbing boundary problem cannot be solved by this method.
Integrating the Fokker-Planck equation (4.2) over timeZ gives a time∞
independent problem for the steady state density P (x, v) =
p(x, v, t) dt.
0
The steady state equation has a complete infinite set of eigenfunctions and
eigenvalues. However, half the eigenfunctions are growing exponentials of
the displacement variable and are ruled out of the expansion. Therefore, the
problem is to find the expansion of a function, which is prescribed only for
half the space v > 0 (4.3), in terms of half the eigenfunctions. Beals and
Protopopescu [9] proved completeness of this half-range expansion problem,
yet the determination of the expansion coefficients remained unresolved for
two additional years.
Titulaer et al. [96, 198, 199, 171, 20, 187] used several numerical approximation methods to solve the steady state problem. The RT and the first
passage time problems were also renamed: the RT problem was renamed
the albedo problem, whereas the first passage time problem was renamed
the Milne problem, after the equivalent problem in transport theory. The
4.1 Introduction
Fokker-Planck equation is often approximated by the Smoluchowski (diffusion) equation. The absorbing boundary condition (4.3) is then replaced
by the absorbing condition p(−xM ) = 0, where xM is the so called Milne
extrapolation length. Attempts to calculate the Milne extrapolation length
include numerical approximations [96, 198, 199, 171, 20, 187], moment closure assumptions [68, 149], and postulates about the form of the flux at the
boundary [150]. These attempts yielded estimates of the Milne extrapolation
length between 1.44 and 1.46.
The half-range expansion problem was solved analytically by Marshall
and Watson (1985) [123], who used the Wiener-Hopf method. Their analysis gave the exact value of the Milne extrapolation length xM = |ζ(1/2)| =
1.46035 . . ., where ζ is the Riemann zeta function. Hagan, Doering and Levermore (1989) [64] found an equivalent (somewhat) simplified solution by
introducing a special complex variable function, and calculated the mean
first passage time (MFPT) for a free Brownian motion initiated away from
two absorbing boundaries. Klosek [103] extended their analysis and calculated the mean RT in the overdamped limit γ 1 in the presence of an
external force field, and in the presence of a second far absorbing boundary.
Marshall and Watson [124] presented a simplified calculation of the MFPT
and deduced the asymptotic distribution of passage times for large t in the
presence of an attracting field. Marshall, Watson and Duck [37] developed a
numerical scheme to calculate the MFPT in the presence of an arbitrary force
field, and presented results for an harmonic oscillator, for which an analytic
solution is unknown.
The distribution of the first passage time (the time-dependent Milne problem) was first determined in [65], where a uniform expansion for both short
and long times was found. The analysis in [65] is based on constructing an
initial time layer of size t0 = 2/γ. The probability to exit before t0 is transcendentally small in γ, as the particle is initiated away from the absorbing
boundary. This technique was extended in [104], where the first passage time
distribution in the presence of an external field was found. These papers contain the solution to Wang and Uhlenbeck’s first passage time problem.
Yet, the RT problem of Wang and Uhlenbeck remained unresolved. Marshall and Watson [123] computed the Laplace transform of the RT distribution in terms of infinite weighted sum of eigenfunctions. However, quoting
[123, p. 3542] “The Laplace inversion ... appears to be out of question.”
The initial time layer method [65] cannot be applied in the RT problem,
because, as seen below, particles are absorbed in arbitrary small times with
55
56
Recurrence Time of the Brownian Motion
non-negligible probabilities.
In this Chapter we solve Wang and Uhlenbeck’s problem by finding the
short time asymptotics of the pdf of the recurrence time. Our approach differs
from that of the above mentioned references in that instead of the traditional
study of the Fokker-Planck equation, we derive an integral equation for the
pdf of the RT. The derivation is based on the concept of unidirectional flux
and on a renewal argument. We construct an asymptotic solution to the
integral equation by developing an appropriate ray method for short times.
We find that the short time asymptotic solution is a superposition of exactly
two rays. The first ray is due to the short time asymptotics of the free
Brownian motion [22] (that does not see the boundary), whereas the second
ray, that has a a different (larger) exponent, is due to the absorbing boundary
at the origin. The second ray is the main result of this Chapter. In principle,
the second ray could have been recovered by re-summation of the Laplace
transformed half range expansion (with respect to time). However, this resummation seems to be unfeasible, as indicated by [123]. The long time
asymptotics of the pdf of the RT, and the small velocity expansion are also
calculated.
A closely related problem is that of the recurrence of the integrated Brownian motion. This problem was studied by McKean [128] and its generalizations were studied by Sinai [173], Goldman [60], Groeneboom et al. [63],
Isozaki and Watanabe [85], Lachal [110] and others. The Wang-Uhlenbeck
RT problem reduces asymptotically to McKean’s winding problem at times
t 1/γ. The short-time asymptotic expansion of McKean’s exact solution
contains the two first terms in the expansion of the solution of the WangUhlenbeck RT problem.
4.2
Short Time Asymptotics of the FokkerPlanck Equation
The phase space trajectories of the Langevin equation
ẍ + γ ẋ + V 0 (x) =
p
2εγ ẇ
(4.4)
are the pair (x(t), v(t)), where ẋ = v. The joint transition probability density
of the pair in the entire phase plane is the solution of the Fokker-Planck
4.2 Short Time Asymptotics of the Fokker-Planck Equation
57
equation [166]
∂p(x, v, t | x0 , v0 )
∂p(x, v, t | x0 , v0 )
= −v
∂t
∂x
∂
+ {[γv + V 0 (x)] p(x, v, t | x0 , v0 )}
∂v
(4.5)
2
+εγ
∂
p(x, v, t | x0 , v0 ) for t > 0, (x, v, x0 , v0 ) ∈ R4 ,
2
∂v
with the initial condition
p(x, v, 0 | x0 , v0 ) = δ(x − x0 )δ(v − v0 ) for (x, v, x0 , v0 ) ∈ R4 .
For example, in the FPE for a free Brownian motion in a constant external
field V 0 (x) = g, the solution is given by [22]
1
exp − GR2 − 2HRS + F S 2 /2(F G − H 2 ) ,
pc (x, v, t | x0 , v0 ) = √
2π F G − H 2
(4.6)
where
R = x − x0 − γ −1 v0 (1 − e−γt ) + gγ −2 γt − 1 + e−γt ,
S = v − v0 e−γt + gγ −1 1 − e−γt ,
(4.7)
and
F = εγ −2 2γt − 3 + 4e−γt − e−2γt ,
G = ε(1 − e−2γt ),
H = εγ −1 (1 − e−γt )2 .
(4.8)
The marginal pdf of the velocity is
[v − v0 e−γt + gγ −1 (1 − e−γt )]2
exp −
,
p(v, t | v0 ) = p
2ε(1 − e−2γt )
2πε(1 − e−2γt )
(4.9)
and the marginal pdf of the displacement is
s
γ2
p(x, t | x0 , v0 ) =
(4.10)
2πε [2γt − 3 + 4e−γt − e−2γt ]
(
)
2
γ 2 [x − x0 − v0 γ −1 (1 − e−γt ) − gγ −2 (1 − e−γt − γt)]
× exp −
.
2ε [2γt − 3 + 4e−γt − e−2γt ]
1
58
Recurrence Time of the Brownian Motion
The solution (4.6) of the free particle problem has the short time asymptotic expansion is (the dependence on x0 , v0 is suppressed)
ψ0 (x) + tψ1 (x, v) + t2 ψ2 (x, v)
pc (x, v, t | x0 , v0 ) ∼ exp −
εγt3
∞
X
×
tn−3 Zn (x, v) for γt 1,
(4.11)
n=1
where the eikonals ψ0 (x), ψ1 (x, v), ψ2 (x, v) are given by
ψ0 (x) = 3(x − x0 )2
ψ1 (x, v) = −3(x − x0 )(v + v0 )
ψ2 (x, v) = v 2 + vv0 + v02 +
and
√
(4.12)
3 2
γ (x − x0 )2 ,
10
γ(x − x0 )(v + v0 ) v02 − v 2
(v0 − v) + γ(x0 − x)
+
+g
.
20ε
4ε
2εγ
(4.13)
In the general case (4.5), the asymptotic structure (4.11) gives the eikonal
equations
3
exp
Z1 (x, v) =
2πεγ
∂ψ0
= 0
∂v
2
∂ψ1
3ψ0 =
∂v
2ψ1 = v
ψ2
∂ψ0
∂ψ1 ∂ψ2
+2
∂x
∂v ∂v
∂ψ1
∂ψ1
∂ 2 ψ1
= v
− [γv + V 0 (x)]
− εγ 2 +
∂x
∂v
∂v
(4.14)
∂ψ2
∂v
2
− 2εγ
∂ψ1 ∂ log Z1
∂v
∂v
∂ψ2
∂ψ2
∂ 2 ψ2
∂ψ2 ∂ log Z1
− [γv + V 0 (x)]
− εγ 2 − 2εγ
∂x
∂v
∂v
∂v
∂v
1 ∂ψ1 ∂Z2
∂ log Z1
−2εγ
− Z2
.
Z1 ∂v
∂v
∂v
−2εγ = v
4.2 Short Time Asymptotics of the Fokker-Planck Equation
The coefficients Zn (x, v) satisfy transport equations, as above.
The first three functions in (4.14) are given by (4.12). The function
Z1 (x, v) is given by
√
3
Z1 (x, v) =
×
(4.15)
2πεγ
γ(x − x0 )(v + v0 ) v02 − v 2 V 0 (x)(v0 − v) V (x) − V (x0 )
+
+
+
.
exp
20ε
4ε
2εγ
2ε
In particular, setting x = x0 = 0 gives ψ0 (0, v) = ψ1 (0, v) = 0, and
√
2
v0 − v 2 V 0 (0)(v0 − v)
3
2
2
ψ2 (0, v) = v + vv0 + v0 , Z1 (x, v) =
exp
+
.
2πεγ
4ε
2εγ
(4.16)
4.2.1
Initial layer?
In the first passage time (Milne) problem, the particle is initiated at x0 , away
from the origin x = 0. The short time behavior of the pdf (4.11) is governed
by the first eikonal ψ0 ,
3x20
p(x = 0, v, t) ∼ exp − 3 .
(4.17)
εγt
Setting t = γ −1 , we find that
3γ 2 x20
p(x = 0, v, γ ) ∼ exp −
,
(4.18)
ε
√
which is exponentially small for x0 γ −1 ε. This observation enabled [65]
to construct an initial time layer of size t0 = 2γ −1 . In [104] a similar initial
layer was used for the first passage time problem of the Brownian motion in
the presence of an external field. The particle undergoes thermalization at
times t γ −1 , its velocity distribution becomes Maxwellian, and its initial
velocity is forgotten.
However, in the RT (albedo) problem, the particle is initiated at the
absorbing boundary x0 = 0. Setting x = x0 = 0 in eqs.(4.12), we find that
both eikonals ψ0 and ψ1 vanish. Therefore, the short time asymptotics is
−1
59
60
Recurrence Time of the Brownian Motion
governed by the first non-vanishing eikonal ψ2 (x = 0, v) = v 2 + vv0 + v02 , so
that
√
2
3
v0 − v
v0 − v 2
p(x = 0, v, t | x0 = 0, v0 ) ∼
+g
(4.19)
exp
2πεγt2
4ε
2εγ
2
v + vv0 + v02
× exp −
.
εγt
In particular, setting t = γ −1 gives
√
3γ
v0 − v 5v 2 + 4vv0 + 3v02
−1
exp g
−
p(x = 0, v, γ | x0 = 0, v0 ) ∼
.
2πε
2εγ
4ε
(4.20)
Clearly, this expression is not exponentially small in γ. In fact, it is O(γ),
indicating that short return times are probable. We conclude that the RT
problem does not give rise to an initial time layer.
4.2.2
The probability distribution of the RT
We have found that the particle is most likely to return to the origin before
it is thermalized. We expect, as in the diffusion case [168], to find a ”beyond
all orders” short time asymptotic expansion. In analogy to the diffusion
case, we refer to Chandrasekhar’s solution (4.6) as a “ray solution”. The ray
solution almost satisfies the absorbing boundary condition (4.3). Indeed, for
γt 1 the pdf
at the boundary (4.19) is approximately normally distributed
v0 εγt
. That is, at short times particles are likely to exit at half
N − ,
2 2
r
εγt
their initial velocity, with a small standard deviation
. Therefore, for
2
v2
γt 0 the absorbing boundary condition (4.3) is almost satisfied (it is
ε
satisfied up to a transcendentally small error). This analysis also shows that
there are two non dimensional parameters in this problem, namely, γt (time
v2
measured in relaxation time) and 0 (initial kinetic energy measured in kB T ).
ε
The RT of a trajectory x(t) is the first time it returns to the origin (or
to any other point),
τ = inf {t : x(t) = 0} .
t>0
4.2 Short Time Asymptotics of the Fokker-Planck Equation
Replacing the pdf by the single ray (4.6) gives an approximation for the
RT distribution at short times. Using the Taylor expansion of the marginal
displacement pdf (4.10) and the large argument asymptotics of the error
function [1], we find the exit time probability distribution function (PDF)
(Fig. 4.1)
Z ∞
p(x, t | x0 , v0 ) dx
Pr {τ > t | v0 } =
x0
Z
∞
1
exp{−R2 /2F } dx
(4.21)
2πF
x0
1
v0 γ −1 (1 − e−γt ) − gγ −2 (γt − 1 + e−γt )
√
= 1 − erfc
2
2F
2
√
εγt
3v0
3v0 g
3v02
∼ 1− √
exp
+
exp −
16ε
4εγ
4εγt
3π v0
εγt
× 1+O
,
v02
=
√
or
2
√
3v0
εγt
3v0 g
3v02
exp
Pr {τ < t | v0 } ∼ √
+
exp −
(4.22)
16ε
4εγ
4εγt
3π v0
εγt
.
× 1+O
v02
We conclude that for short times the pdf of the RT has a beyond all orders
approximation of the form e−α/t , with α given in equation (4.22). Note that
the external force field g appears only as a constant pre-exponential factor—
the beyond all order behavior is driven by diffusion rather than drift.
4.2.3
Extrapolation length
To compare the result (4.22) with the diffusion approximation, we note that
in the diffusion approximation the velocity is not a state variable, therefore
there is no distinction between incoming and outgoing particles. It follows
that a particle initiated at an absorbing boundary is immediately absorbed.
Thus, to obtain non trivial results, the diffusive particle must be initiated
61
62
Recurrence Time of the Brownian Motion
Figure 4.1: Short time asymptotics of the probability distribution function
Pr{τ < t | v0 } (4.21) for different values of initial velocity v0 . The ”beyond
all order” behavior for short times γt 1 becomes noticeable even for short
times.
4.2 Short Time Asymptotics of the Fokker-Planck Equation
63
away from the boundary, say at x0 > 0. In the Smoluchowski approximation
of large damping, the pdf p(x, t) satisfies
∂p
∂ 2 p g ∂p
=D 2 +
,
∂t
∂x
γ ∂x
where D =
(4.23)
ε
is the diffusion coefficient, with an absorbing boundary condiγ
tion
p(x = 0, t) = 0 for all t > 0.
(4.24)
We recall that the solution is found by the method of images, which is a
reflection principle,
1
g2t
g(x − x0 )
p(x, t) = √
− 2
(4.25)
exp −
2γD
4γ D
4πDt
(x − x0 )2
(x + x0 )2
× exp −
− exp −
.
4Dt
4Dt
Integration yields the exit time distribution
Z ∞
Pr{τ > t | x0 } =
p(x, t | x0 ) dx
(4.26)
0
gx0
γx0 − gt
gt + γx0
1
√
√
+ exp
.
erfc
erfc
= 1−
2
γD
2γ Dt
2γ Dt
The large argument asymptotic behavior of the error function [1] gives the
short time asymptotics of the exit time pdf as
√
x0 g
4Dt
x20
x2
Pr{τ < t | x0 } ∼ √
exp
exp −
, for t 0 . (4.27)
2γD
4Dt
4D
πx0
Comparing equations (4.22) and (4.27), we see that choosing
√
x0 = 3 γ −1 v0
(4.28)
results in a similar short time behavior of the Langevin dynamics and its diffusion approximation. Note that the pre-exponential factor is not recovered
by this method. Recovering the pre-exponential factor is possible by using a
different acceleration field g̃ in the Smoluchowski approximation
√
√
3g
3v0 γ 2εγ ln 2
g̃ =
+
− √
.
2
8
3v0
64
Recurrence Time of the Brownian Motion
We conclude that extrapolation length that captures the short time asymptotics is different than that of the steady state problem [123, 103]. The
Smoluchowski approximation fails here twice: 1o near the absorbing boundary the velocity distribution is not Maxwellian (for all t and also for the
steady-state problem), 2o it is doomed to failure for short times γt 1, as
noted by Einstein and Smoluchowski.
4.3
Integral Equation for the pdf of the RT:
Short Time Asymptotics
We have shown that the absorbing boundary condition (4.3) of the FokkerPlanck equation (4.5) is satisfied up to an exponentially small error for short
times by the ray solution (4.6). That is, (4.6) is the first ray. Still, the
exponentially small correction term has to be determined. In the diffusion
(Smoluchowski) limit, the correction term is easily found by the method of
images, and it consists of rays of different eikonals. However, there is no
simple reflection rule for the Fokker-Planck equation with absorbing boundary conditions, so the problem should be tackled differently. The half range
expansion method leads to an infinite sum in the Laplace time coordinate. In
principle, the short time asymptotics could be found by the large argument
asymptotics of the half range expansion series. However, this re-summation
problem seems to be currently out of question [123]. Re-summation should
come as no surprise: solutions of the diffusion equation also have two different representations [168]. Separation of variables gives an infinite series
of time decaying exponentials and space eigenfunctions (e.g., trigonometric
functions in one dimension)
p(x, t) =
∞
X
an e−λn t pn (x).
n=0
This expansion is useful in determining the long time asymptotic behavior.
However, it converges very slowly for short times, when all exponentials
approach 1. Fortunately, the short time behavior is easily determined by
a second representation of the solution, which is obtained by the method
of images, that consists of rays. Here we generalize the concept of rays,
developed for the diffusion equation in [25], to the Fokker-Planck equation.
We introduce a new integral equation for the pdf of the RT, and construct
4.3 Integral Equation for the pdf of the RT: Short Time Asymptotics
its asymptotic solution by the ray method. The asymptotic solution consists
of two rays, which we find explicitly.
4.3.1
Integral equation for the pdf of the RT
The probability density of returning to the origin for the first time with a
given velocity,
f (v, t | v0 ) = Pr {τ = t, v(τ ) = v | x(0) = 0, v(0) = v0 } ,
(4.29)
is called the joint recurrence density. Obviously,
f (v, t | v0 ) = 0 for v > 0, v0 > 0,
and for v < 0, v0 < 0.
(4.30)
Equation (4.30) means that a trajectory that starts at the origin with a
positive velocity v0 returns to the origin for the first time with a negative
velocity, so it cannot return to the origin for the first time with a positive
velocity. It can, however, return to the origin for the second, fourth, and in
general for the 2n-th time with positive velocities.
It follows that for t > 0, v0 > 0, and for all v the recurrence density
satisfies the integral equation
Z t Z 0
vpc (0, v, t | 0, v0 ) = −f (v, t | v0 ) + v
ds
dη f (η, s | v0 )pc (0, v, t − s | 0, η).
0
−∞
(4.31)
Indeed, consider the unidirectional flux density of particles that cross the
origin x = 0 with velocity v at time t. On the one hand, this unidirectional
flux is vpc (0, v, t | 0, v0 ) (see (1.6)). On the other hand, it has two different
contributions. The first contribution, given by f (v, t | v0 ), is due to particles
that return to the origin for the first time exactly at time t with velocity
v. The second contribution is due to particles that returned to the origin
prior to time t. Due to the Markov property of the pair (x(t), v(t)), the
unidirectional flux density of trajectories that cross the origin at time t with
velocity v, given that it started with a positive velocity v0 , is the probability
density f (η, s | v0 ) that it returns to the origin for the first time at τ = s < t
with some negative velocity η, and then it returns to the origin with velocity
v (with probability density p(0, v, t − s | 0, η).
The integral equation (4.31) can be written as
(I − L) f = −vpc ,
(4.32)
65
66
Recurrence Time of the Brownian Motion
where the linear integral operator L is defined as
t
Z
Lf = v
Z
0
dη f (η, s | v0 )pc (0, v, t − s | 0, η).
ds
(4.33)
−∞
0
Obviously, kLk = O(t) < 1 for sufficiently small t, so the Neumann series
expansion gives
f = − I + L + L2 + · · · vpc ,
(4.34)
that is,
f = −vpc (1 + O(L)) ,
as expected.
4.3.2
The Laplace method for v > 0
For v > 0 the integral equation (4.31) reduces to
Z
pc (0, v, t | 0, v0 ) =
t
Z
0
dη f (η, s | v0 )pc (0, v, t − s | 0, η),
ds
(4.35)
−∞
0
because particles cannot return to the origin for the first time with a positive
velocity (4.30). The integral equation (4.35) for γt 1 is asymptotically
(neglecting all non exponentially small terms)
v 2 + vv0 + v02
exp −
εγt
t
0
2
v + vη + η 2
∼
ds
f (η, s| v0 ) exp −
dη.
εγ(t − s)
0
−∞
(4.36)
We look for a solution in the form
ψ(η, v0 )
f (η, s| v0 ) ∼ exp −
.
(4.37)
εγt
Z
Z
Substituting s = tu, the integral takes the form
2
v + vv0 + v02
exp −
∼
εγt
Z 1 Z 0
1 ψ(η, v0 ) v 2 + vη + η 2
du
exp −
+
dη.
εγt
u
1−u
0
−∞
(4.38)
4.3 Integral Equation for the pdf of the RT: Short Time Asymptotics
Asymptotic approximation of the integral is obtained by the Laplace method
(γt is a small parameter). The integral is approximated by the value of
Ψ(η, u; v) =
ψ(η, v0 ) v 2 + vη + η 2
+
u
1−u
(4.39)
at its minimum point (η ∗ , u∗ ) ∈ (−∞, 0] × [0, 1] (v is a parameter). Clearly,
the minimum point depends on the values of v and v0 , that is,
η ∗ = η ∗ (v, v0 ), u∗ = u∗ (v, v0 ).
Since (η ∗ , u∗ ) is a stationary point, it follows that
Ψη (η ∗ , u∗ ; v) =
and
ψη (η ∗ , v0 ) vη ∗ + 2η ∗
+
= 0,
u∗
1 − u∗
ψ(η ∗ , v0 ) v 2 + vη ∗ + η ∗ 2
Ψu (η , u ; v) = −
+
= 0.
u∗ 2
(1 − u∗ )2
∗
∗
(4.40)
(4.41)
In addition, for the integral to have the asymptotic behavior (4.38), necessarily
ψ(η ∗ , v0 ) v 2 + vη ∗ + η ∗ 2
+
= v 2 + vv0 + v02 .
Ψ(η , u ; v) =
∗
∗
u
1−u
∗
∗
(4.42)
Differentiating equation (4.42) with respect to v results in
d
Ψ(η ∗ , u∗ ; v) = Ψη (η ∗ , u∗ ; v)ηv∗ + Ψu (η ∗ , u∗ ; v)u∗v + Ψv (η ∗ , u∗ ; v) = 2v + v0 .
dv
(4.43)
However, equations (4.40), (4.41) imply that Ψη (η ∗ , u∗ ; v) = Ψu (η ∗ , u∗ ; v) =
0. Therefore,
2v + η ∗
= 2v + v0 .
(4.44)
Ψv (η ∗ , u∗ ; v) =
1 − u∗
Eliminating ψ(η ∗ , v0 ) between equations (4.41) and (4.42) gives another relation of η ∗ and u∗ ,
v 2 + vη ∗ + η ∗ 2 = (1 − u∗ )2 (v 2 + vv0 + v02 ).
The two solutions of eqs.(4.44), (4.45) are
vv0
v0
∗
∗
(η , u ) = (v0 , 0), −
,
.
v0 + v v0 + v
(4.45)
67
68
Recurrence Time of the Brownian Motion
The solution (v0 , 0) is ruled out, because v0 > 0 and we require η ∗ < 0
(in fact, this solution represents the original trajectory that is initiated at
v0
vv0
∗
∗
,
time t = 0 with velocity v0 ). Therefore, only (η , u ) = −
v0 + v v0 + v
is possible. We note that, as required, −v0 < η ∗ < 0 and 0 < u∗ < 1,
because v > 0. This means that the main contribution to the probability of
trajectories that cross the origin with a positive velocity v at time t, is due
v0 t
to trajectories that crossed the origin for the first time at time u∗ t =
v0 + v
vv0
∗
with a velocity η = −
.
v0 + v
η ∗ v0
The relations of η ∗ and v are invertible, so that v = − ∗
and u∗ =
η + v0
η ∗ + v0
. Therefore, eq.(4.42) gives
v0
ψ(η ∗ , v0 ) = η ∗ 2 + η ∗ v0 + v02
for all
− v0 < η ∗ < 0.
(4.46)
− v0 < η < 0.
(4.47)
We conclude that (4.37) is actually
2
η + ηv0 + v02
f (η, t | v0 ) ∼ exp −
εγt
for
The evaluation of the integral by the Laplace method also gives the preexponential factor. Expanding
η 2 + ηv0 + v02 η 2 + ηv + v
+
Ψ(η, u; v) =
u
1−u
(4.48)
in Taylor series at the stationary point (η ∗ , u∗ ), gives
1
Ψ(η, u; v) = Ψ(η ∗ , u∗ ; v) + Ψηη (η ∗ , u∗ ; v)(η − η ∗ )2
2
+Ψηu (η ∗ , u∗ ; v)(η − η ∗ )(u − u∗ )
1
+ Ψuu (η ∗ , u∗ ; v)(u − u∗ )2 + · · ·
2
(4.49)
4.3 Integral Equation for the pdf of the RT: Short Time Asymptotics
with the second order derivatives given by
2(v + v0 )2
,
vv0
(v − v0 )(v0 + v)2
Ψηu (η ∗ , u∗ ; v) =
,
vv0
2(v 2 + vv0 + v02 )(v + v0 )2
.
Ψuu (η ∗ , u∗ ; v) =
vv0
Ψηη (η ∗ , u∗ ; v) =
(4.50)
The determinant of the bilinear form (4.49) is
∆ = Ψηη Ψuu − Ψ2ηu = 3
(v + v0 )6
.
(vv0 )2
We look for a solution of the integral equation in the form
2
A(η, v0 )
η + ηv0 + v02
exp −
,
f (η, s | v0 ) ∼
sn
εγs
(4.51)
(4.52)
where A(η, v0 ) and n are to be determined by the integral equation
√
2
2
v0 − v 2
v0 − v
3
v + vv0 + v02
exp
+g
exp −
2πεγt2
4ε
2εγ
εγt
2
Z t Z 0
A(η, v0 )
η + ηv0 + v02
∼
ds
dη
exp −
(4.53)
sn
εγs
0
−∞
√
2
2
3
η − v2
η−v
v + vη + η 2
exp
+g
exp −
.
×
2πεγ(t − s)2
4ε
2εγ
εγ(t − s)
Evaluating the integral by the Laplace method gives
√
2
3
v0 − v 2
v0 − v
exp
+g
=
(4.54)
2πεγt2
4ε
2εγ
√
∗2
3 2πεγ A(η ∗ , v0 )
η − v2
η∗ − v
√
exp
+g
.
2πεγtn ∆ u∗ n (1 − u∗ )2
4ε
2εγ
Therefore, n = 2 and
√
2
3 (v + v0 )3 ∗ 2
v0 − η ∗ 2
v0 − η ∗
∗
∗ 2
A(η , v0 ) =
u (1 − u ) exp
+g
2πεγvv0
4ε
2εγ
√ ∗
2
3η
v0 − η ∗ 2
v0 − η ∗
= −
exp
+g
.
(4.55)
2πεγ
4ε
2εγ
69
70
Recurrence Time of the Brownian Motion
We conclude that for −v0 < v < 0,
√
2
2
v0 − v
v + vv0 + v02
3v
v0 − v 2
+g
exp −
f (v, t | v0 ) = −
exp
2πεγt2
4ε
2εγ
εγt
× [1 + O(γt)] .
(4.56)
Note that for these velocities f (v, t | v0 ) ∼ −vpc (0, v, t | 0, v0 ). In view of
(4.34), this is no mere coincidence, because the ray solution pc (x, v, t | 0, v0 )
satisfies the absorbing boundary condition (4.3) up to exponentially small
error.
4.3.3
Laplace method for v < 0
The preceding analysis shows that for v < 0 the exponent Ψ(η, u; v) has no
interior minima in the semi-infinite strip. Therefore, its single minimum is
attained at the boundary, so a different exponent is recovered. Next, we
compute the terms Ln vpc (n ≥ 0) of the Neumann series (4.34). The first
term is
Z t Z 0
Lvpc = v
ds
ηpc (0, η, s | 0, v0 )pc (0, v, t − s | 0, η) dη.
0
−∞
Evaluating Lvpc by the Laplace method, we see, again, that for v < 0 the
function
η 2 + ηv0 + v02 η 2 + ηv + v 2
+
(4.57)
Ψ(η, u) =
u
1−u
does not have local minima inside the strip −∞ < η < 0, 0 < u < 1.
Therefore, the minimum of Ψ is attained at the boundary of the strip. Since
Ψ → +∞ as η → −∞ or u → 0+ or u → 1− , we conclude that the minimum
is attained at η = 0. This means that trajectories that cross the origin
with an arbitrarily small velocity η = 0− at time τ < t bring in the main
contribution to pdf of the crossing event with a negative velocity at time t.
The time at which this crossing occurs is determined by the minimum of the
function
v02
v2
Ψ(0, u) =
+
.
(4.58)
u
1−u
v0
v0
The stationary condition Ψu = 0 is satisfied by u∗ =
and u∗ =
.
v0 + v
v0 − v
v0
For v < 0, only u∗ =
satisfies 0 < u∗ < 1. Therefore, slow particles
v0 − v
4.3 Integral Equation for the pdf of the RT: Short Time Asymptotics
v0 t
are most probable to cross (again and)
v0 − v
again at time t with a velocity v < 0. The minimum is
that cross the origin at time
Ψ(0, u∗ ) =
v02
v2
+
= (v0 − v)2 .
∗
∗
u
1−u
(4.59)
Therefore, neglecting non exponentially small terms, we have
(v0 − v)2
Lvpc ∼ exp −
εγt
,
(4.60)
which is the second ray. Note that (v0 − v)2 > v 2 + vv0 + v02 for v < 0,
therefore Lvpc vpc . Also Ψη (0, u∗ ) = 0, so near the minimum the Taylor
series of Ψ is
1
Ψ(η, u) = Ψ(0, u∗ ) + Ψηη (0, u∗ )η 2 + Ψηu (0, u∗ )η(u − u∗ )
2
1
+ Ψuu (0, u∗ )(u − u∗ )2 + · · · ,
2
(4.61)
where the second order derivatives are given by
(v0 − v)2
v0 v
(v0 − v)3
=
v0 v
(v0 − v)4
.
= −2
v0 v
Ψηη = −2
Ψηu
Ψuu
(4.62)
The determinant of the bilinear form (4.61) is
∆ = Ψηη Ψuu − Ψ2ηu = 3
(v0 − v)6
>0
(v0 v)2
(4.63)
71
72
Recurrence Time of the Brownian Motion
and Ψηη , Ψuu > 0 (for v < 0), therefore (0, u∗ ) is indeed a minimum. The
Laplace method gives
2
v0 − v
3vt
v0 − v 2
+g
Lvpc =
exp
(2πεγ)2 t4
4ε
2εγ
Z 1
Z 0
du
Ψ(η, u)
×
η exp −
dη
2
2
εγt
0 u (1 − u)
−∞
2
−3vt
v0 − v
v0 − v 2
∼
+g
exp
(2πεγ)2 t4
4ε
2εγ
Ψ(0, u∗ )
√
exp −
εγt
3/2 2πΨuu
×(εγt)
(4.64)
2
∗
∆
u (1 − u∗ )2
r
2
v0 − v 2
v0 − v
−v
(v0 − v)2
=
exp
+g
exp −
.
4π 3 εγv0 t3
4ε
2εγ
εγt
Note that the pre-exponential power of t in Lvpc is 3/2 rather than 2, as in
η
vpc (eq.(4.56)), because the integrated pre-exponential function 2
u (1 − u)2
∗
vanishes at the minimum point η = 0.
Next, we evaluate the term L2 vpc ,
Z 0
Z t
2
(Lηpc )pc (0, v, t − s | 0, η) dη.
(4.65)
ds
L vpc = v
0
−∞
Similar considerations show that
(v0 − v)2
L vpc ∼ exp −
εγt
2
,
(4.66)
because the minimum of
(η − v0 )2 η 2 + vη + v 2
Ψ(η, u) =
+
u
1−u
∗
∗
is attained at the same boundary point (η , u ) = 0,
(4.67)
v0
. The point
v0 − v
(η ∗ , u∗ ) is not stationary, because only Ψu (η ∗ , u∗ ) = 0, while Ψη (η ∗ , u∗ ) =
−3(v0 − v) < 0. Therefore, the only second order derivative needed is
Ψuu (η ∗ , u∗ ) = −2
(v0 − v)4
,
vv0
(4.68)
4.3 Integral Equation for the pdf of the RT: Short Time Asymptotics
and the Laplace method gives
√
2
3
1
v0 − v 2
v0 − v
2
p
L vpc ∼ vt
exp
+g
2πεγt2 4π 3 εγv0 t3
4ε
2εγ
(v0 − v)2
π(εγt)2
1
p
× exp −
(4.69)
3/2
∗
3
εγt
(1 − u∗ )2
−2Ψuu Ψη u
2
r
εγ
v0 − v 2
−1
v0 − v
(v0 − v)2
exp
=
+g
exp −
.
24 −π 3 v03 vt
4ε
2εγ
εγt
Note that L2 vpc has an integrable singularity in v at the origin.
Similarly, the Laplace method gives
2
r
1
v0 − v 2
v0 − v
(v0 − v)2
εγ
3
L vpc ∼
exp
+g
exp −
48 −π 3 v03 vt
4ε
2εγ
εγt
1
= − L2 vpc .
2
(4.70)
The function L2 vpc is an asymptotic eigenfunction of the integral operator L
(with eigenvalue −1/2). Therefore,
n
∞ X
1
2
2
3
(4.71)
L + L + . . . vpc =
−
L2 vpc ∼ L2 vpc .
2
3
n=0
Therefore, the Neumann series (4.34) collapses into
2 2
f ∼ − I + L + L vpc .
3
(4.72)
We conclude that the short time expansion is given by
" √
2
2
v0 − v
3v
v + vv0 + v02
v0 − v 2
−
f (v, t | v0 ) ∼ exp
+g
exp −
4ε
2εγ
2πεγt2
εγt
r
+ −
−v
1
+
4π 3 εγv0 t3 36
r
εγ
−π 3 v03 vt
(4.73)
#
(v0 − v)2
exp −
.
εγt
Equation (4.73) is the beyond all order asymptotic solution of the recurrence
density. The first ray is the unidirectional flux of the free Brownian motion.
73
74
Recurrence Time of the Brownian Motion
Therefore, the first ray is an overestimate of the RT density f , because it
takes into account the unidirectional flux of particles that cross the origin
and come back. Hence, the second ray has a negative contribution to the RT
density as seen in Figure 4.2.
4.4
Mean Number of Returns
The unidirectional probability flux of a Langevin particle, vpc (0, v, t | 0, v0 ),
is the probability that crossed the origin from left to right for v > 0 (from
right to left for v < 0) at time t with velocity v in unit time per unit velocity,
given that the particle
Z was initiated from the origin x = 0 with velocity
0
vpc (0, v, t | 0, v0 ) dv is the probability that crossed
v0 > 0. Therefore, −
−∞
the origin from right to left at time t in unit time. In this Section we calculate
the mean number of times a Langevin trajectory crosses the origin in the time
interval [0, T ]. Note that in the overdamped limit the number of crossings is
infinite for all T > 0, because the Brownian motion is nowhere differentiable
with probability 1 [39]. However, the number of crossings is finite for the
Langevin motion.
We divide the interval [0, T ] into subintervals of size ∆t, where ∆t is so
small such that the number of crossings at time interval of size ∆t is either 0
or 1 (the probability for more than 1 crossing is negligible). The number of
crossings in the j-th time interval is a Bernoulli random variable that takes
Z (j+1)∆t Z 0
vpc (0, v, t | 0, v0 ) dv, and 0
dt
the values 1, with probability −
j∆t
−∞
otherwise. Therefore, in the limit ∆t → 0, the mean number of crossings
from right to left in [0, T ] is
Z T Z 0
E(NRL ) = −
dt
vpc (0, v, t | 0, v0 ) dv.
0
−∞
For short times γT 1, the density pc is given by equation (4.19), and
E(NRL ) is asymptotically
√
Z T
Z 0
3
×
(4.74)
E(NRL ) ∼ −
dt
dv v
2πεγt2
0
−∞
2
2
v0 − v 2
v0 − v
v + vv0 + v02
exp
+g
exp −
.
4ε
2εγ
εγt
4.4 Mean Number of Returns
Figure 4.2: The short time asymptotics of the RT pdf (γ = 1, = 1, v0 =
1, v = −1/4): (a) First ray (red), (b) Second ray (green), (c) Sum of the two
rays (eq.(4.73)) (blue). Note that the contribution of the second ray is negative, and that the particle is most probable to return before the relaxation
time.
75
76
Recurrence Time of the Brownian Motion
Integrating first with respect to t, gives
√
2
Z 0
3v dv
v0 − v 2
v0 − v
E(NRL ) ∼ −
exp
+g
×
2
2
4ε
2εγ
−∞ 2π(v + vv0 + v0 )
2
v + vv0 + v02
exp −
.
(4.75)
εγT
Setting v = −v0 u, we obtain
√
2
Z ∞
3u du
v0 (1 − u2 )
v0 (1 + u)
E(NRL ) ∼
exp
+g
×
2π(u2 − u + 1)
4ε
2εγ
0
2 2
v0 (u − u + 1)
exp −
.
(4.76)
εγT
√
Z ∞
3u du
The integral in eq.(4.76) reduces for v0 = 0 to
, which
2π(u2 − u + 1)
0
diverges, that is, the mean number of returns is infinite for any time interval [0, T ], if the trajectory begins at rest. Next, we find the rate at
which E(NRL ) tends to infinity as v0 → 0. First we consider the case
g = 0 (no accelerationfield), and to facilitate calculations we neglect the
2
2
v (1 − u )
, because it decays much slower than
exponential factor exp 0
4ε
2 2
v (u − u + 1)
exp − 0
for γT 1. Hence,
εγT
√
Z ∞
3u du
exp −z(u2 − u + 1) ,
E(NRL ) ∼
2
2π(u − u + 1)
0
v02
. This integral converges for z = 0 over any finite interval of
εγT
integration, so we replace the lower limit from 0 to 1
√
Z ∞
3u du
2
E(NRL ) ∼
exp
−z(u
−
u
+
1)
+ O(1).
2π(u2 − u + 1)
1
where z =
Furthermore, if the numerator u is replaced by a constant then the improper
integral converges also for z = 0. Therefore,
Z ∞ √
3(u − 1/2) du
2
E(NRL ) ∼
exp
−z(u
−
u
+
1)
+ O(1).
2π(u2 − u + 1)
1
4.4 Mean Number of Returns
77
Setting w = z(u2 − u + 1) yields
√ Z ∞ −w
√
e
3
3
E(NRL ) ∼
dw + O(1) =
Ei(z) + O(1),
4π z
w
4π
(4.77)
where Ei(z) is the exponential integral. The small z asymptotics of the
exponential integral [1] gives
√
3
E(NRL ) ∼ −
log z + O(1).
(4.78)
4π
The mean number of returns to the origin √
at short
times
has a logarithmic
v02
3
singularity at v0 = 0, and it diverges as −
log
. It can be shown
4π
εγT
that the neglected exponential factor contributes only an O(γT ) term, but
does not change the main logarithmic singularity. For each return event from
left to right there is a preceding return event from right to left. Therefore,
the mean number of crossings E(N ) is
√
3
log z + O(1).
(4.79)
E(N ) = E(NRL + NLR ) ∼ −
2π
Expanding the mean number of returns in Taylor series in g gives
E(NRL ) = E(NRL , g = 0) +
∂E(NRL , g = 0)
g + ...
∂g
(4.80)
We have already calculated the first term E(NRL , g = 0) and showed that it
has a logarithmic singularity. The second term
Z ∞ √
∂E(NRL , g = 0)
v0
3u(1 + u) du
=
exp −z(u2 − u + 1)
2
∂g
2εγ 0 2π(u − u + 1)
is calculated in a similar fashion, and we obtain
s !
T
∂E(NRL , g = 0)
,
=O
∂g
εγ
(4.81)
which is of order 1 in v0 . Therefore, the small velocity asymptotics (4.78)
remains even for small acceleration fields.
78
Recurrence Time of the Brownian Motion
4.5
Long Time Asymptotics
We find the long time asymptotic behavior of the recurrence pdf by solving
the integral equation (4.31) asymptotically for γt 1. We show that the
recurrence pdf is to leading order
g0 (v, v0 )
1
.
f (v, t | v0 ) =
+
O
t3/2
t2
The t−3/2 decay law should come as no surprise, because it is also the tail
asymptotics of the first passage time of hitting a boundary away from the
point of initiation for the Brownian motion.
We rewrite the integral equation (4.31) as
Z 0
vpc (0, v, t | 0, v0 ) = −f (v, t | v0 ) + v
dηf (η, · | v0 ) ∗ pc (0, v, · | 0, η), (4.82)
−∞
where
Z
f (η, · | v0 ) ∗ pc (0, v, · | 0, η) =
t
f (η, s | v0 ) ∗ pc (0, v, t − s | 0, η) ds
0
is the time convolution. The Laplace transform of the integral equation (4.82)
with respect to t is
Z 0
ˆ
v pˆc (0, v, θ | 0, v0 ) = −f (v, θ | v0 ) + v
dη fˆ(η, θ | v0 )pˆc (0, v, θ | 0, η), (4.83)
−∞
where θ is the Laplace time coordinate.
We find the long time behavior γt 1 by ignoring the exponentially
decaying terms of the fundamental solution pc of eq. (4.6) and eqs.(4.7),
(4.8). Therefore, introducing transcendentally small errors,
R ∼ x − x0 − γ −1 v0 ,
S ∼ v,
(4.84)
where we have assumed g = 0 for the free Brownian motion, and
F ∼ εγ −2 (2γt − 3),
G ∼ ε,
H ∼ εγ −1 .
(4.85)
Thus, for γt 1
2
γ
v
(v0 + v)2
√
pc (0, v, t | 0, v0 ) ∼
exp −
exp −
, (4.86)
2ε
2ε(2γt − 4)
2πε 2γt − 4
4.5 Long Time Asymptotics
79
that is, the density is almost Maxwellian, as expected. Furthermore, eq.
(4.86) gives
2
1
v
pc (0, v, t | 0, v0 ) =
1+O
.
exp −
2ε
γt
2πε 2γt
γ
√
(4.87)
Hence, the small θ asymptotics of the Laplace transform pˆc is
2r γ
θ
1
v
1+O
.
pˆc (0, v, θ | 0, v0 ) = √
exp −
2ε
θ
γ
2 2πε
(4.88)
In view of eq.(4.88), we look for
s a solution of the integral equation (4.83) in
θ
the form of a power series in
γ
s
fˆ(v, θ | v0 ) = f0 (v, v0 ) + f1 (v, v0 )
θ
θ
+ f2 (v, v0 ) + . . .
γ
γ
(4.89)
The series (4.89) starts from a θ-independent term, because f (v, t | v0 ) is a,
possibly defective, pdf and so
Z
0
Z
−∞
Z
0
f (η, t | v0 ) ds =
dη
0<
∞
fˆ(η, 0 | v0 ) dη ≤ 1.
−∞
0
r γ
Equating the leading order singular terms in eq.(4.83), which are O
,
θ
gives
Z 0
f0 (η, v0 ) dη = 1,
(4.90)
−∞
which means that the free particle is recurrent. Equating the O(1) terms
gives
2
v
Z 0
v exp −
2ε
√
f0 (v, v0 ) =
f1 (η, v0 ) dη.
(4.91)
2 2πε
−∞
80
Recurrence Time of the Brownian Motion
We integrate equation (4.91) and obtain by equation (4.90)
Z
0
f0 (v, v0 ) dv
2
v
Z 0 v exp −
Z 0
2ε
√
=
f1 (η, v0 ) dη
dv
2 2πε
−∞
−∞
Z 0
1
= − √
f1 (η, v0 ) dη.
2 2π −∞
1 =
−∞
(4.92)
Therefore,
2
v
v exp −
2ε
f0 (v, v0 ) = −
.
ε
r θ
, and we obtain
Similarly, f1 (v, v0 ) is determined by the O
γ
2
v
v
exp
−
√
2ε
f1 (v, v0 ) = 2 2π
.
ε
(4.93)
(4.94)
We conclude that the half-integer power series (4.89) is self consistent.
1
d ˆ
f ∼ √ is singular for θ = 0. Therefore, the mean time of
Note that
dθ
θ
return is infinite
Z ∞
d
tf (v, t | v0 ) dt = − fˆ(v, 0 | v0 ) = ∞,
dθ
0
d ˆ
1
f ∼ √ means
dθ
θ
that the asymptotic behavior of tf (·, t | ·) for large t is the inverse Laplace
1
1
1
transform of √ , which is proportional to √ . That is, tf (·, t | ·) ∼ √ , or
t
t
θ
more precisely
f1 (v, v0 )
f (v, t | v0 ) ∼ − p
,
(4.95)
2 πγt3
which is the Gambler’s ruin paradox [51]. Moreover,
4.6 Small v asymptotics
81
which, by eq.(4.94) is
2
v
v exp −
2ε
p
f (v, t | v0 ) ∼ −
,
ε 2γt3
for γt 1.
(4.96)
Note that (4.96) means that at long times the return probability is to leading
order independent of the initial velocity v0 , because the initial velocity is
thermalized for times longer than relaxation time.
4.6
Small v asymptotics
The Laplace method of Section 4.3 assumed that γt is a small parameter,
smaller than all other parameters. This assumption does not hold for small
velocities at fixed time t, that is, for v 2 /ε γt. Therefore, the asymptotic
expansion (4.73) holds for small γt, but it is not uniform in v 2 /ε. In this
section we construct an asymptotic solution of the integral equation (4.31)
for v 2 /ε γt 1.
For γt 1 we have
2
v + vv0 + v02
1
.
pc (0, v, t | 0, v0 ) ∼ 2 exp −
t
εγt
Therefore, the left hand side (LHS) of the integral equation (4.31) is O(v).
We show below that the integral term of the right hand side (RHS) of (4.31)
is singular in v. Specifically,
Z 0
Z t
v
dη
pc (0, v, t − s | 0, η) ds = O(1),
(4.97)
−∞
Z
0
0
Z
v
t
ηdη
−∞
Z
0
v
n
Z
η dη
−∞
pc (0, v, t − s | 0, η) ds = O(v log v),
(4.98)
pc (0, v, t − s | 0, η) ds = O(v),
(4.99)
0
t
n ≥ 2.
0
Note that the right hand side terms of (4.31) have the same sign, hence, their
leading order term in v cannot cancel out. It follows that f is not O(1), for
otherwise equation (4.97) gives that the RHS is O(1), whereas the LHS is
82
Recurrence Time of the Brownian Motion
O(v). Similarly, f is not O(v), because eq.(4.98) gives an O(v log v) leading
order term that cannot be balanced. We conclude that
f (v, t | v0 ) ∼ O(v 2 ),
which, by eq.(4.99), gives balanced O(v) terms on both the RHS and the
LHS.
For example, we verify eq. (4.97)
Z
0
v
Z
t
pc (0, v, t − s | 0, η) ds
dη
−∞
0
0
t
2
v + vη + η 2
ds
exp −
∼ v
dη
2
εγ(t − s)
−∞
0 (t − s)
Z 0
dη
v 2 + vη + η 2
= vεγ
exp −
2
2
εγt
−∞ v + vη + η
2
Z ∞
du
v (1 + u + u2 )
= εγ
exp −
= O(1),
1 + u + u2
εγt
0
Z
Z
(4.100)
because the last integral converges for v = 0. Equations (4.98)-(4.99) are
obtained in a similar fashion. The entire asymptotic series of (4.99) is (for
n ≥ 2)
0
Z
n
v
t
Z
pc (0, v, t − s | 0, η) ds ∼
η dη
−∞
(n)
where aj
Z
0
2
Z
(4.101)
j=1
t
pc (0, v, t − s | 0, η) ds ∼
η dη
−∞
(n)
aj v j + bn v n log v,
and bn are coefficients. For example, for n = 2
0
v
∞
X
0
∞
X
(2)
aj v j + b2 v 2 log v.
(4.102)
j=1
P
j
If we look for a solution in the form of merely a power series f ∼ ∞
n=2 fj v ,
then the logarithmic terms, such as v 2 log v, cannot be balanced. This term
can be balanced if we look for a solution that also includes powers of logarithms. In particular, the v 2 log v term can be balanced by a v 3 log v term.
However, this term also produces a v log v term. Therefore, we include terms
4.6 Small v asymptotics
83
in the form v n log v in our asymptotic series. Similarly, the v 3 log v term produces a v 3 log2 v term, that can be balanced by a v 4 log2 term. The procedure
is now clear. We conclude that the asymptotic form of the solution is
f∼
∞ X
n−2
X
n=2 m=0
anm v n logm v.
(4.103)
Chapter 5
Narrow Escape in Three
Dimensions
The contents of this chapter were submitted for publication [177]
A Brownian particle with diffusion coefficient D is confined to a bounded
domain of volume V in R3 by a reflecting boundary, except for a small absorbing window. The mean time to absorption diverges as the window shrinks,
thus rendering the calculation of the mean escape time a singular perturbation problem. We construct an asymptotic approximation for the case of an
elliptical window of large semi axis a V 1/3 and show that the mean escape
V
time is Eτ ∼
K(e), where e is the eccentricity of the ellipse; and K(·) is
2πDa
the complete elliptic integral of the first kind. In the special case of a circular
V
, which was dehole the result reduces to Lord Rayleigh’s formula Eτ ∼
4aD
rived by heuristic considerations. For the special case
of a spherical domain,
a V
a
R
we obtain the asymptotic expansion Eτ =
1 + log + O
.
4aD
R
a
R
This problem is important in understanding the flow of ions in and out of
narrow valves that control a wide range of biological and technological function.
5.1
Introduction
We consider the exit problem of a Brownian motion from a bounded domain, whose boundary is reflecting, except for a small absorbing window.
5.1 Introduction
The mean first passage time to the absorbing window (MFPT), Eτ , is the
solution of a mixed Neumann-Dirichlet boundary value problem (BVP) for
the Poisson equation, known as the corner problem, which has singularity
at the boundary of the hole [33, 105, 106]. The MFPT grows to infinity
as the window size shrinks to zero, thus rendering its calculation a singular
perturbation problem for the mixed BVP, which we call the narrow escape
problem. The narrow escape problem has been considered in the literature
in only a few special cases, beginning with Lord Rayleigh (in the context
of acoustics), who found the flux through a small hole by using a result of
Helmholtz [69]. He stated [147] (p.176) “Among different kinds of channels
an important place must be assigned to those consisting of simple apertures
in unlimited plane walls of infinitesimal thickness. In practical applications
it is sufficient that a wall be very thin in proportion to the dimensions of the
aperture, and approximately plane within a distance from aperture large in
proportion to the same quantity.” More recently, Rayleigh’s result was shown
to fit the MFPT obtained from Brownian dynamics simulations [62]. Another
result was presented in [78], where a two-dimensional narrow escape problem
was considered and whose method is generalized here. A related problem is
that of escape from a domain, whose boundary is reflecting, except for an
absorbing sphere, disjoint from the reflecting part of the boundary [144, and
references therein]. It differs from the narrow escape problem in that there
is no singularity at the boundary and there is no boundary layer. The mixed
boundary value problems of classical electrostatics (e.g., the electrified disk
problem [88]), elasticity (punch problems), diffusion and conductance theory,
hydrodynamics, and acoustics were solved, by and large, for special geometries by separation of variables. In axially symmetric geometries this method
leads to a dual series or to integral equations that can be solved by special
techniques [184, 49, 50, 116, 192]. The special case of asymptotic representation of the solution of the corner problem for small Dirichlet and large
Neumann boundaries was not done for general domains. The first attempt
in this direction seems to be [78]. The narrow escape problem does not seem
to fall within the theory of large deviations [36]. It is different from Kolmogorov’s exit problem [126] of a diffusion process with small noise from an
attractor of the drift (e.g., a stable equilibrium or limit cycle) in that the
narrow escape problem has no large and small coefficients in the equation.
The singularity of Kolmogorov’s problem is the degeneration of a second order elliptic operator into a first order operator in the limit of small noise,
whereas the singularity of the narrow escape problem is the degeneration of
85
86
Narrow Escape in Three Dimensions
the mixed BVP to a Neumann BVP on the entire boundary. There exist
precise asymptotic expansions of Eτ for Kolmogorov’s exit problem, including error estimates (see, e.g., [66], [54]), which show that the MFPT grows
exponentially with decreasing noise. In contrast, the narrow escape time
grows algebraically rather than exponentially, as the window shrinks. Our
first main result is a derivation of the leading order term in the expansion of
the MFPT of a Brownian particle with diffusion coefficient D, from a general
domain of volume V to an elliptical hole of large semi axis a that is much
smaller than V 1/3 ,
V
K(e),
(5.1)
2πDa
where e is the eccentricity of the ellipse, and K(·) is the complete elliptic
integral of the first kind. In the special case of a circular hole (5.1) reduces
to
V
Eτ ∼
.
(5.2)
4aD
Equation (5.1) shows that the MFPT depends on the shape of the hole, and
not just on its area. This result was known to Lord Rayleigh [147], who
considered the problem of the electrified disk (which he knew was equivalent
to finding the flow of an incompressible fluid through a channel and to the
problem of finding the conductance of the channel), who reduced the problem
to that of solving an integral equation for the flux density through the hole.
The solution of the integral equation, which goes back to Helmholtz [69] and
is discussed in [116], is proportional to (a2 −ρ2 )−1/2 in the circular case, where
ρ is the distance from the center of the hole [88, 184, 49]. Note that equations
(5.1) and (5.2) are leading order approximations and do not contain an error
estimate. We prove (5.1) by using the singularity properties of Neumann’s
function for three-dimensional domains, in a manner similar to that used in
[78] for two-dimensional problems. The leading order term is the solution of
Helmholtz’s integral equation [69]. Our second main result is a derivation
of the second term and error estimate for a ball of radius R with a small
circular hole of radius a in the boundary,
a a
R
V
1 + log + O
.
(5.3)
Eτ =
4aD
R
a
R
Eτ ∼
Equation (5.3) contains both the second term in the asymptotic expansion of
the MFPT and an error estimate. We use Collins’ method [26, 27] of solving
5.2 General 3D Bounded Domain
dual series of equations and expand the resulting solutions for small ε = a/R.
The estimate of the error term, which turns out to be O(ε log ε), seems to
be a new result. An error estimate for eq.(5.1) for a general domain is still
an open problem. We conjecture that it is O(ε log ε), as is the case for the
ball. If the absorbing window touches a singular point of the boundary, such
as a corner or cusp, the singularity of the Neumann function changes and so
do the asymptotic results. In three dimensions the class of isolated singularities of the boundary is much richer than in the plane, so the methods of
[179] cannot be generalized in a straightforward manner to three dimensions.
We postpone the investigation of the MFPT to windows at isolated singular
points in three dimensions to a future paper. In Section 5.2 we derive a leading order approximation to the MFPT for a general domain with a general
small window. The leading order term is expressed in terms of a solution
to Helmholtz’s integral equation, which is solved explicitly for an elliptical
window. In Section 5.3 we obtain two terms in the asymptotic expansion of
the MFPT from a ball with a circular window and an error estimate. Finally,
we present a summary and list some applications in Section 5.4. This is the
first chapter in a series of three, the second of which considers the narrow
escape problem from a bounded simply connected planar domain, and the
third of which considers the narrow escape problem from a bounded domain
with boundary with corners and cusps on a two-dimensional Riemannian
manifold.
5.2
General 3D Bounded Domain
A Brownian particle diffuses freely in a bounded domain Ω ⊂ R3 , whose
boundary ∂Ω is sufficiently smooth. The trajectory of the Brownian particle,
denoted x(t), is reflected at the boundary, except for a small hole ∂Ωa , where
it is absorbed. The reflecting part of the boundary is ∂Ωr = ∂Ω − ∂Ωa . The
lifetime of the particle in Ω is the first passage time τ of the Brownian particle
from any point x ∈ Ω to the absorbing boundary ∂Ωa . The MFPT,
v(x) = E[τ | x(0) = x],
is finite under quite general conditions [166]. As the size (e.g., the diameter)
of the absorbing hole decreases to zero, but that of the domain remains finite,
we assume that the MFPT increases indefinitely. A measure of smallness can
87
88
Narrow Escape in Three Dimensions
be chosen as the ratio between the surface area of the absorbing boundary
and that of the entire boundary,
ε=
|∂Ωa |
1,
|∂Ω|
(see, however, a pathological example in Appendix B.3). The MFPT v(x)
satisfies the mixed boundary value problem [166]
∆v(x) = −
1
,
D
v(x) = 0,
∂v(x)
= 0,
∂n(x)
for x ∈ Ω,
(5.4)
for x ∈ ∂Ωa ,
(5.5)
for x ∈ ∂Ωr ,
where D is the diffusion coefficient. According to our assumptions v(x) → ∞
as the size of the hole decreases to zero, e.g., as ε → 0, except in a boundary
layer near ∂Ωa . Our purpose is to find an asymptotic approximation to v(x)
in this limit.
5.2.1
The Neumann function and integral equations
To calculate the MFPT v(x), we use the Neumann function N (x, ξ) (see
[78], [144]), which is a solution of the boundary value problem
∆x N (x, ξ) = −δ(x − ξ),
1
∂N (x, ξ)
= −
,
∂n(x)
|∂Ω|
for x, ξ ∈ Ω,
(5.6)
for x ∈ ∂Ω, ξ ∈ Ω,
and is defined up to an additive constant. The Neumann function has the
form [55]
1
N (x, ξ) =
+ vS (x, ξ),
(5.7)
4π|x − ξ|
5.2 General 3D Bounded Domain
89
where vS (x, ξ) is a regular harmonic function of x ∈ Ω and of ξ ∈ Ω. Green’s
identity gives
Z
[N (x, ξ)∆v(x) − v(x)∆N (x, ξ)] dx =
Ω
∂v(x(S))
∂N (x(S), ξ)
=
N (x(S), ξ)
− v(x(S))
dS
∂n
∂n
∂Ω
Z
Z
∂v(x(S))
1
=
N (x(S), ξ)
dS +
v(x(S)) dS.
∂n
|∂Ω| ∂Ω
∂Ω
Z
On the other hand, equations (5.4) and (5.6) imply that
Z
Z
1
[N (x, ξ)∆v(x) − v(x)∆N (x, ξ)] dx = v(ξ) −
N (x, ξ) dx,
D Ω
Ω
hence
Z
1
v(ξ) −
N (x, ξ) dx =
D Ω
Z
Z
∂v(x(S))
1
N (x(S), ξ)
v(x(S)) dS.
dS +
∂n
|∂Ω| ∂Ω
∂Ω
(5.8)
Note that the second integral on the right hand side of eq.(5.8) is an additive
constant. Setting
Z
1
v(x(S)) dS,
(5.9)
C=
|∂Ω| ∂Ω
we rewrite eq.(5.8) as
Z
Z
1
∂v(x(S))
v(ξ) =
N (x, ξ) dx +
N (x(S), ξ)
dS − C,
D Ω
∂n
∂Ωa
(5.10)
which is an integral representation of v(ξ). We define the boundary flux
density
∂v(x(S))
g(S) =
,
(5.11)
∂n
choose ξ ∈ ∂Ωa , and use the boundary condition (5.5) to obtain the equation
Z
Z
1
0=
N (x, ξ) dx +
N (x(S), ξ)g(S) dS − C,
(5.12)
D Ω
∂Ωa
90
Narrow Escape in Three Dimensions
for all ξ ∈ ∂Ωa . Equation (5.12) is an integral equation for g(S) and C.
To construct an asymptotic approximation to the solution, we note that the
first integral in equation (5.12) is a regular function of ξ on the boundary.
Indeed, due to symmetry of the Neumann function, we have from (5.6)
Z
N (x, ξ) dx = −1 for ξ ∈ Ω
∆ξ
(5.13)
Ω
and
∂
∂n(ξ)
Z
N (x, ξ) dx = −
Ω
|Ω|
|∂Ω|
for ξ ∈ ∂Ω.
(5.14)
Equation (5.13) and the boundary condition (5.14) are independent of the
hole ∂Ωa , so they define the integral as a regular function, up to an additive
constant, also independent of ∂Ωa . The assumption that for all x ∈ Ω,
away from ∂Ωa , the MFPT v(x) increases to infinity as the size of the hole
decreases and eq.(5.9) imply that C → ∞ as as the size of the hole decreases
to zero. This means that for ξ ∈ ∂Ωa the second integral in eq.(5.12) must
also become infinite in this limit, because the first integral is independent of
∂Ωa . Therefore, the integral equation (5.12) is to leading order
Z
N (x(S), ξ))g0 (S) dS = C0
for ξ ∈ ∂Ωa ,
(5.15)
∂Ωa
where g0 (S) is the first asymptotic approximation to g(S) and C0 is the first
approximation to the constant C. Furthermore, only the singular part of the
Neumann function contributes to the leading order, so we obtain the integral
equation
Z
g0 (x)
1
dSx = C0 ,
(5.16)
2π ∂Ωa |x − y|
where C0 is a constant, which represents the first approximation to the mean
first passage time (MFPT). Note that the singularity of the Neumann function at the boundary is twice as large as it is inside the domain, due to the
contribution of the regular part (the “image charge”) and therefore the factor
1
1
of equation (5.7) was replaced by
. In general, the integral equation
4π
2π
(5.16) has no explicit solution, and should be solved numerically.
5.2 General 3D Bounded Domain
5.2.2
91
Elliptic hole
When the hole ∂Ωa is an ellipse, the solution of the integral equation (5.16)
is known [147], [116]. Specifically, assuming the ellipse is given by
x2 y 2
+ 2 = 1,
a2
b
z = 0,
the solution is
(b ≤ a),
g̃0
g0 (x) = r
x2 y 2
1− 2 − 2
a
b
,
(5.17)
where g̃0 is a constant (to be determined below). The proof, originally given
in [69], is reproduced in Appendix B.2. To determine the value of the constant
g̃0 , we use the compatibility condition
Z
|Ω|
,
(5.18)
g0 (x) dSx =
D
∂Ωa
obtained from the integration of eq.(5.4) over Ω. Using the value
Z
Z
a
dx
g0 (x) dSx =
∂Ωa
Z
−a
q
2
b 1− x2
g̃0 dy
a
q
2
−b 1− x2
a
r
x2 y 2
1− 2 − 2
a
b
= 2πabg̃0
(5.19)
and the compatibility condition (5.18), we obtain
g̃0 =
|Ω|
.
2πDab
(5.20)
Hence, by equation (B.6), the leading order approximation to C is
1
C0 =
2π
Z
∂Ωa
g0 (x)
|Ω|
dSx =
K(e),
|x − y|
2πDa
(5.21)
where K(·) is the complete elliptic integral of the first kind, and e is the
eccentricity of the ellipse,
r
b2
e = 1 − 2.
(5.22)
a
92
Narrow Escape in Three Dimensions
In other words, the MFPT from a large cavity of volume |Ω| through a small
elliptic hole is to leading order
Eτ (a, b) ∼
|Ω|
K(e).
2πDa
(5.23)
π
For example, in the case of a circular hole, we have e = 0 and K(0) = , so
2
that
1
|Ω|
=O
,
(5.24)
Eτ (a, a) ∼
4Da
ε
provided
|Ω|2/3
= O(1) for ε 1.
|∂Ω|
Equation (5.24) was used in [62], [32]. If the mouth of the channel is not
circular, the MFPT is different. Equation (5.24) indicates that a Brownian
particle that tries to leave the domain “sees” finer details in the geometry
of the hole and the domain than just the quotient of the surface areas. The
additional geometric features contained in the MFPT are illustrated by the
two interesting limits e 1, where the ellipse is almost circular, and 1 − e 1, where the ellipse is squeezed. In the case e 1, we use the expansion of
the complete elliptic integral of the first kind [1]
(
)
2
2
3
1
1
·
3
1
·
3
·
5
π
1+
e2 +
e4 +
e6 + · · · . (5.25)
K(e) =
2
2
2·4
2·4·6
In the second limit 1 − e 1, we find from the asymptotic behavior [1]
16
1
lim K(e) − log
=0
(5.26)
e→1
2
1−e
that
|Ω|
16
log
,
4πa
1−e
The area of the hole is given by
Eτ ∼
for 1 − e 1.
√
S = πab = πa2 1 − e2 ,
(5.27)
(5.28)
or equivalently
a=
S 1/2
π 1/2 (1 − e2 )1/4
,
(5.29)
5.3 Explicit Computations for the Sphere
93
and the MFPT has the asymptotic form
√
4
Eτ ∼
5.3
2 |Ω| (1 − e)1/4
16
√
log
,
1−e
4 πS
for 1 − e 1.
(5.30)
Explicit Computations for the Sphere
The analysis of Section 5.2 is not easily extended to the computation, or even
merely the estimation of the next term in the asymptotic approximation of
the MFPT. The explicit results for the particular case of escape from a ball
through a small circular hole gives an idea of the order of magnitude of the
second term and the error in the asymptotic expansion of the MFPT. If
the domain Ω is a ball, the method of [184, 49, 50, 26, 27] can be used to
obtain a full asymptotic expansion of the MFPT. We consider the motion of
a Brownian particle inside a ball of radius R. The particle is reflected at the
ε
sphere, except for a small cap of radius a = εR and surface area 4πR2 sin2 ,
2
where it exits the ball. We assume ε 1. The MFPT v(r, θ, φ) satisfies the
mixed boundary value problem for Poisson’s equation in the ball [166],
∆v(r, θ, φ) = −1,
v(r, θ, φ)
for r < R,
0 ≤ θ ≤ π,
0 ≤ φ < 2π,
= 0,
for 0 ≤ θ < ε,
0 ≤ φ < 2π,
∂v(r, θ, φ) = 0,
∂r
r=R
for ε ≤ θ ≤ π,
0 ≤ φ < 2π,
(5.31)
r=R
The diffusion coefficient has been chosen to be D = 1. Due to the cylindrical
symmetry of the problem, the solution is independent of the angle φ, that is,
v(r, θ, φ) = v(r, θ), so the system (5.31) can be written as
∆v(r, θ) = −1, for r < R, 0 ≤ θ ≤ π,
v(r, θ)
= 0, for 0 ≤ θ < ε,
r=R
∂v(r, θ) = 0, for ε ≤ θ ≤ π,
∂r r=R
94
Narrow Escape in Three Dimensions
where the Laplacian is given by
1 ∂
1
∂
∂v
2 ∂v
∆v(r, θ) = 2
r
+ 2
sin θ
.
r ∂r
∂r
r sin θ ∂θ
∂θ
The function f (r, θ) =
R2 − r 2
is the solution of the boundary value problem
6
∆f = −1,
f for r < R,
= 0.
r=R
In the decomposition v = u + f , the function u(r, θ) satisfies the mixed
Dirichlet-Neumann boundary value problem for the Laplace equation
∆u(r, θ) = 0,
u(r, θ)
= 0,
for r < R,
0 ≤ θ ≤ π,
for 0 ≤ θ < ε,
(5.32)
r=R
∂u(r, θ) R
=
,
∂r
3
r=R
for ε ≤ θ ≤ π.
Separation of variables suggests that
u(r, θ) =
∞
X
n=0
an
r n
R
Pn (cos θ),
(5.33)
where Pn (cos θ) are the Legendre polynomials, and the coefficients {an } are
to be determined from the boundary conditions
∞
X
u(r, θ)
=
an Pn (cos θ) = 0, 0 ≤ θ < ε,
(5.34)
r=R
∂u(r, θ) ∂r n=0
=
r=R
∞
X
n=1
nan Pn (cos θ) =
R2
,
3
ε ≤ θ ≤ π.
(5.35)
Equations (5.34), (5.35) are dual series equations of the mixed boundary
value problem at hand, and their solution results in the solution of the bound-
5.3 Explicit Computations for the Sphere
95
ary value problem (5.32). Dual series equations of the form
∞
X
an Pn (cos θ) = 0,
for 0 ≤ θ < ε,
(5.36)
n=0
∞
X
(2n + 1)an Pn (cos θ) = G(θ),
for ε ≤ θ ≤ π
(5.37)
n=0
are solved in [184, eqs.(5.5.12)-(5.5.14), (5.6.12)]. However, the dual series
equations (5.36)-(5.37) are different from equations (5.34)-(5.35). The factor
2n + 1 that appears in equation (5.37) is replaced by n in equation (5.35).
What seems as a slight difference turns out to make our task much harder.
The factor 2n + 1 fits much more easily into the infinite sums (5.36)-(5.37),
because it is the normalization constant of the Legendre polynomials.
5.3.1
Collins’ method
The solution of dual relations of the form (5.35) (see [184, (5.6.19)-(5.6.20)])
is discussed in [26, 27]. Specifically, assume that for given functions G(θ)
and F (θ) we have the representation
∞
X
−m
(1 + Hn )bn Tm+n
(cos θ) = F (θ),
for
0 ≤ θ < ε,
n=0
∞
X
−m
(2n + 2m + 1)bn Tm+n
(cos θ) = G(θ),
for ε < θ ≤ π,
n=0
−m
where Tm+n
are Ferrer’s associated Legendre polynomials [48, 5] and {Hn }
is a given series that is O(n−1 ) as n → ∞. Then for m = 0, we have
∞
X
(1 + Hn )bn Pn (cos θ) = F (θ),
for 0 ≤ θ < ε,
(5.38)
for ε < θ ≤ π.
(5.39)
n=0
∞
X
n=0
(2n + 1)bn Pn (cos θ) = G(θ),
96
Narrow Escape in Three Dimensions
Setting a0 = b0 , an =
∞
X
2n + 1
bn , n ≥ 1 in equations (5.34)-(5.35) results in
2n
(1 + Hn )bn Pn (cos θ) = 0,
for 0 ≤ θ < ε,
(5.40)
n=0
∞
X
(2n + 1)bn Pn (cos θ) =
n=0
2R2
+ b0 ,
3
for ε ≤ θ ≤ π.
(5.41)
Equations (5.40)-(5.41) are equivalent to (5.38)-(5.39) with H0 = 0, Hn =
2R2
1
, n ≥ 1, F (θ) = 0, and G(θ) =
+ b0 . Collins’ method of solution
2n
3
consists in finding an integral equation for the function
∞
X
(2n + 1)bn Pn (cos θ), for 0 ≤ θ < ε,
h(θ) =
n=0
so that
Z
Z
1 ε
1 π
bn =
h(α)Pn (cos α) sin α dα +
G(α)Pn (cos α) sin α dα.
2 0
2 ε
Substituting into equation (5.38), with F (θ) ≡ 0, we find for 0 ≤ θ < ε that
Z
∞
X
1 ε
(1 + Hn )Pn (cos α)Pn (cos θ) sin α dα
0 =
h(α)
2 0
n=0
1
+
2
5.3.2
Z
π
G(α)
ε
∞
X
(1 + Hn )Pn (cos α)Pn (cos θ) sin α dα.
(5.42)
n=0
The asymptotic expansion
To facilitate the calculations, we consider first the case Hn = 0 for all n.
Then we will show that the leading order term obtained for this case is the
same as that for the case Hn 6= 0. In the latter case, we obtain the first
correction to the leading order term and an estimate on the remaining error.
The leading order term when Hn ≡ 0
We will now sum the series (5.42) in the case Hn ≡ 0. First, we recall
Mehler’s integral representation for the Legendre polynomials [1, 117],
√ Z θ
cos(n + 21 )u du
2
√
Pn (cos θ) =
,
(5.43)
π 0
cos u − cos θ
5.3 Explicit Computations for the Sphere
97
and the identity [184]
∞
√ X
1
H(α − u)
2
Pn (cos α) cos n +
u= √
,
2
cos u − cos α
n=0
(5.44)
where H(x) is the Heaviside unit step function. Then we obtain for u < θ <
ε < α,
Z
∞
X
1 π
G(α)
Pn (cos α)Pn (cos θ) sin α dα =
2 ε
n=0
√ Z θ
Z
∞
X
cos(n + 12 )u du
1 π
2
√
G(α)
Pn (cos α)
sin α dα
=
2 ε
π 0
cos u − cos θ
n=0
Z θ
Z π
1
du
G(α) sin α dα
√
√
=
.
(5.45)
2π 0
cos u − cos α
cos u − cos θ ε
Similarly,
1
2
ε
Z
h(α)
0
1
=
2π
∞
X
Pn (cos α)Pn (cos θ) sin α dα =
n=0
θ
Z
0
ε
Z
du
√
cos u − cos θ
u
h(α) sin α dα
√
.
cos u − cos α
(5.46)
Hence,
Z
θ
du
√
cos u − cos θ
0
Z
−
0
θ
ε
Z
h(α) sin α dα
√
=
cos u − cos α
u
du
√
cos u − cos θ
Z
ε
π
G(α) sin α dα
√
.
cos u − cos α
(5.47)
Equation (5.47) means that the Abel transforms [197] of two functions are
the same, so that
Z ε
Z π
G(α) sin α dα
h(α) sin α dα
√
√
=−
,
(5.48)
cos u − cos α
cos u − cos α
ε
u
because the Abel transform is uniquely invertible. Equation (5.48) is an
Abel-type integral equation, whose solution is given by
Z ε
Z π
1 d
sin u du
G(α) sin α dα
√
√
h(θ) sin θ =
,
(5.49)
π dθ θ
cos u − cos α
cos θ − cos u ε
98
Narrow Escape in Three Dimensions
or
Z
2 d
h(θ) = −
sin θ dθ
ε
θ
H(u) sin u du
√
,
cos θ − cos u
(5.50)
where
H(u) = −G(u, ε),
(5.51)
and
1
G(u, ε) =
2π
Z
ε
π
G(θ) sin θ dθ
√
.
cos u − cos θ
The dual integral equations (5.40)-(5.41) define G(θ) =
(5.52)
2R2
+ b0 , so that
3
Z π 2
1
2R
sin θ dθ
G(ψ, φ) =
+ b0 √
2π φ
3
cos ψ − cos θ
π
2
1p
2R
+ b0
cos ψ − cos θ =
3
π
θ=φ
2
2R
ψ p
1 √
=
+ b0
2 cos − cos ψ − cos φ ,
3
π
2
(5.53)
for ψ < φ.
In particular, setting n = 0 in equation (5.42) and using equation (5.50),
gives
b0
ε
π
2R2
+ b0 sin α dα
3
0
ε
2
√ Z ε
ψ
2R
ε
=
2
H(ψ) cos dψ +
+ b0 cos2 .
2
3
2
0
1
=
2
Z
1
h(α) sin α dα +
2
Z
(5.54)
5.3 Explicit Computations for the Sphere
Integrating equation (5.53), we obtain
√ Z ε
ψ
G(ψ, ε) cos dψ =
(5.55)
2
2
0
2
√ Z ε
√
2R
2
ψ p
ψ
=
+ b0
2 cos − cos ψ − cos ε cos dψ
3
π 0
2
2
=
=
2R2
2
Z sin ε
+ b0
2
2R
4
2 r s ds
3
(ε + sin ε) −
+ b0
π
3
π 0
ε
sin2 − s2
2
2R2
2
+ b0
2R
ε
3
(ε + sin ε) −
+ b0 sin2 .
π
3
2
Combining equations (5.54) and (5.55) gives
|Ω| a 2R2 π
π
−1 =
+ O(1) =
1+O
,
ε + sin ε
3
2ε
4a
R
(5.56)
3
4πR
where |Ω| =
is the volume of the ball, and a = Rε is the radius of the
3
hole.
2R2
b0 =
3
The case Hn 6= 0
The asymptotic expression (5.56) for b0 , was derived under the simplifying
assumption that Hn ≡ 0. However, we are interested in the value of b0 which
is produced by the solution of the dual series equations (5.40)-(5.41), where
99
100
Narrow Escape in Three Dimensions
Hn =
1
. We sum the series (5.42) by the identities
2n
1
2
Z
∞
X
ε
h(α)
0
Hn Pn (cos α)Pn (cos θ) sin α dα
n=0
√ Z α
√ Z
cos(n + 12 )v dv 2 θ cos(n + 12 )u du
2
√
√
sin α dα
h(α)
Hn
π
π
cos
v
−
cos
α
cos
u
−
cos
θ
0
0
0
n=0
Z ε
Z α
Z θ
1
dv
K(u, v) du
√
√
=
h(α) sin α dα
2π 0
cos v − cos α 0
cos u − cos θ
0
Z ε
Z ε
Z θ
h(α) sin α dα
1
du
√
√
K(u, v) dv
=
,
(5.57)
2π 0
cos v − cos α
cos u − cos θ 0
v
1
=
2
Z
∞
X
ε
where
∞
2X
1
1
K(u, v) =
Hn cos n +
u cos n +
v
π n=0
2
2
1
cos 12 (v + u)
log 2 sin (v + u)
= −
2π
2
1
cos 12 (v − u)
log 2 sin (v − u)
−
2π
2
+
(5.58)
v+u−π
1
v−u−π
1
sin (v + u) +
sin (v − u).
4π
2
4π
2
Similarly,
1
2
π
Z
G(α)
ε
Hn Pn (cos α)Pn (cos θ) sin α dα
(5.59)
n=0
1
=
2π
Z
1
2π
Z
=
∞
X
π
Z
α
G(α) sin α dα
ε
0
0
θ
√
du
cos u − cos θ
Z
dv
√
cos v − cos α
π
0
Z
G(α) sin α dα
ε
θ
Z
0
K(u, v) du
√
cos u − cos θ
α
√
K(u, v) dv
.
cos v − cos α
5.3 Explicit Computations for the Sphere
101
Substituting equations (5.45), (5.46), (5.57), and (5.59) into equation (5.42)
yields
Z ε
Z θ
h(α) sin α dα
1
du
√
√
0 =
2π 0
cos u − cos α
cos u − cos θ u
Z ε
Z ε
Z θ
1
h(α) sin α dα
du
√
√
+
K(u, v) dv
2π 0
cos v − cos α
cos u − cos θ 0
v
Z π
Z θ
G(α) sin α dα
1
du
√
√
+
2π 0
cos u − cos α
cos u − cos θ ε
Z θ
Z π
Z α
1
du
K(u, v) dv
√
√
+
G(α) sin α dα
,
2π 0
cos v − cos α
cos u − cos θ ε
0
which is again an Abel-type integral equation. Inverting the Abel transform
[197], we obtain
Z ε
Z ε
Z ε
1
h(α) sin α dα
1
h(α) sin α dα
√
√
0 =
+
K(u, v) dv
2π u
cos u − cos α 2π 0
cos v − cos α
v
(5.60)
Z π
Z π
Z α
1
G(α) sin α dα
K(u, v) dv
1
√
√
+
G(α) sin α dα
+
.
2π ε
cos u − cos α 2π ε
cos v − cos α
0
Setting
Z ε
h(α) sin α dα
1
√
,
(5.61)
H(u) =
2π u
cos u − cos α
we invert the Abel transform (5.61) to obtain
Z ε
2 d
sin uH(u) du
√
h(θ) = −
.
(5.62)
sin θ dθ θ
cos θ − cos u
Writing
J(u) = H(u) + G(u, ε),
(5.63)
equation (5.60) becomes
Z
J(u) +
ε
K(u, v)J(v) dv = M (u),
(5.64)
0
where the free term M (u) is given by
Z π
M (u) = −
K(u, v)G(v, v) dv.
ε
Equation (5.64) is a Fredholm integral equation for J.
(5.65)
102
Narrow Escape in Three Dimensions
The second term and the remaining error: L2 estimates
Equations (5.54), (5.55), and (5.63) give that
√
Z ε
2R2
2R2
π
2π
u
b0 +
=
+
J(u) cos du,
3
3 ε + sin ε ε + sin ε 0
2
(5.66)
where J is the solution of the Fredholm equation (5.64). In this section we
show that
√
Z ε
u
2R2
1
2π
J(u) cos du = b0 +
ε log + O(ε) ,
ε + sin ε 0
2
3
ε
therefore the last term in eq.(5.66) should be considered a small correction
R2 π
to the leading order term
, obtained in Section 5.3.2. This confirms the
3 ε
intuitive results of [62], [32] and gives an estimate on the error term. Due to
the logarithmic singularity of the function K(u, v) (see (5.58)) the operator
K, defined by
Z
ε
Kf (u) =
K(u, v)f (v) dv,
(5.67)
0
maps L2 [0, ε] into L2 [0, ε]. In Appendix B.1 we derive the estimate
√
30
1
ε log ,
kKk2 ≤
2π
ε
(5.68)
for ε 1. Better estimates can be found; however we settle for this rough
estimate that suffices for our present purpose.
Estimate of kJk2
In terms of the operator K, equation (5.64) can be written as
J = M − KJ.
(5.69)
kJk2 ≤ kM k2 + kKJk2 ≤ kM k2 + kKk2 kJk2 ,
(5.70)
The triangle inequality yields
which together with the estimate (5.68) gives
1
kM k2
≤ 1 + ε log
kM k2
kJk2 ≤
1 − kKk2
ε
for ε 1.
(5.71)
5.3 Explicit Computations for the Sphere
103
Estimate of kM k2
We proceed to find an estimation for kM k2 . First, we prove that the kernel
satisfies the identity
Z π
v
K(u, v) cos dv = 0, for all u.
(5.72)
2
0
Indeed, by changing the order of summation and integration, we obtain
Z π
Z π
∞
u
v
1 X cos n+1
1
v
2
K(u, v) cos dv =
cos n +
v cos dv
2
π n=1
n
2
2
0
0
Z
∞
u π
1 X cos n+1
2
(cos(n + 1)v + cos nv) dv
=
2π n=1
n
0
= 0.
Equations (5.53), (5.65), and (5.72) imply that
√ 2
Z ε
v
2 2R
M (u) =
+ b0
K(u, v) cos dv.
π
3
2
0
The estimate (5.68) gives
√ 2
√ 2
√
2 2R
15 2R
1
kM k2 ≤
+ b0 kKk2 ε ≤ 2
+ b0 ε3/2 log .
π
3
π
3
ε
Combining the estimates (5.71) and (5.75), we obtain for ε 1
2
4 2R2
2R
1
3/2
kJk2 ≤ 2
+ b0 ε log =
+ b0 O(ε3/2 log ε).
π
3
ε
3
(5.73)
(5.74)
(5.75)
(5.76)
The second term and error estimate
The Cauchy-Schwartz inequality implies that
√
Z
2
2π ε
u 2R
1
J(u) cos du ≤
+ b0 ε log ,
ε + sin ε 0
2
3
ε
(5.77)
for ε 1, which together with (5.66) gives
b0 =
πR2
|Ω|
(1 + O(ε log ε)) =
(1 + O(ε log ε)) .
3ε
4a
(5.78)
104
Narrow Escape in Three Dimensions
To obtain the explicit expression for the term O(ε log ε), we write the Fredholm integral equation (5.64) as
(I + K)J = M.
(5.79)
The estimate (5.68) implies that kKk2 < 1 for sufficiently small ε, hence
J = M + O (kKk2 kM k2 ) .
(5.80)
Thus, using equation (5.74) and the estimates (5.68) and (5.75), we write the
last term in equation (5.66) as
Z ε
Z ε
u
u
J(u) cos du =
M (u) cos du + O (εkKk2 kM k2 ) =
(5.81)
2
2
0
0
√ Z ε Z ε
u
v
2
2R2
2
3
K(u, v) cos cos du dv + O ε log ε .
b0 +
π
3
2
2
0
0
Equation (5.58) gives the double integral as
Z εZ ε
u
v
1
1
K(u, v) cos cos du dv = ε2 log + O(ε2 ),
2
2
π
ε
0
0
hence
√
2π
ε + sin ε
Z
0
ε
u
J(u) cos du =
2
2R2
b0 +
3
1
ε log + O(ε) .
ε
Now it follows from equation (5.66) that
|Ω|
1
b0 =
1 + ε log + O(ε) .
4a
ε
5.3.3
(5.82)
The MFPT
Using the explicit expression (5.82), we obtain the MFPT from the center of
the ball as
R2
|Ω|
1
R2
v
= u
+
= b0 +
=
1 + ε log + O(ε) .
(5.83)
6
6
4a
ε
r=0
r=0
This is also the averaged MFPT for a uniform initial distribution,
Z 2π
Z π
Z R
1
|Ω|
1
2
Eτ =
dφ
sin θ dθ
v(r, θ)r dr =
1 + ε log + O(ε) .
|Ω| 0
4a
ε
0
0
5.4 Summary and Applications
5.4
Summary and Applications
The narrow escape problem for a Brownian particle leads to a singular perturbation problem for a mixed Dirichlet-Neumann (corner) problem with
large Neumann part and small Dirichlet part of the boundary. The corner
problem, that arises in classical electrostatics (e.g., the electrified disk), elasticity (punch problems), diffusion and conductance theory, hydrodynamics,
acoustics, and more recently in molecular biophysics, was solved hitherto
mainly for special geometries. In this chapter, we have constructed a leading
order asymptotic approximation to the MFPT in the narrow escape problem
for a general smooth domain and have derived a second term and an error
estimate for the case of a sphere. Our derivation makes Lord Rayleigh’s
qualitative observation into a quantitative one. Our leading order analysis of
the general case uses the singularity property of the Neumann function for a
general domain in R3 . The special case of the sphere is analyzed by a method
developed by Collins and yields a better result. A different approach to the
calculation of the MFPT would be to use singular perturbation techniques.
The vanishing escape time at the boundary would then be matched to the
large outer escape time of order ε−1 by constructing a boundary layer near
the boundary. The analysis of the MFPT to a small window at an isolated
singular point of the boundary is postponed to a future paper. Brownian
motion through narrow regions controls flow in many non-equilibrium systems, from fluidic valves to transistors and ion channels, the protein valves
of biological membranes [73]. Indeed, one can view an ion channel as the
ultimate nanovalve—nearly picovalve—in which macroscopic flows are controlled with atomic resolution. In this context, the narrow escape problem
appeared in the calculation of the equilibration time of diffusion between
two chambers connected by a capillary [32]. The equilibration time is the
reciprocal of the first eigenvalue of the Neumann problem in this domain,
which depends on the MFPT of a Brownian motion in each chamber to the
narrow connecting channel. The first eigenfunction is constructed by piecing
together the eigenfunctions of the narrow escape problem in each chamber
and in the channel so that the function and the flux are continuous across
the connecting interfaces. It was assumed in [32] that the flux profile in
the connecting hole was uniform. The structure of the flux profile, which is
proportional to (a2 − ρ2 )−1/2 , has been observed by Rayleigh in 1877 [147].
Rayleigh first assumed a radially uniform profile of flux and then refined the
profile of flux going through the channel, allowing it to vary with the radial
105
106
Narrow Escape in Three Dimensions
distance from the center of the cross section of the channel, so as to minimize
the kinetic energy. A calculation of the equilibration time was carried out
in [98] by solving the same problem, and gave a result that differs from that
of [147], which was obtained by heuristic means, by less than two percent.
A different approximation, based on the Fourier-Bessel representation in the
pore, was derived in [50]. Another application of the narrow escape problem
concerns ionic channels [73], and particularly particle simulations of the permeation process [83, 84, 29, 200, 189] that capture much more detail than
continuum models. Up to now, computer simulations are inefficient because
an ion takes so long even to enter a channel and then so many of the ions
return from where they came. From the present analysis, it becomes clear
why ions take so long to enter the channel. According to (5.2) the mean time
between arrival of ions at the channel is
1
Eτ
=
,
(5.84)
τ̄ =
N
4DaC
where N is the number of ions in the simulation and C is their concentration.
A coarse estimate of τ̄ at the biological concentration of 0.1Molar, channel
radius a = 20Å, diffusion coefficient D = 1.5 × 10−9 m2 /sec is τ̄ ≈ 1nsec. In
a Brownian dynamics simulation of ions in solution with time step which is
10 times the relaxation time of the Langevin equation to the Smoluchowski
(diffusion) equation at least 1000 simulation steps are needed on the average
for the first ion to arrive at the channel. It should be taken into account that
most of the ions that arrive at the channel do not cross it [47].
The narrow escape problem comes up in problems of the escape from a
domain composed of a big subdomain with a small hole, connected to a thin
cylinder (or cylinders) of length L. If ions that enter the cylinder do not
return to the big subdomain, the MFPT to the far end of the cylinder is
the sum of the MFPT to the small hole and the MFPT to the far end of
the narrow cylinder. The latter can be approximated by a one-dimensional
problem with one reflecting and one absorbing endpoint. If the domain has
a volume V , the approximate expression for the MFPT is
L2
V
+
.
(5.85)
4εD 2D
This method can be extended to a domain composed of many big subdomains
with small holes connected by narrow cylinders. The case of one sphere of
4πR3
volume V =
, with a small opening of size ε connected to a thin cylinder
3
Eτ ≈
5.4 Summary and Applications
107
of length L is relevant in biological micro-structures, such as dendritic spines
in neurobiology. Indeed, the mean time for calcium ion to diffuse from the
spine head to the parent dendrite through the neck controls the spine-dendrite
coupling [80]. This coupling is involved in the induction of processes such
as synaptic plasticity [118]. Formula (5.85) is useful for the interpretation of
experiments and for the confirmation of the diffusive motion of ions from the
spine head to the dendrite.
Another significant application of the narrow escape formula is to provide
a new definition of the forward binding rate constant in micro-domains [79].
Indeed, the forward chemical constant is really the flux of particles to a
given portion of the boundary, depending on the substrate location. Up
to now, the forward binding rate was computed using the Smoluchowski
formula, which corresponds to the absorption flux of particles in a given
sphere immersed in an infinite medium. The formula applies when many
particles are involved. But to model chemical reactions in micro-structures,
where a bounded domain contains only a few particles that bind to a given
number of binding sites, the forward binding rate,
kforward =
1
,
τ̄
has to be computed with τ̄ given in eq.(5.84).
Chapter 6
Narrow Escape: The Circular
Disk
The contents of this chapter were submitted for publication [178]
We consider Brownian motion in a circular disk Ω, whose boundary ∂Ω is
reflecting, except for a small arc, ∂Ωa , which is absorbing. As ε = |∂Ωa |/|∂Ω|
decreases to zero the mean time to absorption in ∂Ωa , denoted Eτ , becomes
infinite. The narrow escape problem is to find an asymptotic expansion of
Eτ for ε 1. We find the first two terms in the expansion and an estimate of the error. The results are extended in a straightforward manner
to planar domains and two-dimensional Riemannian manifolds that can be
mapped conformally onto the disk. Our results improve the previously de
|Ω|
1
rived expansion for a general smooth domain, Eτ =
log + O(1) ,
Dπ
ε
(D is the diffusion coefficient) in the case of a circular disk. We find that
the mean first passage time from
the center of the disk is E[τ | x(0) = 0] =
2
1
1
R
log + log 2 + + O(ε) . The second term in the expansion is needed
D
ε
4
in real life applications, such as trafficking of receptors on neuronal spines,
1
because log is not necessarily large, even when ε is small. We also find the
ε
singular behavior of the probability flux profile into ∂Ωa at the endpoints of
∂Ωa , and find the value of the flux near the center of the window.
6.1 Introduction
6.1
Introduction
The expected lifetime of a Brownian motion in a bounded domain, whose
boundary is reflecting, except for a small absorbing portion, increases indefinitely as the absorbing part shrinks to zero. The narrow escape problem is to
find an asymptotic expansion of the expected lifetime of the Brownian motion in this limit. The narrow escape problem in three dimensions has been
studied in the first chapter of this series [177], where is was converted to a
mixed Dirichlet-Neumann boundary value problem for the Poisson equation
in the domain. This is a well known problem of classical electrostatics (e.g.,
the electrified disk problem [88]), elasticity (punch problems), diffusion and
conductance theory, hydrodynamics, and acoustics [184, 49, 50, 116, 192]. It
dates back to Helmholtz [69] and Lord Rayleigh [147] and has been extensively studied in the literature for special geometries.
The study of the two-dimensional narrow escape problem began in [78]
in the context of receptor trafficking on biological membranes [14], where a
leading order expansion of the expected lifetime was constructed for a general smooth planar domain. In this chapter we present a thorough analysis
of the narrow escape problem for the circular disk and note that our calculations apply in a straightforward manner to any simply connected domain
in the plane that can be mapped conformally onto the disk. According to
Riemann’s mapping theorem [122], this covers all simply connected planar
domains whose boundary contains at least one point. The same conclusion holds for the narrow escape problem on two-dimensional Riemannian
manifolds that are conformally equivalent to a circular disk. The biological
problem of receptor trafficking on membranes is locally planar, but globally
it is a problem on a Riemannian manifold. The narrow escape problem of
non-smooth domains that contain corners or cusp points at their boundary
is treated in the third part of this series [179], where the conformal mapping
method is demonstrated.
The specific mathematical problem can be formulated as follows. A Brownian particle diffuses freely in a disk Ω, whose boundary ∂Ω is reflecting,
except for a small absorbing arc ∂Ωa . The ratio between the arclength of
the absorbing boundary and the arclength of the entire boundary is a small
parameter
|∂Ωa |
1.
ε=
|∂Ω|
The mean first passage time to ∂Ωa , denoted Eτ , becomes infinite as ε → 0.
109
110
Narrow Escape: The Circular Disk
The asymptotic expansion of Eτ for ε 1 was considered for the particular
case when ∂Ωa is a disjoint component of ∂Ω in [144, and references therein].
This case differs from the case at hand in that the absorption probability
flux density in the former is regular, while in the latter it is singular. It was
shown in [78] that Eτ for the narrow escape problem in a general planar
domain Ω has the asymptotic form
1
|Ω|
log + O(1) ,
(6.1)
Eτ =
Dπ
ε
where |Ω| is the area of Ω, and D is the diffusion coefficient. This leading
order asymptotics has the drawback that log ε can be O(1) when ε 1.
Thus the second term in the expansion is needed. For the particular case of
a circular disk an approximate value for the correction was given in [78]. In
contrast, the asymptotics of Eτ for a three dimensional ball of radius R with
an absorbing window of radius εR is [177]
|Ω|
[1 + O(ε log ε)] ,
4DεR
so the leading order term is much larger than the correction term if ε is
small. The difference in the asymptotic form of Eτ stems from the different
singularities of the Neumann function in two and three dimensions: it is
logarithmic in two dimensions and has a pole in three dimensions.
Our computations are based on the mixed boundary value techniques of
[184]. They reveal the singularity of the absorption flux in the absorbing
arc ∂Ωa . Specifically, the singularity is (ε2 − s2 )−1/2 , where s is the (dimensionless) arclength measured from the center of ∂Ωa , and attains the values
s = ±ε at the endpoints.
The exit time vanishes at the absorbing boundary, and is small near
the absorbing boundary, but it attains large and almost constant values of
1
order log inside the domain. We show that this “jump” occurs in a small
ε
1
boundary layer of size O ε log
. We calculate the average exit time, where
ε
the averaging is against a uniform initial distribution in the disk, the time
to exit from the center, and the maximum mean exit time, attained at the
antipodal point to the center of the absorbing window.
The mean first passage time (MFPT) from the center of the disk is
R2
1
1
E[τ | x(0) = 0] =
log + log 2 + + O(ε) ,
(6.2)
D
ε
4
Eτ =
6.2 Solution of a Mixed Boundary Value Problem
111
the MFPT, averaged with respect to an initial uniform distribution in the
disk is
R2
1
1
Eτ =
log + log 2 + + O(ε) ,
(6.3)
D
ε
8
and the maximal value of the MFPT is attained on the circumference, at the
antipodal point to the center of the hole,
R2
1
max E[τ | x] = E[τ | r = 1, θ = 0] =
log + 2 log 2 + O(ε) .
(6.4)
x∈Ω
D
ε
The boundary layer analysis of Eτ can be applied to the approximation
of the first eigenfunction and eigenvalue of the mixed Neumann-Dirichlet
boundary value problem with a small Dirichlet window on the boundary. This
problem arises in the construction of the first eigenfunction and eigenvalue
of the Neumann problem in a domain that consists of two domains (e.g.,
circular disks) connected by a narrow channel [62], [32].
Specifically, it is easy to see that
∞
X
1
1
Eτ =
∼
λ
λ0
n=0 n
(6.5)
where 0 < λ0 λ1 < · · · are the eigenvalues of the mixed problem and
the MFPT is also averaged with respect to the initial point. The first eigenfunction u0 of the mixed problem is differs from the first eigenfunction of the
Neumann problem, which is v0 = 1, only in a boundary layer about the small
window. Thus u0 is a small perturbation (in L2 norm) of v0 = 1. It follow
that u0 /λ0 differs from Eτ only in the boundary layer.
6.2
Solution of a Mixed Boundary Value Problem
In non-dimensional variables the narrow escape problem concerns Brownian
motion inside the unit disk, whose boundary is reflecting but for a small
absorbing arc of length 2ε (see Fig.6.1). In polar coordinates x = (r, θ) the
MFPT
v(r, θ) = E[τ | x(0) = (r, θ)],
112
Narrow Escape: The Circular Disk
Figure 6.1: A circular disk of radius R. The arclength of the absorbing
boundary (dashed line) is 2εR. The solid line indicates the reflecting boundary.
6.2 Solution of a Mixed Boundary Value Problem
113
is the solution to the mixed Neumann-Dirichlet inhomogeneous boundary
value problem (see, e.g. [166])
∆v(r, θ) = −1,
v(r, θ)
r < 1,
for 0 ≤ θ < 2π,
= 0,
for |θ − π| < ε,
∂v(r, θ) = 0,
∂r r=1
for |θ − π| > ε,
r=1
(6.6)
which is reduced by the substitution
u=v−
1 − r2
4
(6.7)
to the mixed Neumann-Dirichlet problem for the Laplace equation
∆u(r, θ) = 0,
u(r, θ)
for r < 1,
0 ≤ θ < 2π,
= 0,
for |θ − π| < ε,
∂u(r, θ) 1
=
,
∂r
2
r=1
for |θ − π| > ε.
(6.8)
r=1
We adapt the method of [184] to the solution of (6.8). Separation of
variables suggests that
∞
a0 X
u(r, θ) =
+
an rn cos nθ,
2
n=1
(6.9)
where the coefficients {an } are to be determined by the boundary conditions
u(r, θ)
∞
r=1
a0 X
=
+
an cos nθ = 0,
2
n=1
∞
X
∂u(r, θ) 1
=
nan cos nθ = ,
∂r
2
r=1
n=1
for π − ε < θ ≤ π, (6.10)
for 0 ≤ θ < π − ε.
(6.11)
114
Narrow Escape: The Circular Disk
We identify this problem with problem (5.4.4) in [184], where general functions appear on the right hand sides of equations (6.10) (6.11). Due to the
invertibility of Abel’s integral operator, the equation
∞
a0 X
1
+
an cos nθ = cos θ
2
2
n=1
π−ε
Z
√
θ
h1 (t) dt
,
cos θ − cos t
(6.12)
defines h1 (t) uniquely for 0 ≤ t < π − ε. The coefficients are given by
Z π−ε
h (t) dt
1
√ 1
cos nθ cos θ dθ
2
cos θ − cos t
θ
0
Z t
Z
cos n + 21 θ + cos n − 12 θ
1 π−ε
√
h1 (t) dt
=
dθ.
π 0
cos θ − cos t
0
2
=
π
an
The integral
Z
π−ε
√ Z u
cos n + 12 θ
2
√
Pn (cos u) =
dθ,
π 0
cos θ − cos u
(6.13)
(6.14)
is Mehler’s integral representation of representation of the Legendre polynomial [1]. It follows that
1
an = √
2
Z
π−ε
h1 (t)[Pn (cos t) + Pn−1 (cos t)] dt,
(6.15)
0
for n > 0, and
2
a0 =
π
Z
π−ε
Z
h1 (t) dt
0
0
t
√
cos 21 θ
cos θ − cos t
√ Z
dθ = 2
π−ε
h1 (t) dt.
(6.16)
0
Integration of (6.11) gives
∞
X
1
an sin nθ = θ,
2
n=1
for 0 ≤ θ < π − ε.
(6.17)
Changing the order of summation and integration yields
Z
0
π−ε
∞
1 X
1
h1 (t) √
[Pn (cos t) + Pn−1 (cos t)] sin nθ dt = θ.
2
2 n=1
(6.18)
6.2 Solution of a Mixed Boundary Value Problem
115
Using [184, eq.(2.6.31)],
∞
cos 12 θH(θ − t)
1 X
√
[Pn (cos t) + Pn−1 (cos t)] sin nθ = √
,
cos t − cos θ
2 n=1
(6.19)
we obtain
Z
0
θ
√
θ
h1 (t) dt
,
=
2 cos 12 θ
cos t − cos θ
for 0 ≤ θ < π − ε.
(6.20)
The solution of the Abel-type integral equation (6.20) is given by
h1 (t) =
1 d
π dt
t
Z
0
u
2
√
du.
cos u − cos t
u sin
(6.21)
Together with (6.16) this gives
u
√ Z π−ε
u sin
2
2
√
a0 =
du.
π 0
cos u + cos ε
(6.22)
We expect the function u(r, θ), closely related to the MFPT, to be almost
constant in the disk, except for a boundary layer near the absorbing arc. The
value of this constant is a0 , because all other terms of expansion (6.9) are
oscillatory.
6.2.1
Small ε asymptotics
The results of the previous section are independent of the value of ε. Here
we find the asymptotic of a0 for ε 1. Substituting
r
s=
cos u + cos ε
2
(6.23)
116
Narrow Escape: The Circular Disk
in the integral (6.22) yields
a0
4
=
π
cos(ε/2)
Z
0
r
ε
arccos s2 + sin2
2
r
ds
2 ε
2
s + sin
2
r
ε
arcsin s2 + sin2
4
1
2
q
r
= 2
ds −
ds
π
ε
2 ε
2
0
0
2
s + sin 2
s2 + sin
2
r
ε
Z cos(ε/2) arcsin s2 + sin2
ε 4
ε
2
r
− 2 log sin −
= 2 log 1 + cos
ds
2
2 π 0
2 ε
2
s + sin
2
Z 1
ε
4
arcsin s
= −2 log + 2 log 2 −
ds + O(ε)
2
π 0
s
Z
cos(ε/2)
= −2 log
Z
Z
cos(ε/2)
ε
+ O(ε),
2
(6.24)
1
π
arcsin s
ds = log 2. The substitution (6.23) turns out to be
s
2
0
extremely useful in evaluating the integrals appearing here.
because
6.2.2
Expected lifetime
Now, that we have the asymptotic expansion of a0 (eq.(6.24)), the evaluation of expected lifetime (MFPT to the absorbing boundary ∂Ωa ) becomes
possible. Setting r = 0 in equations (6.7) and (6.9), we obtain the expression
(6.2) for MFPT from the center of the disk.
Averaging (6.8) with respect to a uniform initial distribution in Ω gives
1
Eτ =
π
=
Z
2π
Z
dθ
0
1
Z
v(r, θ)r dr =
0
a0 1
ε 1
+ = − log + ,
2
8
2 8
0
1
1
r3
a0 +
r−
dr
2
2
(6.25)
6.2 Solution of a Mixed Boundary Value Problem
117
as asserted in eq.(6.3).
The maximal value of the MFPT is attained at the point r = 1, θ = 0,
∂u
which is antipodal to the center of the absorbing arc. At this point
= 0,
∂θ
as can be seen by differentiating expansion (6.9) term by term. Setting r = 1
and θ = 0, we find that
∞
vmax
a0 X
= u(1, 0) =
+
an .
2
n=1
(6.26)
The evaluation of the maximal exit time is not as straightforward as the
previous evaluated MFPTs, because one needs to calculate the infinite sum in
(6.26). This calculation is done in Appendix C.1, where we find (eq.(C.12))
vmax = log
1
+ 2 log 2 + O(ε),
ε
as asserted in equation (6.4).
6.2.3
Boundary layers
1
We see that the maximal exit time is only vmax − vcenter = log 2 −
=
4
.4431471806 . . . longer than its value at the center of the disk. In other words,
the variance along the radius θ = 0, 0 ≤ r ≤ 1 is very small. However, in the
opposite direction θ = π, 0 ≤ r ≤ 1, we expect a much different behavior.
1
In particular, the MFPT is decreasing from a value of vcenter ≈ log at the
ε
center of the disk to v(1, π) = 0 at the center of ∂Ωa . The calculation of the
exit time
∞
1 − r 2 a0 X
vray (r) ≡ v(r, θ = π) =
+
+
an (−r)n ,
4
2
n=1
(6.27)
is similar to that of √
the maximal exit time and is done in Appendix C.2. For
ε 1 and 1 − r ε, we find the asymptotic form (eq.(C.20))
vray (r) = − log
ε
1 − r2
+ 2 log(1 − r) +
− log(1 + r2 ) + q(r) + O(ε), (6.28)
2
4
where q(r) is a smooth function in the interval [0, 1] (eqs.(C.18)-(C.19)).
Clearly, this asymptotic expansion does not hold all the way through to the
118
Narrow Escape: The Circular Disk
absorbing arc at r = 1, where the boundary condition requires vrayr
(r = 1) =
ε
0. Instead, the boundary condition is almost satisfied at r = 1 −
2
r r ε
ε
ε
vray 1 −
= − log + 2 log
+ O(ε) = O(ε),
(6.29)
2
2
2
In other words, the asymptotic series (6.28) is the outer expansion [10].
√
We proceed to construct the boundary layer for 1 − r ε. Setting
δ = 1 − r, we have the identities
1
1
1 − r2
=
δ − δ2,
4
2
4
1 − 2r cos ε + r2 = 4 sin2
ε
(1 − δ) + δ 2 .
2
The exact form of the MFPT along the ray, eq.(C.15), gives the expansion
vray (δ) =
δ
a0 δ
+
2 4 sin ε
2
(6.30)
r
−
Z
δ
π sin
ε
2
0
cos(ε/2)
arccos
s2
s2 + sin2
ε 2
s ds
2
ε 3/2
+ sin
2
2
+O
δ2
ε
.
Evaluating the integral in eq.(6.30),
r
ε
Z cos(ε/2) arccos s2 + sin2 s2 ds
2
3/2
ε
0
s2 + sin2
2
i
πh
ε
ε
ε
+ log 2 + O(ε), (6.31)
= − log sin + cos − log 1 + cos
2
2
2
2
we obtain the boundary layer structure
δ
δ2
vray (δ) =
+ O δ,
.
ε
ε
(6.32)
6.2 Solution of a Mixed Boundary Value Problem
ε
yields
2
ε
vray (δ0 ) = − log + O(ε log2 ε),
2
119
In particular, setting δ0 = −ε log
(6.33)
which is the value of the outer
solution. We conclude that the width of the
1
boundary layer is O ε log
. Furthermore, the flux at the center of the
ε
hole is given by
∂vray ∂vray 1
fluxcenter =
(6.34)
=−
= − + O(1).
∂r r=1
∂δ δ=0
ε
6.2.4
Flux profile
Next, we calculate the profile of the flux on the absorbing arc. Differentiating
expansion (6.9) gives the flux as
∞
∂v(r, θ) ∂u(r, θ) 1
1 X
nan cos nθ,
(6.35)
f (θ) =
=
− =− +
∂r r=1
∂r r=1 2
2 n=1
for π − ε < θ ≤ π. Using equation (6.15) for the coefficients, we have
Z π−ε
∞
X
1
1
√
f (θ) = − +
n[Pn (cos t) + Pn−1 (cos t)] cos nθ
h1 (t) dt
2
2 0
n=1
Z π−ε
∞
X
1
1 d
= − +√
h1 (t) dt
[Pn (cos t) + Pn−1 (cos t)] sin nθ.
2
2 dθ 0
n=1
Since θiπ − εit, equation (6.19) implies
Z
1
h1 (t) dt
d
θ π−ε
√
f (θ) = − +
cos
.
2 dθ
2 0
cos t − cos θ
(6.36)
The evaluation of this integral is not immediate and is given in Appendix
C.3. We find that (eq.(C.37))
!
∞
2
α2
1 X (2n+1 (n + 1)!) 2
(2n n!)2
f (α) = − √
−
(1 − α2 )n+1/2
α −
(2n + 2)!
(2n + 1)!
ε 1 − α2 ε n=0
∞ π X (2n)!
(2n + 2)!(2n + 2) 2
−
−
α (1 − α2 )n + O(1), (6.37)
2ε n=0 (2n n!)2
(2n+1 (n + 1)!)2
120
Narrow Escape: The Circular Disk
π−θ
, |α| < 1. The flux has a singular part, represented by the
ε
half-integer powers of (1 − α2 ), and a remaining regular part (the integer
α2
powers.) The first term, − √
, is the most singular one, because it
ε 1 − α2
becomes infinite as |α| → 1. In other words, the flux is infinitely large near
the boundary of the hole. The splitting of the solution into singular and
regular parts is common in the theory of elliptic boundary value problems in
domains with corners (see e.g., [105, 106, 33]).
The value of the flux at the center of the hole is to leading order
where α =
∞
1X
f (0) = −
ε n=0
(2n n!)2
π (2n)!
−
2 (2n n!)2 (2n + 1)!
1
=− ,
ε
(6.38)
in agreement with (6.34) (thanks Maple for calculating the infinite sum.)
The size of the boundary layer is varying with θ proportionally to 1/f (θ).
The singularity at the end points of the hole indicate that the layer shrinks
there to zero. Therefore, the boundary layer is shaped
√ as a small cap bounded
by the absorbing arc and (more or less) the curve 1 −α2 (see
Fig.6.2). In
1
particular, the MFPT on the reflecting boundary is O log
, even when
ε
taken arbitrarily close to the absorbing boundary. The singularity of the flux
near the endpoints indicates that the diffusive particle prefers to exit near
the endpoints rather than through the center of the hole.
The expansion (C.37) is useful in approximating the flux near the endpoints (α = ±1), where few terms are needed. However, it is slowly converging near the center of the hole, where a power series in α2 should be used
instead
∞
X
f (α) =
fn α2n + O(1),
(6.39)
n=0
where the coefficients fn are O(ε−1 ). Equations (6.38) and (6.34) indicate
1
that f0 = − . All other coefficients can be found in a similar fashion. We
ε
conclude that near the center (α 1) we have
1
α2
.
(6.40)
f (α) = − + O 1,
ε
ε
6.2 Solution of a Mixed Boundary Value Problem
Figure 6.2: The boundary layer, indicated by “BL”, is the area bounded
by the absorbing boundary (dashed
line)
and the solid arc. Outside the
1
boundary layer the MFPT is O log
.
ε
121
Chapter 7
Narrow Escape: Riemann
Surfaces and Non-Smooth
Domains
The contents of this chapter were submitted for publication [179]
We consider Brownian motion in a bounded domain Ω on a two-dimensional
Riemannian manifold (Σ, g). We assume that the boundary ∂Ω is smooth
and reflects the trajectories, except for a small absorbing arc ∂Ωa ⊂ ∂Ω.
As ∂Ωa is shrunk to zero the expected time to absorption in ∂Ωa becomes
infinite. The narrow escape problem consists in constructing an asymptotic
expansion of the expected lifetime, denoted Eτ , as ε = |∂Ωa |g /|∂Ω|
g → 0. We
1
|Ω|g
log + O(1) .
derive a leading order asymptotic approximation Eτ =
Dπ
ε
The order 1 term can be evaluated for simply connected domains on a sphere
by projecting stereographically on the complex plane and mapping conformally on a circular disk. It can also be evaluated for domains that can be
mapped conformally onto an annulus. This term is needed in real life ap1
plications, such as trafficking of receptors on neuronal spines, because log
ε
is not necessarily large, even when ε is small. Ifthe absorbing window is
|Ω|g
1
located at a corner of angle α, then Eτ =
log + O(1) , if near a
Dα
ε
cusp, then Eτ grows algebraically, rather than logarithmically. Thus, in
the domain bounded between two tangent circles, the expected lifetime is
7.1 Introduction
|Ω|
Eτ = −1
(d − 1)D
7.1
123
1
+ O(1) .
ε
Introduction
In many applications it is necessary to find the mean first passage time
(MFPT) of a Brownian particle to a small absorbing window in the otherwise
reflecting boundary of a given bounded domain. This is the case, for example,
in the permeation of ions through protein channels of cell membranes [73],
and in the trafficking of AMPA receptors on nerve cell membranes [78], [14].
While the first example is three dimensional the second is two dimensional,
which leads to very different results. In this chapter we consider the two
dimensional case.
In the first two parts of this series of chapters, we considered the narrow escape problem in three dimensions [177] and in the planar circular disk
[178]. The leading order asymptotic behavior of the MFPT is different in
the three and two dimensional cases; it is proportional to the relative size
of the reflecting and absorbing boundaries in three dimensions, but in two
dimensions it is proportional to the logarithm of this quotient. The difference in the orders of magnitude is the result of the different singularities of
Neumann’s function for Laplace’s equation in the two cases.
While the second term in the asymptotic expansion of the MFPT in
three dimensions is much smaller then the first one, it is not necessarily so
in two dimensions, because of the slow growth of the logarithmic function.
It is necessary, therefore, to find the second term in the expansion in the
two-dimensional case. This term was found for the case of a planar circular
disk in [178], and can therefore be found for all simply connected domains
in the plane that can be mapped conformally onto the disk. Similarly, it
can be found for simply connected domains on two-dimensional Riemannian
manifolds that can be mapped conformally on the planar disk. For example,
the sphere with a circular cap cut off can be projected stereographically onto
the disk, and so the second term for the narrow escape problem for such
domains can be found.
The specific mathematical problem can be formulated as follows. A Brownian particle diffuses freely in a bounded domain Ω on a two-dimensional
Riemannian manifold (Σ, g). The boundary ∂Ω is reflecting, except for a
small absorbing arc ∂Ωa . The ratio between the arclength of the absorbing
124
Narrow Escape: Riemann Surfaces and Non-Smooth Domains
boundary and the arclength of the entire boundary is a small parameter,
ε=
|∂Ωa |g
1.
|∂Ω|g
The MFPT to ∂Ωa , denoted Eτ , becomes infinite as ε → 0.
In this chapter we calculate the first term in the asymptotic expansion
of Eτ for a general smooth bounded domain on a general two-dimensional
Riemannian manifold. We find the second term for an annulus of two concentric circles, with a small hole located on its inner boundary. This result
is generalized in a straightforward manner to domains that are conformally
equivalent to the annulus.
The calculation of the second term involves the solution of the mixed
Dirichlet-Neumann problem for harmonic functions in Ω. While in the three
dimensional case this is a classical problem in mechanics, diffusion, elasticity
theory, hydrodynamics, and electrostatics [184, 49, 50], the two dimensional
problem did not draw as much attention in the literature.
First, we consider the problem of narrow escape on two dimensional manifolds, and derive the leading order asymptotic approximation
1
|Ω|g
log + O(1)
for ε 1.
(7.1)
Eτ =
Dπ
ε
This generalizes the result of [78] from general smooth planar domains to
general domains on general smooth two-dimensional Riemannian manifolds.
The second term in the asymptotic expansion is found for the 2-sphere
2
x + y 2 + z 2 = R2 . The calculation is made possible by the stereographic
projection that maps the Riemann sphere onto a circular disk, a problem
that was solved in [178]. The boundary in this case is a spherical cap of
central angle δ at the north pole, where ε is the ratio between the absorbing
arc and the entire boundary circle. We find that the MFPT, averaged with
respect to an initial uniform distribution, is given by
1
1
1
|Ω|g
2
2
log + 2 log + 3 log 2 − + O(ε, δ log δ, δ log ε) , (7.2)
Eτ =
2πD
δ
ε
2
where |Ω|g = 4πR2 is the surface area of the sphere. Note that there are two
small parameters that control the behavior of the MFPT in this problem. The
small ε contributes as equation (7.1) predicts, whereas the small δ parameter
contributes half as much.
7.1 Introduction
The second case that we consider is that of narrow escape from an annulus, whose boundary is reflecting, except for a small absorbing arc on the
inner circle. Specifically, the annulus is the domain R1 < r < R2 , with all reflecting boundaries except for a small absorbing window located at the inner
circle (see Fig. 7.1). The inversion w = 1/z transforms this case into that of
R1
< 1, the MFPT,
the absorbing boundary on the outer circle. Setting β =
R2
averaged with respect to a uniform initial distribution, can be written as
Eτ =
(R22
−
R12 )
1
1 R22
1 1
2
log + log 2 + 2β +
log − R22 + O(ε, β 4 )R22 .
2
ε
21−β
β 4
Also in this case we find two small parameters, the ε contribution belongs to a
singular perturbation problem with a boundary layer solution and an almost
constant outer solution with singular fluxes near the edges of the window,
whereas the δ contribution is just the singularity of Green’s function at the
origin–a problem with a regular flux. This result is generalized to a sphere
with two antipodal circular caps removed. We find that for β 1 the
maximum exit time is attained near the south pole, as expected. This result
can be generalized to manifolds that can be mapped conformally onto the
said domain.
The asymptotic expansion of the MFPT to a non-smooth part of the
boundary is different. We consider two types of singular boundary points:
corners and cusps. If the absorbing arc is located at a corner of angle α, the
MFPT is
1
|Ω|g
(7.3)
log + O(1) .
Eτ =
Dα
ε
For example, the MFPT from a rectangle with sides a and b to an absorbing
window of size ε at the corner (α = π/2, see Figure 7.2), is
ε
2|Ω|
a
2 πb
2
4
Eτ =
,
log + log +
+ 2β + O
,β
π
ε
π 6a
a
where |Ω| = ab and β = e−πb/a . The calculation of the second order term
turns out to be similar to that in the annulus case. The pre-logarithmic
|Ω|g
factor
is the result of the different singularity of the Neumann function
Dα
125
126
Narrow Escape: Riemann Surfaces and Non-Smooth Domains
at the corner. It can be obtained by either the method of images, or by
the conformal mapping z 7→ z π/α that flattens the corner. In the vicinity
of a cusp α → 0, therefore the asymptotic expansion (7.3) is invalid. We
1
find that near a cusp the MFPT grows algebraically fast as λ , where λ
ε
is the order of the cusp. Note that the MFPT grows faster to infinity as
the boundary is more singular. The change of behavior from a logarithmic
growth to an algebraic one expresses the fact that entering a cusp is a rare
Brownian event. For example, the MFPT from the domain bounded between
two tangent circlesto a smallarc at the common point (see Figure 7.4) is
1
|Ω|
+ O(1) , where d < 1 is the ratio of the radii. This
Eτ = −1
(d − 1)D ε
result is obtained by mapping the cusped domain conformally onto the upper
half plane. The singularity of the Neumann function is transformed as well.
The leading order term of the asymptotics can be found for any domain that
can be mapped conformally to the upper half plane.
In three dimensions the class of isolated singularities of the boundary is
much richer than in the plane. The results of [177] cannot be generalized
in a straightforward way to windows located near a singular point or arc of
the boundary. We postpone the investigation of the MFPT to windows at
isolated singular points in three dimensions to a future paper.
As a possible application of the present results, we mention the calculation
of the diffusion coefficient from the statistics of the lifetime of a receptor in
a corral on the surface of a neuronal spine [14].
7.2
Asymptotic Approximation to the MFPT
on a Riemannian Manifold
We denote by x(t) the trajectory of a Brownian motion in a bounded domain
Ω on a two-dimensional Riemannian manifold (Σ, g). For a domain Ω ⊂ Σ
with a smooth boundary ∂Ω (at least C 1 ), we denote by |Ω|g the Riemannian
surface area of Ω and by |∂Ω|g the arclength of its boundary, computed with
respect to the metric g. The boundary ∂Ω is partitioned into an absorbing
arc ∂Ωa and the remaining part ∂Ω − ∂Ωa is reflecting for the Brownian
trajectories. We assume that the absorbing part is small, that is,
ε=
|∂Ωa |g
1,
|∂Ω|g
7.2 Asymptotic Approximation to the MFPT on a Riemannian Manifold
however, Σ and Ω are independent of ε; only the partition of the boundary
∂Ω into absorbing and reflecting parts varies with ε.
The first passage time τ of the Brownian motion from Ω to ∂Ωa has a
finite mean and we define
u(x) = E[τ | x(0) = x].
The function u(x) satisfies the mixed Neumann-Dirichlet boundary value
problem (see for example [127], [166])
D∆g u(x) = −1 for x ∈ Ω
∂u(x)
= 0 for x ∈ ∂Ω − ∂Ωa
∂n
u(x) = 0 for x ∈ ∂Ωa ,
(7.4)
(7.5)
(7.6)
where ∆g is the Laplace-Beltrami operator on Σ and D is the diffusion coefficient. Obviously, u(x) → ∞ as ε → 0, except for x in a boundary layer
near ∂Ωa .
7.2.1
Expression of the MFPT using the Neumann
function
We consider the Neumann function defined on Σ by
∆g N (x, y) = −δ(x − y) +
∂N (x, y)
= 0,
∂n
1
,
|Ω|g
for x, y ∈ Ω
(7.7)
for x ∈ ∂Ω, y ∈ Ω.
The Neumann function N (x, y) is defined up to an additive constant and is
symmetric [55]. The Neumann function exists for the domain Ω, because the
compatibility condition is satisfied (i.e., both sides of eq.(7.7) integrate to 0
over Ω due to the boundary condition). The Neumann function N (x, y) is
constructed by using a parametrix H(x, y) [6] ,
H(x, y) = −
h(d(x, y))
log d(x, y),
2π
(7.8)
127
128
Narrow Escape: Riemann Surfaces and Non-Smooth Domains
where d(x, y) is the Riemannian distance between x and y and h(·) is a
regular function with compact support, equal to 1 in a neighborhood of y.
As a consequence of the construction N (x, y) − H(x, y) is a regular function
on Ω.
To derive an integral representation of the solution u, we multiply eq.(7.4)
by N (x, y), eq.(7.7) by u(x), integrate with respect to x over Ω, and use
Green’s formula to obtain the identity
Z
I
1
∂u(x(S))
dSg = −
u(x) dVg + u(ξ)
N (x(S), ξ)
∂n
|Ω|g Ω
∂Ω
Z
(7.9)
− N (x, ξ) dVg .
Ω
The integral
1
Cε =
|Ω|g
Z
u(x) dVg
(7.10)
Ω
is an additive constant and the flux on the reflecting boundary vanishes, so
we rewrite eq.(7.9) as
Z
Z
∂u(x(S))
dSg ,
(7.11)
u(ξ) = Cε + N (x, ξ) dVg +
N (x(S), ξ)
∂n
Ω
∂Ωa
where S is the coordinate of a point on ∂Ωa , and dSg is arclength element
on ∂Ωa associated with the metric g. Setting
f (S) =
∂u(x(S))
,
∂n
and choosing ξ ∈ ∂Ωa in eq.(7.11), we obtain
Z
Z
0 = Cε + N (x, ξ) dVg +
N (x(S), ξ)f (S) dSg .
Ω
(7.12)
∂Ωa
The first integral in eq.(7.11) is a constant (independent of ε), because due
to the symmetry of N (x, y) eq.(7.7) gives the boundary value problem
Z
∆ξ
N (x, ξ) dVg = 0 for ξ ∈ Ω
Ω
∂
∂n(ξ)
Z
N (x, ξ) dVg = 0 for ξ ∈ ∂Ω,
Ω
7.2 Asymptotic Approximation to the MFPT on a Riemannian Manifold
whose solution is any constant. Changing the definition of the constant Cε ,
equation (7.11) can be written as,
Z
u(ξ) =
N (x(S), ξ)f (S) dSg + Cε ,
(7.13)
∂Ωa
and both f (S) and Cε are determined by the absorbing condition (7.6)
Z
0=
N (x(S), ξ)f (S) dSg + Cε for ξ ∈ ∂Ωa .
(7.14)
∂Ωa
Equation (7.14) has been considered in [78] for a domain Σ ⊂ R2 as an
integral equation for f (S) and Cε .
Actually, the boundary coordinate S can be chosen as arclength on ∂Ωa ,
denoted s. Under the regularity assumptions of the boundary, the normal
derivative f (s) is a regular function, but develops a singularity as ξ(s) approaches the corner boundary of ∂Ωa in ∂Ω [105]. Both can be determined
from the representation (7.13), if all functions in eq.(7.14) and the boundary
are analytic. In that case the solution has a series expansion in powers of
arclength on Ωa . The method to compute Cε follows the same step as in [78].
7.2.2
Leading order asymptotics
Under our assumptions, u(ξ) → ∞ as ε → 0 for any fixed ξ ∈ Ω, so that
eq.(7.10) implies that Cε → ∞ as well. It follows from eq.(7.14) that the
integral in (7.14) decreases to −∞.
An origin 0 ∈ ∂Ωa is fixed and the boundary
∂Ω is parameterized
by
1
1
(x(s), y(s)).
We
rescale s so that
∂Ω = (x(s), y(s)) : − 2 < s ≤ 2 and
x − 12 , y − 12 = x 21 , y 12 . We assume that the functions x(s) and
y(s) are real analytic in the interval 2|s| < 1 and that the absorbing part of
the boundary ∂Ωa is the arc
∂Ωa = {(x(s), y(s)) : |s| < ε} .
The Neumann function can be written as
N (x, ξ) = −
1
log d(x, ξ) + vN (x, ξ),
2π
for x ∈ Bδ (ξ),
(7.15)
where Bδ (ξ) is a geodesic ball of radius δ centered at ξ and vN (x; ξ) is a
regular function. We consider a normal geodesic coordinate system (x, y)
129
130
Narrow Escape: Riemann Surfaces and Non-Smooth Domains
at the origin, such that one of the coordinates coincides with the tangent
coordinate to ∂Ωa . We choose unit vectors e1 , e2 as an orthogonal basis in
the tangent plane at 0 so that for any vector field X = x1 e1 + x2 e2 , the
metric tensor g can be written as
gij = δij + ε2
X
2
akl
ij xk xl + o(ε ),
(7.16)
kl
where |xk | ≤ 1, because ε is small. It follows that for x, y inside the geodesic
ball or radius ε, centered at the origin, d(x, y) = dE (x, y) + O(ε2 ), where
dE is the Euclidean metric. We can now use the computation given in the
Euclidean case in [78]. To estimate the solution of equation (7.14), we recall
that when both x and ξ are on the boundary, vN (x, ξ) becomes singular
(see [55, p.247, eq.(7.46)]) and the singular part gains a factor of 2, due to
the singularity of the “image charge”. Denoting by ṽN the new regular part,
equation (7.14) becomes
log d(x(s), ξ(s0 ))
0
f (s0 ) S(ds0 ) = Cε ,
ṽN (x(s ); ξ(s)) −
π
0
|s |<ε
Z
(7.17)
where S(ds0 ) is the induced measure element on the boundary, and x =
(x(s), y(s)), ξ = (ξ(s), η(s)). Now, we expand the integral in eq.(7.17), as in
[78],
0
log d(x(s), ξ(s )) = log
p
(x(s0 )
−
ξ(s))2
+
(y(s0 )
−
η(s))2
1 + O(ε )
2
and
S(ds)f (s) =
∞
X
j=0
fj sj ds,
ṽN (x(s0 ); ξ(s))S(ds0 ) =
∞
X
vj (s0 )sj ds0 (7.18)
j=0
for |s| < ε, where vj (s0 ) are known coefficients and fj are unknown coefficients, to be determined from eq.(7.17). To expand the logarithmic term
in the last integral in eq.(7.17), we recall that x(s0 ), y(s0 ), ξ(s), and η(s) are
analytic functions of their arguments in the intervals |s| < ε and |s0 | < ε,
7.2 Asymptotic Approximation to the MFPT on a Riemannian Manifold
respectively. Therefore
Z ε
(s0 )n log d(x(s), ξ(s0 )) ds0 =
(7.19)
−ε
ε
Z
(s0 )n log
p
(x(s0 ) − ξ(s))2 + (y(s0 ) − η(s))2 1 + O(ε2 ) ds0 =
−ε
ε
Z
(s0 )n log |s0 − s| 1 + O (s0 − s)2
1 + O(ε2 ) ds0 .
−ε
We keep in Taylor’s expansion of log {|s0 − s| (1 + O ((s0 − s)2 ))} only the
leading term, because higher order terms contribute positive powers of ε to
the series
Z
ε
log(s − s0 )2 ds0 = 4ε (log ε − 1) + 2
−ε
∞
X
j=1
1
s2j
.
(2j − 1)j ε2j−1
(7.20)
For even n ≥ 0, we have
Z ε
(s0 )n log(s − s0 )2 ds0 =
−ε
4
εn+1
εn+1
log ε −
n+1
(n + 1)2
−2
∞
X
s2j
j=1
εn−2j+1
,
j(n − 2j + 1)
(7.21)
whereas for odd n, we have
∞
X
s2j+1 εn−2j
(s ) log(s − s ) ds = −4
.
2j
+
1
n
−
2j
−ε
j=1
Z
ε
0 n
0 2
0
(7.22)
Using the above expansion, we rewrite eq.(7.17) as
Z
ε
0 =
−ε
∞
X
j=0
(
∞
0
X
−1
2
0
2
2
log |s − s| 1 + O (s − s) (1 + O(ε )) +
vj (s0 )sj
π
j=0
fj s0j ds0 + Cε ,
)
×
131
132
Narrow Escape: Riemann Surfaces and Non-Smooth Domains
and expand in powers of s. At the leading order, we obtain
X ε2p+1
ε2p+1
ε (log ε − 1) f0 +
log ε −
f2p =
2p + 1
(2p + 1)2
p
Z
π ε
v0 (s0 ) ds0 + Cε .
2 −ε
(7.23)
Equation (7.23) and
1
2
Z
ε
f (s) S(ds) =
−ε
X ε2p+1
f2p
(2p
+
1)
p
determine the leading order term in the expansion of Cε . Indeed, integrating
eq.(7.4) over the domain Ω, we see that the compatibility condition gives
ε
Z
f (s) S(ds) = −|Ω|g ,
(7.24)
−ε
Z
ε
and using the fact that
v0 (s0 )S(ds0 ) = O(ε), we find that the leading
−ε
order expansion of Cε in eq.(7.23) is
|Ω|g
1
Cε =
log + O(1)
π
ε
for ε 1.
(7.25)
If the diffusion coefficient is D, eq.(7.13) gives the MFPT from a point x ∈ Ω,
outside the boundary layer, as
1
|Ω|g
log + O(1)
E[τ | x] = u(x) =
πD
ε
7.3
for ε 1.
(7.26)
The Annulus Problem
We consider a Brownian particle that is confined in the annulus R1 < r < R2 .
The particle can exit the annulus through a narrow opening of the inner circle
(see Fig.7.1). The MFPT v(x) satisfies
7.3 The Annulus Problem
Figure 7.1: An annulus R1 < r < R2 . The particle is absorbed at an arc
of length 2εR1 (dashed line) at the inner circle. The solid lines indicate
reflecting boundaries.
133
134
Narrow Escape: Riemann Surfaces and Non-Smooth Domains
∆v = −1,
for R1 < r < R2 ,
∂v
= 0,
∂r
for r = R2 ,
∂v
= 0,
∂r
for r = R1 , |θ − π| > ε,
v = 0,
for r = R1 , |θ − π| < ε.
(7.27)
R12 − r2
The function w =
is a solution of the Dirichlet problem for eq.(7.27)
4
in the exterior domain of the inner circle r > R1 . More specifically, it satisfies
the boundary value problem
∆w = −1,
for R1 < r < R2 ,
∂w
1
= − R1 ,
∂r
2
for r = R1 ,
1
∂w
= − R2 ,
∂r
2
for r = R2 ,
w = 0,
for r = R1 .
(7.28)
The function u = v − w satisfies
∆u = 0,
for R1 < r < R2 ,
∂u
1
=
R2 ,
∂r
2
for r = R2 ,
∂u
1
=
R1 ,
∂r
2
for r = R1 , |θ − π| > ε,
u = 0,
for r = R1 , |θ − π| < ε.
(7.29)
Separation of variables produces the solution
n
n ∞ a0 X
r
R2
r
u(r, θ) =
+
an
+ bn
cos nθ + α log
, (7.30)
2
R
r
R
2
1
n=1
7.3 The Annulus Problem
135
where an , bn and α are to be determined by the boundary conditions. Differentiating with respect to r yields
"
n−1
n−1 #
∞
∂u X
an
r
bn R2 R2
α
=
n
− 2
cos nθ + .
(7.31)
∂r
R2 R2
r
r
r
n=1
Setting r = R2 gives
"∞
#
1
1 X
R2 =
n (an − bn ) cos nθ + α ,
2
R2 n=1
(7.32)
1
therefore, an = bn and α = R22 , and we have
2
n n ∞
a0 X
r
r
R2
1 2
an
u(r, θ) =
+
+
cos nθ + R2 log
. (7.33)
2
R
r
2
R
2
1
n=1
The boundary conditions at r = R1 become
n n ∞
R1
a0 X
R2
+
+
an
cos nθ =
2
R
R
1
2
n=1
" n−1 #
∞
n+1
X
R1
R2
−
nan
cos nθ =
R
R
1
2
n=1
the dual series equations
0,
for |θ − π| < ε,
R2
R22 − R12 ,
2R1
for |θ − π| > ε.
Setting
cn =
R1
R2
"
R2
R1
n+1
−
R1
R2
n−1 #
an ,
for
n ≥ 1,
(7.34)
and c0 = a0 converts the dual series equations to
∞
c0 X cn
+
cos nθ = 0,
2
1 + Hn
n=1
∞
X
n=1
ncn cos nθ =
for π − ε < θ < π,
1 2
(R − R12 ),
2 2
(7.35)
for 0 < θ < π − ε, (7.36)
2β 2n
R1
for n ≥ 1, and H0 = 0, with β =
< 1. Note that
2n
1+β
R2
Hn = O(β 2n ) which tends to zero exponentially fast (much faster than the
n−1 decay required for the Collins method [26, 27], see also [177]).
where Hn = −
136
Narrow Escape: Riemann Surfaces and Non-Smooth Domains
The case Hn ≡ 0 was solved in [178]. We now try to find the correction
of that result due to the non vanishing Hn . As in [178] the equation
∞
c0 X cn
+
cos nθ =
2
1 + Hn
n=1
Z
h (t) dt
θ π−ε
√ 1
cos
2 θ
cos θ − cos t
for 0 < θ < π − ε
defines the function h1 (θ) uniquely for 0 < θ < π − ε, the coefficients are
given by
Z
1 + Hn π−ε
h1 (t) [Pn (cos t) + Pn−1 (cos t)] dt,
(7.37)
cn = √
2
0
and
√ Z
c0 = 2
π−ε
h1 (t) dt.
(7.38)
0
Integrating equation (7.36) gives
∞
X
cn sin nθ =
n=1
1 2
R2 − R12 θ,
2
for 0 < θ < π − ε.
(7.39)
Substituting eq.(7.39) in equation (7.37), changing the order of summation
and integration, while using [184, eq.(2.6.31)],
∞
cos 1 θH(θ − t)
1 X
√
,
[Pn (cos t) + Pn−1 (cos t)] sin nθ = √ 2
cos t − cos θ
2 n=1
we obtain for 0 < θ < π − ε,
Z θ
Z π−ε
h1 (t)
(R22 − R12 ) θ
√
dt +
Kβ (θ, t)h1 (t) dt =
,
θ
cos t − cos θ
0
0
2 cos
2
where the kernel Kβ is
Kβ (θ, t) = √
1
∞
X
(7.40)
(7.41)
Hn (Pn (cos t) + Pn−1 (cos t)) sin nθ
θ
2 cos n=1
2
√
θ
= −2 2(1 + cos t) sin β 2 + O(β 4 ).
(7.42)
2
7.3 The Annulus Problem
137
The infinite sum in eq.(7.42) is approximated by its first term, while using
the first two Legendre polynomials P0 (x) = 1, P1 (x) = x. Using Abel’s
inversion formula applied to equation (7.41), we find that
u
Z
Z π−ε
u sin
R22 − R12 d t
2
√
K̃β (t, s)h1 (s) ds =
h1 (t) −
du, (7.43)
π
dt
cos
u
−
cos t
0
0
where the kernel K̃β is
1 d
K̃β (t, s) = −
π dt
Z
0
t
K (u, s) sin u
√β
du
cos u − cos t
(7.44)
√
Z t sin u sin u
2(1
+
cos
s)
d
2
√ 2
du + O(β 4 ).
= β2
π
dt 0 cos u − cos t
The substitution
r
s=
gives
Z
0
t
cos u − cos t
2
u
sin sin u
π
t
√ 2
du = √ sin2 ,
2
cos u − cos t
2
(7.45)
(7.46)
therefore,
s
sin t + O(β 4 ).
(7.47)
2
Equation (7.43) is a Fredholm integral equation of the second kind for h1 ,
of the form
(I − K̃β )h = z,
(7.48)
K̃β (t, s) = 2β 2 cos2
where
R2 − R12 d
z(t) = 2
π
dt
Therefore, we h can be expanded as
Z
0
t
u
2
√
du.
cos u − cos t
u sin
h = z + K̃β z + K̃β2 z + . . . ,
(7.49)
√
which converges in L2 . Since c0 = 2hh, 1i (eq.(7.38)), we find an asymptotic
expansion of the form
i
√ h
c0 = 2 hz, 1i + hK̃β z, 1i + . . . .
(7.50)
138
Narrow Escape: Riemann Surfaces and Non-Smooth Domains
The leading order term of this expansion was calculated in [178]. We now
estimate the error term hK̃β z, 1i, which is also the O(β 2 ) correction. Integrating by parts and changing the order of integration yields
R2 − R12
sin t
K̃β z(t) = 2β 2 2
π
= β
=
2
2 R2
√
− R12
sin t
π
π
Z
0
π
Z
u
2
√
du
cos
u
−
cos s
0
u
u sin du
2
√
cos u − cos s
s
d
cos2 ds
2
ds
Z
sin s ds
0
0
s
Z
s
u sin
2β 2 (R22 − R12 ) sin t.
(7.51)
Therefore,
hK̃β z, 1i =
√
2β
2
(R22
√
sin t dt = 2 2β 2 (R22 − R12 ).
(7.52)
1
2
4
2 log + 2 log 2 + 4β + O(ε, β ) .
ε
(7.53)
−
R12 )
Z
π
0
We conclude that
c0 =
(R22
−
R12 )
The MFPT averaged with respect to a uniform initial distribution is
c0 1 R24
R2 1 2
Eτ =
+
log
− R
2
2 R22 − R12
R1 4 2
1
2
2
2
= (R2 − R1 ) log + log 2 + 2β
ε
2
1 R2
1 1 2
+
log
− R + O(ε, β 4 )R22 .
2 1 − β2
β 4 2
(7.54)
Note that there are two different logarithmic contributions to the MFPT.
The “narrow escape” small parameter ε contributes
|Ω|g
1
log ,
π
ε
(7.55)
as expected from the general theory (equation (7.1)), whereas the parameter
β contributes
|Ω|g
1
log .
(7.56)
2π
β
7.4 Domains with Corners
139
These asymptotics differ by a factor 2, because they account for different singular behaviors. The asymptotic expansion (7.55) comes out from a singular
perturbation problem with singular flux near the edges, boundary layer and
an outer solution, whereas the asymptotics (7.56) is an immediate result of
the singularity of the Neumann function, with a regular flux.
The maximum exit time is attained at the antipode point of the center of
the hole at the outer circle. Indeed, eq. (7.33) indicates that the maximum is
attained for θ = 0. To determine its location along the cord R1 < r < R2 , we
note that Hn → 0 as β → 0. Therefore, in this limit, eqs. (7.35)-(7.36) are
equivalent to the circular disk problem. The logarithmic
term of eq. (7.33)
1
for r = R2 . Therefore, for
is monotonic increasing with r, and is O log
β
β 1 the maximum is attained at the outer circle r = R2 , which is also
the farthest point from the hole. Note that (7.54) is valid, with the obvious
modifications, for any domain that is conformally equivalent to the annulus.
7.4
Domains with Corners
Consider a Brownian motion in a rectangle Ω = (0, a) × (0, b) of area ab. The
boundary is reflecting except the small absorbing segment ∂Ωa = [a − ε, a] ×
{b} (see Fig. 7.2). The MFPT v(x, y) satisfies the boundary value problem
∆v = −1, (x, y) ∈ Ω,
v = 0, (x, y) ∈ ∂Ωa ,
∂v
= 0, (x, y) ∈ ∂Ω − ∂Ωa .
∂n
The function f =
∆f
f
∂f
∂n
∂f
∂n
(7.57)
b2 − y 2
satisfies
2
= −1, (x, y) ∈ Ω,
= 0, (x, y) ∈ ∂Ωa ,
= 0,
= −b,
(x, y) ∈ {0} × [0, b] ∪ {a} × [0, b] ∪ [0, a] × {0},
(x, y) ∈ [0, a − ε] × {b},
(7.58)
140
Narrow Escape: Riemann Surfaces and Non-Smooth Domains
Figure 7.2: Rectangle of sizes a and b with a small absorbing segment of size
ε at the corner.
7.4 Domains with Corners
141
therefore, the function u = v − f satisfies
∆u =
u =
∂u
=
∂n
∂u
=
∂n
A solution for u
0,
0,
(x, y) ∈ Ω,
(x, y) ∈ ∂Ωa ,
0,
(x, y) ∈ {0} × [0, b] ∪ {a} × [0, b] ∪ [0, a] × {0},
b,
(x, y) ∈ [0, a − ε] × {b}.
(7.59)
in the form of separation of variables is
∞
a0 X
πny
πnx
an cosh
u(x, y) =
+
cos
,
2
a
a
n=1
(7.60)
where the coefficients an are to be determined by the boundary conditions
at y = b
∞
πnb
πnx
a0 X
an cosh
+
cos
= 0, x ∈ (a − ε, a),
u(x, b) =
2
a
a
n=1
∞
∂u
πX
πnb
πnx
(x, b) =
nan sinh
cos
= b,
∂y
a n=1
a
a
πnb
, we have
a
∞
c0 X cn
+
cos nθ = 0,
2
1 + Hn
n=1
x ∈ (0, a − ε).
(7.61)
Setting cn = an sinh
∞
X
n=1
ncn cos nθ =
ab
,
π
π − δ < θ < π,
0 < θ < π − δ,
(7.62)
πε
πnb
where δ =
and Hn = tanh
− 1, n ≥ 1. Note that Hn = O (β 2n )
a
a
πb
for β = exp −
< 1. The rectangle problem and annulus problem (eq.
a
(7.36)) are almost mathematically equivalent, and equation (7.53) gives the
value of c0
2ab
1
2
4
c0 =
2 log + 2 log 2 + 4β + O(δ, β )
π
δ
ε
4ab
a
2
2
4
=
log + log + 2β + O
,β
.
(7.63)
π
ε
π
a
142
Narrow Escape: Riemann Surfaces and Non-Smooth Domains
Figure 7.3: A small opening near a corner of angle α.
The error term due to O(β 4 ) is generally small. For example, in a square
a = b and β = e−π so that β 4 ≈ 3 × 10−6 . The MFPT averaged with respect
to a uniform initial distribution is
ε
2ab
a
2 πb
c 0 b2
2
4
+
=
log + log +
+ 2β + O
,β
Eτ =
.
(7.64)
2
3
π
ε
π 6a
a
The leading order term of the MFPT is
a
2|Ω|
log ,
π
ε
(7.65)
which is twice as large than (7.55). The general result (7.1) was proved for
a domain with smooth boundary (at least C 1 ). However, in the rectangle
example, the small hole is located at the corner. The additional factor 2 is
the result of the different singularity of the Neumann function at the corner,
which is 4 times larger than that of the Green function. At the corner there
are 3 image charges — the number of images that one sees when standing
near two perpendicular mirror plates. In general, for a small hole located at
a corner of an opening angle α (see Fig. 7.3), the MFPT is to leading order
|Ω|
1
Eτ =
log + O(1) .
(7.66)
Dα
ε
This result is a consequence of the method of images for integer values of
π
π
. For non-integer
we use the complex mapping z 7→ z π/α that flattens
α
α
7.5 Domains with Cusps
143
1
the corner. The upper half plane Neumann function log z is mapped to
π
1
log z and the analysis of Section 7.2 gives (7.66).
α
To see that the area factor |Ω| remains unchanged under the conformal
mapping f : (x, y) 7→ (u(x, y), v(x, y)), we note that this factor is a consequence of the compatibility condition, that relates the area to the integral
Z
|Ω|
∆(x,y) w dx dy = − ,
D
Ω
where w(x, y) = E[τ | x(0) = x, y(0) = y] satisfies ∆(x,y) w = −1/D. The
Laplacian transforms according to
∆(x,y) w = (u2x + u2y )∆(u,v) w,
by the Cauchy-Riemann equations and the Jacobian of the transformation is
J = u2x + u2y . Therefore,
Z
Z
∆(u,v) w du dv.
∆(x,y) w dx dy =
Ω
f (Ω)
This means that the compatibility condition of Section 7.2 remains unchanged and gives the area of the original domain.
7.5
Domains with Cusps
Here we find the leading order term of the MFPT for small holes located
near a cusp of the boundary. A cusp is a singular point of the boundary.
As α = 0 at the cusp, one expects to find a different asymptotic expansion
than (7.66). As an example, consider the Brownian motion inside the domain
bounded between the circles (x − 1/2)2 + y 2 = 1/4 and (x − 1/4)2 + y 2 = 1/16
(see Figure 7.4). The conformal mapping z 7→ exp{πi(1/z − 1)} maps this
domain onto the upper half plane. Therefore, the MFPT is to leading order
|Ω| 1
Eτ =
+ O(1) .
(7.67)
D ε
This result can also be obtained by mapping the cusped domain to the unit
circle. The absorbing boundary is then transformed to an exponentially
144
Narrow Escape: Riemann Surfaces and Non-Smooth Domains
Figure 7.4: The point (0, 0) is a cusp point of the dotted domain bounded
between the two circles. The small absorbing arc of length ε is located at
the cusp point.
7.6 Diffusion on a 2-Sphere
145
small arc of length exp{−π/ε} + O(exp{−2π/ε}), and equation (7.67) is
recovered. If the ratio between the two radii is d < 1, then the conformal
map that
between the two circles to the upper half plane
maps the domain
πi
is exp
(1/z − 1) (for d = 1/2 we arrive at the previous example),
d−1 − 1
so the MFPT is to leading order
|Ω|
1
Eτ = −1
+ O(1) .
(7.68)
(d − 1)D ε
1
The MFPT tends algebraically fast to infinity, much faster than the O log
ε
behavior near smooth or corner boundaries. The MFPT for a cusp is much
larger because it is more difficult for the Brownian motion to enter the cusp
than to enter a corner. The MFPT (7.68) can be written in terms of d instead
of the area. Substituting |Ω| = πR2 (1 − d2 ), we find
πR2 d(1 + d) 1
+ O(1) ,
(7.69)
Eτ =
D
ε
where R is the radius of the outer circle. Note that although the area of Ω
is a monotonically decreasing function of d, the MFPT is a monotonically
increasing function of d and tends to a finite limit as d → 1.
Similarly, one can consider different types of cusps and find that the leading order term for the MFPT is proportional to 1/ελ , where λ is a parameter
that describes the order of the cusp, and can be obtained by the same technique of conformal mapping.
7.6
7.6.1
Diffusion on a 2-Sphere
Small absorbing cap
Consider a Brownian motion on the surface of a 2-sphere of radius R [141],
described by the spherical coordinates (θ, φ)
x = R sin θ cos φ,
y = R sin θ sin φ,
z = R cos θ.
The particle is absorbed when it reaches a small spherical cap. We center
the cap at the north pole, θ = 0. Furthermore, the FPT to hit the spherical
146
Narrow Escape: Riemann Surfaces and Non-Smooth Domains
cap is independent of the initial angle φ, due to rotational symmetry. Let
v(θ) be the MFPT to hit the spherical cap. Then v satisfies
∆M v = −1,
(7.70)
where ∆M is the Laplace-Beltrami operator [141] of the 2-sphere. This
Laplace-Beltrami operator ∆M replaces the regular plane Laplacian, because
the diffusion occurs on a manifold [141, and reference therein]. For a function
v independent of the angle φ the Laplace-Beltrami operator is (see Appendix
D.1)
∆M v = R−2 (v 00 + cot θ v 0 ) .
(7.71)
The MFPT also satisfies the boundary conditions
v 0 (π) = 0,
v(δ) = 0,
(7.72)
where δ is the opening angle of the spherical cap. The solution of the boundary value problem (7.71), (7.72) is given by
v(θ) = 2R2 log
sin(θ/2)
.
sin(δ/2)
(7.73)
Not surprisingly, the maximum of the MFPT is attained at the point θ = π
with the value
1
δ
2
2
2
(7.74)
vmax = v(π) = −2R log sin = 2R log + log 2 + O(δ ) .
2
δ
The MFPT, averaged with respect to a uniform initial distribution, is
Z π
1
Eτ =
v(θ) sin θ dθ
δ δ
2
2 cos
2
log sin(δ/2) 1
2
= −2R
+
cos2 (δ/2)
2
1
1
2
2
= 2R log + log 2 − + O(δ log δ) .
(7.75)
δ
2
Both the average MFPT and the maximum MFPT are
|Ω|g
1
log + O(1) ,
τ=
2π
δ
(7.76)
7.6 Diffusion on a 2-Sphere
147
where |Ω|g = 4πR2 is the area of the 2-sphere. This asymptotic expansion
is the same as for the planar problem of an absorbing circle in a disk. The
result is two times smaller than the result (7.1) that holds when the absorbing boundary is a small window of a reflecting boundary. The factor two
difference is explained by the aspect angle that the particle “sees”. The two
problems also differ in that the “narrow escape” solution is almost constant
and has a boundary layer near the window, with singular fluxes near the
edges, whereas in the problem of puncture hole inside a domain the flux is
regular and there is no boundary layer (the solution is simply obtained by
solving the ODE).
7.6.2
Mapping of the Riemann sphere
We present a different approach for calculating the MFPT for the Brownian
particle diffusing on a sphere. We may assume that the radius of the sphere
is 1/2, and use the stereographic projection that maps the sphere into the
plane [74]. The point Q = (ξ, η, ζ) on the sphere (often called the Riemann
sphere)
ξ 2 + η 2 + (ζ − 1/2)2 = (1/2)2
is projected to a plane point P = (x, y, 0) by the mapping
x=
ξ
,
1−ζ
y=
η
,
1−ζ
r 2 = x2 + y 2 =
ζ
,
1−ζ
(7.77)
and conversely
ξ=
x
,
1 + r2
η=
y
,
1 + r2
ζ=
r2
.
1 + r2
(7.78)
The stereographic projection is conformal and therefore transforms harmonic
functions on the sphere harmonic functions in the plane, and vice versa. However, the stereographic projection is not an isometry. The Laplace-Beltrami
operator ∆M on the sphere is mapped onto the operator (1 + r2 )2 ∆ in the
plane (∆ is the Cartesian Laplacian). The decapitated sphere is mapped
onto the interior of a circle of radius
δ
rδ = cot .
2
(7.79)
148
Narrow Escape: Riemann Surfaces and Non-Smooth Domains
Therefore, the problem for the MFPT on the sphere is transformed into the
planar Poisson radial problem
∆V = −
1
,
(1 + r2 )2
for r < rδ ,
(7.80)
subject to the absorbing boundary condition
V (r = rδ ) = 0,
(7.81)
where
V (r) = v(θ).
The solution of this problem is
1
V (r) = log
4
1 + rδ2
1 + r2
.
(7.82)
Transforming back to the coordinates on the sphere, we get
v(θ) =
sin(θ/2)
1
log
.
2
sin(δ/2)
(7.83)
As the actual radius of the sphere is R rather than 1/2, multiplying eq.(7.83)
by (2R)2 , we find that (7.83) is exactly (7.73).
7.6.3
Small cap with an absorbing arc
Consider again a Brownian particle diffusing on a decapitated 2-sphere of
radius 1/2. The boundary of the spherical cap is reflecting but for a small
window that is absorbing (see Fig. 7.5). We calculate the mean time to
absorption. Using the stereographic projection of the preceding subsection,
we obtain the mixed boundary value problem
∆v = −
v(r, φ)
1
,
(1 + r2 )2
for r < rδ ,
= 0,
for |φ − π| < ε,
∂v(r, φ) = 0,
∂r r=rδ
for |φ − π| > ε.
0 ≤ φ < 2π,
r=rδ
(7.84)
7.6 Diffusion on a 2-Sphere
Figure 7.5: A sphere of radius R without a spherical cap at the north pole
of central angle δ. The particle can exit through an arc seen at angle 2ε.
149
150
Narrow Escape: Riemann Surfaces and Non-Smooth Domains
The function
1
w(r) = log
4
1 + rδ2
1 + r2
is the solution of the all absorbing boundary problem eq.(7.82), so the function u = v − w satisfies the mixed boundary value problem
∆u = 0,
r < rδ ,
for 0 ≤ φ < 2π,
u(r, φ)
= 0, for |φ − π| < ε,
r=rδ
∂u(r, φ) rδ
=
, for |φ − π| > ε.
∂r
2(1 + rδ2 )
r=rδ
(7.85)
Scaling r̃ = r/rδ , we find this mixed boundary value problem to be that of
a planar disk [178], with the only difference that the constant 1/2 is now
rδ2
. Therefore, the solution is given by
replaced by
2(1 + rδ2 )
i
2rδ2 h
ε
a0 = −
log
+
O(ε)
.
(7.86)
1 + rδ2
2
Transforming back to the spherical coordinate system, the MFPT is
v(θ, φ) =
sin θ/2
1
log
2
sin δ/2
(7.87)
∞
n
h
i
X
ε
δ
cot (θ/2)
− cos2
log + O(ε) +
an
cos nφ.
2
2
cot
(δ/2)
n=1
The MFPT, averaged over uniformly distributed initial conditions on the
decapitated sphere, is
2
1 log sin(δ/2) 1
2 δ
+ cos
log + O(ε) .
Eτ = −
+
(7.88)
2
cos2 (δ/2)
2
2
ε
Scaling the radius R of the sphere into (7.88), we find that for small ε and δ
the averaged MFPT is
1
1
1
2
2
2
Eτ = 2R log + 2 log + 3 log 2 − + O(ε, δ log δ, δ log ε) . (7.89)
δ
ε
2
7.6 Diffusion on a 2-Sphere
151
There are two different contributions to the MFPT. The ratio ε between the
absorbing arc and the entire boundary brings in a logarithmic contribution
to the MFPT, which is to leading order
|Ω|g
1
log .
π
ε
However, the central angle δ gives an additional logarithmic contribution, of
the form
|Ω|g
1
log .
2π
δ
The factor 2 difference in the asymptotic expansions is the same as encountered in the planar annulus problem.
The MFPT for a particle initiated at the south pole θ = π is
v(π) = −2R2 log sin
i
δh
ε
δ
− 4R2 cos2
log + O(ε)
2
2
2
1
1
2
2
= 2R log + 2 log + 3 log 2 + O(ε, δ log δ, δ log ε) .(7.90)
δ
ε
2
We also find the location (θ, φ) for which the MFPT is maximal. The sta∂v
tionarity condition
= 0 implies that φ = 0, as expected (the opposite
∂φ
φ-direction to the center of the window). The infinite sum in equation (7.87)
is O(1). Therefore, for δ 1, the MFPT is maximal near the south pole
θ = π. However, for δ = O(1), the location of the maximal MFPT is more
complex.
Finally, we remark that the stereographic projection also leads to the
determination of the MFPT for diffusion on a 2-sphere with a small hole as
discussed above, and an all reflecting spherical cap at the south pole. In this
case, the image for the stereographic projection is the annulus, a problem
solved in Section 7.3.
Chapter 8
Narrow Escape at Short Times
In Chapters 5-7 we calculated the mean first passage time (MFPT) to a
narrow hole. In this Chapter we calculate the short time asymptotics of
the survival probability. In particular, we calculate the initial location from
which it is most difficult to exit at short times, that is, the initial location
that maximizes the survival probability. We find that this location can be
different than the location where the MFPT is maximal.
8.1
Absorbing Disk
First, we consider Brownian motion in a circular disk of radius R whose
boundary is absorbing. We show the intuitively obvious result that at short
times it is hardest to exit from the center of the disk. The survival probability
is
Z
Pr{τ > t | x0 = x} =
p(y, t | x) dy,
(8.1)
Ω
where the transition pdf p(y, t | x) is the solution of the initial boundary
value problem
∂p(y, t | x)
= D∆y p(y, t | x) for |x|, |y| < R,
∂t
p(y, t | x) = 0 for |y| = R,
p(y, 0 | x) = δ(y − x),
|x| < R,
for |x|, |y| < R,
(8.2)
8.1 Absorbing Disk
153
where D is the diffusion coefficient. The pdf p(y, t | x) has a short time
asymptotic expansion of the form [25, 168]
∞
X
Si2 (x, y)
Zi (x, y, t),
(8.3)
p(y, t | x) =
exp −
4Dt
i=0
where the eikonals Si (x, y) satisfy the eikonal equation |∇y Si |2 = 1, whose
characteristics are line segments, called rays. The pre-exponential factors
have regular expansions in powers of t,
Zi (x, y, t) =
∞
X
(n)
Zi (x, y)tn−1 ,
(8.4)
n=0
(n)
where Zi (x, y) satisfy transport equations. In particular, the first eikonal
is S0 (x, y) = |x − y|, which is the length of the ray connecting x and y,
1
and Z0 (x, y, t) =
. Therefore, the leading order short time asymptotics
4πt
of the density p(y, t | x) is
1
|x − y|2
p(y, t | x) ∼
exp −
.
(8.5)
4πDt
4Dt
Due to rotational symmetry, we can assume that the initial point is lying
on the x-axis, x = (x, 0), x ≥ 0. Introducing polar coordinates with origin
at x, we write y = (ξ, η), ξ = x + r cos θ and η = r sin θ. The integral (8.1)
takes the form
Z 2π
Z rmax (θ)
r2
1
dθ
exp −
r dr
Pr{τ > t | x0 = x} =
4πDt 0
4Dt
0
Z 2π
rmax (θ)2
1
exp −
dθ,
(8.6)
= 1−
2π 0
4Dt
p
where rmax (θ) = −x cos θ + R2 − x2 sin2 θ is the solution of the equation
R2 = ξ 2 + η 2 = (x + r cos θ)2 + (r sin θ)2 .
The integral (8.6) is of Laplace type with small parameter t. Therefore, the
main contribution to this integral comes from the point of minimum of rmax ,
min rmax (θ) = R − x.
0≤θ<2π
(8.7)
154
Narrow Escape at Short Times
Hence,
Z
0
2π
rmax (θ)2
(R − x)2
exp −
dθ ∼ C exp −
,
4Dt
4Dt
(8.8)
00
at the minimum. It is most
where C > 0 is a constant that depends on rmax
difficult to exit when the survival probability is maximal, that is, when the
exponential function (8.8) is smallest, which happens for x = 0. Therefore,
at short times it is mostly difficult to exit from the center of the disk, and
the survival probability is given by
R2
Pr{τ > t | x0 = (0, 0)} ∼ 1 − exp −
for t 1.
(8.9)
4Dt
The above calculation takes into account only the contribution of the first
ray. Can the other rays of the short time expansion (8.3) alter the starting
point from which it is hardest to exit at short times? To answer this question,
we consider the second eikonal S1 (x, y) = |x − x0 | + |x0 − y|, which is the
length of a ray the emanates from x, hits the boundary once at x0 (x, y), and
ends at y. The law of reflection [25] is that the angle of incidence equals the
1
angle of reflection. The transport function Z1 (x, y, t) = −Z0 (x, y, t) = −
4πt
is chosen so as to satisfy the absorbing boundary condition at the boundary
point y = x0 . In such a case S1 (x, x0 ) = S0 (x, x0 ). However, the sum of the
first two rays satisfies the boundary condition only on a part of the boundary.
For example, the boundary condition is satisfied at y = (R, 0), but not at
the antipodal point y = (−R, 0), where the error is exponentially small, and
will be corrected by the next (third) eikonal.
The integral of the second eikonal is evaluated using the Laplace method
as well. Therefore, the entire contribution to the integral comes from the
minimum of the second eikonal
min S1 (x, y) = R − x,
y
(8.10)
which is obtained at the boundary point y that connects the center of the
disk and x. Thus
2
Z
(R − x)2
S1 (x, y
= −C exp −
,
(8.11)
Z1 exp
4Dt
4Dt
Ω
with C > 0. We conclude that the contribution of the second eikonal to the
expansion (8.6) is of the same order and the same sign as the exponentially
8.2 Narrow Escape from a Disk
155
small remainder of the first eikonal. Therefore, the location of the point
from which it is hardest to exit at short times is not altered, because the
contributions of all
eikonals (three and above) are exponentially
higher order
2
(R − x)
.
smaller than exp −
4Dt
The survival probability given that the trajectory was initiated from the
center of the disk is calculated explicitly, because the first and second rays
satisfy the boundary condition on the entire boundary
1
|y|2
(2R − |y|)2
p(y, t | (0, 0)) =
exp −
− exp −
.
(8.12)
4πt
4Dt
4Dt
Therefore, the survival probability is
Z R
1
r2
(2R − r)2
Pr {τ > t | (0, 0)} =
exp −
− exp −
r dr
2Dt 0
4Dt
4Dt
(2R)2
= 1 − exp −
4Dt
√
R
R 4πDt
2R
− erf √
+
erf √
2Dt
4Dt
4Dt
R2
∼ 1 − 2 exp −
.
(8.13)
4Dt
Comparing eqs.(8.9) and (8.13), we note that, indeed, the second eikonal
contributes the same as the first one.
8.2
Narrow Escape from a Disk
Next, we consider diffusion in a disk, whose boundary is reflecting, but for a
small absorbing arc. We show again the intuitive result, that at short times
it is hardest to exit from the point which is farthest from the hole, that is,
from the antipodal point to the center of the hole.
Clearly, if the entire boundary were reflecting, the diffusing particle could
not exit, so its survival probability is
Pr{τ > t | x} = 1,
for all t > 0,
|x| < R.
(8.14)
156
Narrow Escape at Short Times
The survival probability is also the integral of the density (eq. (8.1)), which
is an infinite sum of rays (eq. (8.3)), therefore, the integral over all reflected
rays is exactly 1. We also note that the contributions of the reflected rays
are all positive, because they have to satisfy the no flux Neumann boundary
condition.
However, in the case at hand, a small part of the boundary is absorbing.
Therefore, the contribution of rays that hit the absorbing boundary an odd
number of times is negative, whereas the contribution of rays that hit the
boundary an even number of times is positive. Therefore, to leading order,
)
( 2
Z
Si(x,y ) (x, y)
dy, (8.15)
Pr{τ > t | x} ∼ 1 − 2 Zi(x,y ) (x, y, t) exp
4Dt
Ω
where i(x, y) is the index of the minimal ray that emanates from x, hits the
absorbing boundary once (and can hit the reflecting part of the boundary
any number of times), and finally gets to y. This ray was already counted
with a positive (instead of a negative) sign in the first term 1, and therefore
it is subtracted twice in eq.(8.15). Evaluating the integral in eq.(8.15) by
the Laplace method, shows that the main contribution to the integral comes
from y which minimizes Si(2 x,y )) (x, y). Clearly, the shortest ray is obtained
when y is in the absorbing part of the boundary. Therefore,
ρ(x)2
,
(8.16)
Pr{τ > t | x} ∼ 1 − C exp −
4Dt
where ρ(x) is the distance of x from the absorbing boundary, and C > 0 is
the Laplace method constant. It follows that the point x from which it is
hardest to exit is the farthest point from the absorbing boundary. In a disk,
it is the antipodal point to the center of the hole. We note that the above
considerations hold also for a general convex domain, because all points can
be seen from all points.
8.3
Narrow Escape in an Annulus
Consider an annulus between two concentric circles of radii R1 R2 , centered at O, with all reflecting boundary, except for a small absorbing arc
at the outer circle (see Fig. 7.1). We show that the point from which it
is hardest to exit at short times is the antipodal point to the center of the
8.3 Narrow Escape in an Annulus
absorbing hole in the inner circle. Although this point is not the farthest
point from the hole (the antipodal point of the hole on the outer circle is the
one, almost twice as much in distance), it is most difficult to exit from there.
This result is somewhat counterintuitive, because the antipodal point on the
outer circle is not only the farthest point, but also the point from which the
MFPT is maximal (see Chapter 7).
The analysis of the previous section shows that the survival probability
Pr{τ > t | x} is asymptotically determined by the shortest ray that connect
the point x to the absorbing
boundary that hits the reflecting boundary any
√
number of times. If 2R1 < R2 , then the length of the shortest eikonal that
connects
the center of the hole and its antipodal point on the outer circle is
√
2 2R2 (see Fig. 8.1). However, the shortest ray that connects the center of
the hole and its antipodal point at the inner circle turns out to be longer.
Therefore, it is more difficult to exit from the antipodal point on the inner
circle, rather than the antipodal point on the outer circle. The calculation
of the length of the ray is geometric, and is similar to the calculations of
trajectories of billiard balls in the field of quantum chaos (see, e.g. [19]).
Note that a trajectory of a billiard ball inside a disk is tangent to an inscribed
circle. Therefore, the ray that emanated from the center of the hole and
finally hits its antipodal point on the inner circle must hit the inner circle
before it hits the outer circle (for otherwise it will never hit the inner circle).
Specifically, consider a ray emanating from a point A1 on the outer circle
and reflected once in the inner circle, at a point B1 , to a point A2 on the
outer circle (see Fig. 8.2). The ray is reflected from the inner circle for the
n-th time at Bn to the point An on the outer circle. We set α = ∠A1 OB1 ,
so ∠A1 OBn = (2n − 1)α. The condition of hitting the antipodal point at
the inner circle is
(2n − 1)α = π mod 2π,
(8.17)
R1
π
− arcsin
(see Fig. 8.3). For β =
with the constraint 0 ≤ α ≤
R2
2
R1
R1
R1
1, we have arcsin
≈
1. Therefore, the minimal soR2
R2
R2
π
lution is obtained with α =
and n = 2. The length of this ray is
3
p
R1
2
2
3 R2 − R2 R1 + R1 = 3R2 1 + O
(see Fig. 8.4). This ray is
R2
√
longer than the ray that hits the outer circle, because 3 > 2 2. Note
that the two rays are equal in length, if 9 (1 − β + β 2 ) = 8, which holds for
157
158
Narrow Escape at Short Times
√
5
π
π
1
= .12732200 . . .. Also 1.047 . . . = < −arcsin β0 = 1.443 . . . ,
β0 = −
2 6
3
2
so this ray is permissible. For β > β0 it is once again the hardest to exit
from the antipodal point on the outer boundary.
8.3 Narrow Escape in an Annulus
Figure 8.1: The length of the shortest ray that connects
the center of the
√
hole and its antipodal point on the outer circle is 2 2R2 .
159
160
Narrow Escape at Short Times
Figure 8.2: The geometry of a billiard ball in an annulus: the accumulative
angle after n hits of the inner circle is (2n − 1)α mod 2π.
8.3 Narrow Escape in an Annulus
Figure 8.3: A billiard ball would hit the antipodal point, only if it hits
π
the inner circle before it hits the outer circle. Therefore α ≤ αmax = −
2
R1
arcsin
.
R2
161
162
Narrow Escape at Short Times
Figure 8.4: The shortest billiard trajectory that connects the center
ofpthe hole and its antipodal point on the inner circle is of length
3 R22 − R2 R1 + R12 .
Part II
Non-Equilibrium Statistical
Mechanics of Interacting
Particles
164
The most common continuum model of ion permeation is the coupled
Poisson-Nernst-Planck (PNP) system of equations [43], also known as the
drift-diffusion equation in the physics of semiconductors [170]. In this model,
ions are described as point charges, while both short range (e.g., size effects)
and long range (electric forces) interactions are neglected. Thus, phenomena such as blocking and selectivity are not expected to be recovered by
this model. Recently, an alternative model, derived at the molecular level,
was shown to lead to a slightly different equation [167], the conditional-PNP
equations, where conditioned densities (pair correlation function) and potentials replace the unconditioned quantities of the PNP equations. In Chapter
9 we outline the infinite hierarchy of equations for the correlation functions
in systems out of equilibrium. However, there are two missing ingredients to
make this hierarchy well posed. The first is a boundary condition for the pair
correlation function, derived from the stochastic differential equations that
describe the physical microscopic model. The second is a closure relation
that relates high order correlation functions to correlation functions of lower
order. Even when both ingredients are at hand, solving the C-PNP system
numerically in a finite or infinite domain is computationally prohibitive, because it is a problem in high dimensions (each particle is three-dimensional).
The numerical challenge exceeds the scope of this dissertation.
The diffusion of interacting particles in confined geometries is dominated
by the pair correlation function of a non-equilibrium system. In Chapter 10,
we derive the Kirkwood superposition approximation to a closure relation
for systems at equilibrium in the thermodynamic limit, using a variational
formulation. We define the entropy of the triplet correlation function and
show that the Kirkwood closure brings the entropy to its maximum. This
approach leads to a new interpretation of the Kirkwood closure relation,
usually explained by probabilistic considerations of dependence and independence of particles. The Kirkwood closure is generalized to systems of
finite volume at equilibrium by computing the pair correlation function in
finite domains. Closure relations for high order correlation functions are also
found by the variational approach. In particular, maximizing the entropy of
quadruplets leads to the high order closure used in the BGY2 (Born-GreenYvon) equations, which are a pair of integral equations for the triplet and
pair correlation functions. A new direction will be the generalization of this
approach to non-equilibrium systems, such as ions crossing a channel.
One of the fundamental problems in the theory of ionic solutions is the
question of how far the electrostatic field extends in a disordered system of
165
charges. This question is important if we were to use closure relations in a
charged system, such as Kirkwood’s superposition approximation. We develop a new theory for this problem, based on a statistical approach and
on large deviations theory. In Chapter 11 we define the concept of a totally disordered and electroneutral infinite system of charges and evaluate the
probability distribution of their electric potentials and fields. We show that
in one dimension the electric field is always small, of the order of the field of
a single charge. If the system is electroneutral, the spatial variations in potential do not exceed those that can be produced by a single charge. In two
and three dimensions, the electric field in similarly disordered electroneutral systems is usually small, with small variations. Interestingly, in two
and three dimensional systems, the electric potential is usually very large,
even though the electric field is not, because large amounts of energy are
needed to put together a typical disordered configuration of charges in two
and three dimensions, but not in one dimension. If the system is locally
electroneutral—as well as globally electroneutral—the potential is usually
small in all dimensions.
An additional problem concerning the electric field in an ionic solution
is that of the compatibility of the electric current carried by the diffusing
ions and Maxwell’s equations. We resolve this problem by applying the
Ramo-Shockley theorem, which leads to the intuitively obvious result that the
current carried by the ionic flux is identical to the electric current measured
by the electrodes.
Finally, in Chapter 12, we combine the results of the previous chapters to
derive boundary conditions for the pair correlation function, both for infinite
systems in steady state and for finite systems carrying a quasi steady state
diffusion current.
Chapter 9
Ionic Diffusion Through
Confined Geometries: From
Langevin Equations to Partial
Differential Equations
The contents of this chapter were published in [138]
Ionic diffusion through and near small domains is of considerable importance in molecular biophysics in applications such as permeation through
protein channels and diffusion in a domain near the charged active sites of
macromolecules. The validity of standard continuum-type descriptions of the
motion of interacting ions in domains of nanoscale geometry and charge distribution in and near a channel are domain is questionable. The description
in terms of density of only a few ions in such a domain may be overstretching continuum mechanics beyond its underlying assumptions. In particular,
non-equilibrium diffusion of interacting particles has not been adequately
captured by continuum theories to this day. More specifically, although the
standard machinery of equilibrium statistical mechanics contains microscopic
details it is not applicable, because these our system is usually not in equilibrium. Concentration gradients and the presence of an external applied
potential drive a non-vanishing stationary current through the system, rendering it a non-equilibrium system. A molecular level study of a stochastic
molecular model for the diffusive motion of interacting particles in an external field of force and a derivation of effective partial differential equations
9.1 Introduction
and their boundary conditions that describe the stationary non-equilibrium
system began in [167]. The interactions can include electrostatic, LennardJones, and other pairwise forces. The main result of [167] is the derivation
of a diffusion equation for the single ion density, coupled to the joint density of two ions, but not for the pair or higher order densities. Also the
boundary conditions that these densities satisfy were not derived so far. The
analysis of this chapter yields a new type of Poisson-Nernst-Planck equations, that involve conditional and unconditional charge densities and potentials. The conditional charge densities are the non-equilibrium analogs of
the well-studied pair correlation functions of equilibrium statistical physics.
Our proposed theory is an extension of equilibrium statistical mechanics of
simple fluids to stationary non-equilibrium problems. The proposed system
of equations differs from the standard Poisson-Nernst-Planck system in two
important aspects. First, the force term depends on conditional densities and
thus on the finite size of ions, and second, it contains the dielectric boundary
force on a discrete ion near dielectric interfaces. Recently, various authors
(see citations below) have shown that both of these terms are important for
diffusion through confined geometries, in the context of ion channels. The
problems of deriving the missing boundary conditions and decoupling the
equations for higher order densities are studied in later chapters of this part.
9.1
Introduction
With ongoing advances in technology and experimental techniques, the physical systems that are either studied or designed become smaller and smaller,
nowadays reaching the nanometer and nanosecond length and time scales,
respectively. In this chapter we focus on a particular type of systems, containing a nanoscale (nearly picoscale) pore connected to two large reservoirs
of electrolyte solutions. Two examples of such systems, which are of great importance in molecular biophysics and biotechnological applications, are ionic
permeation through protein channels [43], [73] and through carbon nanotubes
[91].
These nanoscale systems exhibit a range of phenomena not encountered in
larger macroscopic systems. For example, due to the confined geometry of the
pore through which the fluid flows, ions typically cannot pass by each other,
leading to single filing phenomena, complex relations between unidirectional
currents, and other nonlinear phenomena in mixtures [73]. The confined
167
168
From Langevin Equations to Partial Differential Equations
geometry enhances the importance of individual ion-ion interactions, which
are thought to be the cause of the high selectivity of these nanopores to
specific ionic species [58].
The function of these systems depends not only on the nanoscale atomic
details and geometry of the pore, but also on experimentally controlled
macroscopic variables, such as the applied electrostatic potential and the
surrounding bath concentrations. Therefore, the function of these systems
involves many different time and length scales, from the femto-second and
Angström scales for the motion of single water molecules till the microsecond time scale at which current measurements are made and micro- to millimeter distances at which measuring devices are placed. One final important
feature worth mentioning is that these systems almost always function away
from equilibrium, as does almost all biology. These nanopores separate regions of different concentrations and an applied voltage is usually present
across them. Therefore, these systems can be viewed as nanodevices, with
well defined input and output relations that are the purpose of the device.
Many of these biological systems are of interest because they are highly
sensitive, with atomic scale details controlling macroscopic flows. Mutations
or modifications to the atomic structure of the nanopore lead to significant
changes in its characteristics. Thus, the theoretical analysis of such systems
necessarily resides at the interface between continuum and discrete atomic
molecular physics, presenting new challenges in non-equilibrium statistical
physics. On the one hand, due to the importance of confined geometry,
discrete ion-ion interactions and non-uniform dielectric coefficient, standard
continuum equations of electrolyte solutions, such as Poisson-Boltzmann
(PB), Poisson-Fokker-Planck, or Poisson-Nernst-Planck (PNP) are not valid
[167, 132, 28]. On the other hand, due to the wide range of time and length
scales involved — mixing nanoscale and continuum — direct molecular dynamics simulations are impractical for the study of the macroscopic function
of these systems. Regretfully, even coarse grained Brownian Dynamics (BD)
simulations are not always the appropriate tool to study such systems. For
example, it is not possible to study the effects of a 10µM concentration of
calcium ions in a 100 mM Na+ Cl− electrolyte in a simulation that contains
only a few hundred ions, even though the biological effects of such trace concentrations are often of overwhelming importance [2]. Therefore, there is a
need for a hierarchy of models each valid in its own scale, and connections
between them.
While the standard PNP system may not be valid in confined geometries,
9.1 Introduction
it is important to note the advantages of a continuum description of nanoscale
systems. The computational complexity of continuum descriptions, that typically involve the solution of a system of coupled partial differential equations,
is often orders of magnitude lower than that of corresponding BD or MD
simulations. Another significant advantage of continuum descriptions over
BD and MD simulations is that they easily accommodate non-equilibrium
boundary conditions for the macroscopic electrostatic potential and ionic
concentrations. Simulations in the chemical tradition have difficulty with
such conditions (see however [89]). The goal, then, is to derive continuum
equations for non-equilibrium systems that include molecular details, absent
in the PNP theory. We note that for the corresponding equilibrium problem,
the theory of equilibrium statistical mechanics of simple fluids incorporates
particle-particle interactions by the Bogolyubov-Born-Green-Kirkwood-Yvon
(BBGKY) hierarchy [7], [12]. This hierarchy, coupled with a closure relation,
has proven successful in the calculation of a number of important macroscopic
properties of electrolytes from their underlying molecular interactions [158].
The main problem discussed in this chapter is thus how to generalize the
theory of equilibrium statistical mechanics of simple fluids, which is based
on the equilibrium Boltzmann distribution, to stationary non-equilibrium
problems. To this end, we note that one of the key missing elements in
the equilibrium theory is the dynamics of the moving particles. The Boltzmann distribution defines the probability of configurations but does not include a description of how individual particles move in the system. In nonequilibrium problems there is usually a net flow of particles. It is not possible
to compute net fluxes from equilibrium theories. In this chapter, following
[167], we present a molecular model of permeation, based on diffusive motion of ions, and propose a mathematical averaging procedure that results in
a hierarchy of Poisson and Nernst-Planck type equations containing conditional and unconditional charge densities. The proposed conditional system,
called C-PNP, includes molecular details such as excluded volume effects and
the dielectric force on a discrete ion that are absent in the standard PNP
system. Recently, various authors have shown that both of these terms are
important for diffusion through confined geometries, in the context of ion
channels [58, 133, 136, 30, 119, 59]. The C-PNP system, along with a closure
relation and boundary conditions, provides a theoretical framework for the
study of non-equilibrium diffusing systems. The C-PNP system reduces to
the BBGKY hierarchy when equilibrium boundary conditions are given, thus
generalizing equilibrium statistical mechanics to stationary non-equilibrium
169
170
From Langevin Equations to Partial Differential Equations
problems. Application of C-PNP to ion channels can be expected to predict
blocking and possibly selectivity.
This chapter is organized as follows. In Section 9.2 we describe the
standard Poisson-Boltzmann and Poisson-Nernst-Planck theories and discuss their limitations in confined geometries. Then, in Section 9.3, we briefly
review the theory of equilibrium statistical mechanics of simple fluids, and
its limited applicability to non-equilibrium systems. The main results of this
chapter are described in Sections 9.4 and 9.5, where we formulate a nonequilibrium stochastic molecular model based on trajectories and present
a mathematical derivation of the corresponding continuum equations. We
conclude in Section 9.6 with summary and discussion.
9.2
Standard Continuum Treatments and
their Failure in Confined Systems
We consider the setup shown in Figure 2.1. A rigid nanopore connects two
reservoirs of electrolyte solutions. The system is kept in a stationary nonequilibrium condition by an external feedback mechanism that maintains
average concentrations cL and cR in the left and right reservoirs, respectively,
and an average applied voltage V across them. The problem at hand is to
compute the stationary ionic current through the nanopore as a function of
the parameters of the system: the geometry, dielectric coefficients, and fixed
charge distribution of the pore, the atomic characteristics of the ions (e.g.,
radii, ion-ion interaction forces, and their diffusion coefficients), and as a
function of the experimentally controlled macroscopic variables, cL , cR and
V.
Throughout this chapter we consider modeling approaches at the level
of the primitive model [7, 72], that do not consider explicitly the individual
molecules of the solvent, but rather treat them as a continuous structureless
dielectric, and as a viscous and noisy medium. Some of the limitations of this
approach are discussed in the summary section. For simplicity, we consider
an ionic solution with only two ionic species, with positive and negative
charges, such as Na+ Cl− , although this assumption can easily be relaxed.
Two of the most common mean-field continuum theories of ionic solutions are the equilibrium Poisson-Boltzmann (PB) and the non-equilibrium
Poisson-Nernst-Planck (PNP) theories [7], [12]. Both PB and PNP assume
9.2 Standard Continuum Treatments
171
a constitutive relation of the form
ezp
J p (x) = −Dp ∇cp (x) + cp (x)
∇ψ(x)
kB T
between the local averaged particle flux of the positive ionic species at location x, called J p (x), their concentration cp (x), and a potential of mean field
ψ(x). In this equation kB is Boltzmann’s constant, T is temperature, Dp is
the diffusion coefficient of the positive ions and zp is their valence. A similar
equation for the flux of the negative ions is also assumed. The equilibrium
PB theory assumes that the local flux vanishes everywhere,
J p (x) = J n (x) = 0
(9.1)
while the PNP theory assumes conservation of flux, e.g. the Nernst-Planck
equation
∇ · J p (x) = ∇ · J n (x) = 0.
(9.2)
The simplicity of these two theories lies in their crude approximation for
the potential of the mean field. Both PB and PNP assume that the potential
ψ(x) satisfies Poisson’s equation with the average concentrations cp (x) and
cn (x),
∇ · [ε(x)∇ψ(x)] = −e zp cp (x) − zn cn (x) + ρfixed (x)
(9.3)
where ε(x) is the location dependent dielectric coefficient and ρfixed is the
concentration of fixed charges in the system. Stated differently, the electrostatic field exerted on an ion at location x is approximated by the gradient
of the mean electrostatic potential.
The PB theory is often the starting point of modern electrochemistry,
with the famous Debye-Hückel theory of ionic shielding [12], [13]. The PNP
theory has also found numerous applications in the study of electrolyte transport, plasma physics and in modeling of semiconductor devices [12, 170].
Both of these theories represent the ions as a continuous charge distribution in an ambient mean potential field, defined by the Poisson equation
(9.3). Both theories neglect effects due to the finite size of ions and due to
the discrete nature of their charge. The limitations of the PB approach, even
in bulk electrolyte solutions with moderate to high concentrations, are well
known [12], [72], [13]. Recently, the failure of PB and PNP in confined geometries has also been shown both in simulations [132], [28] and in theoretical
treatments [167].
172
From Langevin Equations to Partial Differential Equations
9.3
Equilibrium Statistical Mechanics of Simple Fluids
The preceding discussion indicates that the problem at hand is how to include molecular details in non-equilibrium continuum type equations. It is
important to note that molecular details can be included in continuum models with satisfactory results. For example, the theory of equilibrium statistical
mechanics of simple fluids yields continuum type equations with molecular
detail, and has proven successful in the computation of many macroscopic
quantities [12], [158], some of which (e.g., activity of ions) are difficult to determine in MD or BD simulations. We now briefly describe the key elements
in equilibrium theory and its limitations for non-equilibrium systems. For
simplicity we consider the canonical (NVT) ensemble formulation.
The classical description of the statistical physics of interacting particles
(at the level of the primitive model) starts with a finite system of N particles
in a finite volume V at a fixed temperature T with the N particles at locations
x1 , ..., xN . It is usually assumed that the potential of the forces acting on
the ions, U (x1 , x2 , ..., xN ), is explicitly known, and is a sum of pair radial
interactions,
X
U (x1 , x2 , ..., xN ) =
Ui,j (|xi − xj |),
i<j
with the role of forces at the boundary and electrostatic interactions with
boundary charges being often ignored. The configurational partition function
for this system is defined as
Z
QN (β) =
Z
−βU
···
e
VN
N
Y
dxi ,
i=1
where β = 1/kB T . The main assumption of the theory is that the probability
of a given configuration of the N particles follows the Boltzmann distribution,
p(x1 , x2 , ..., xN ) =
e−βU (x1 ,...xN )
.
QN
(9.4)
Since the number of particles in a given system is usually of the order of Avogadro’s number, one must consider a reduced description of the system; thus
9.3 Equilibrium Statistical Mechanics of Simple Fluids
173
sometimes the marginal densities of only one or two particles are considered.
By definition, the average physical density at location x1 is given by
Z
Z
N
Y
ρ(x1 ) = N · · ·
p(x1 , ..., xN )
dxi .
(9.5)
V N −1
i=2
In equilibrium statistical mechanics, it is often customary to consider the
limit of very large systems, namely N, |V | → ∞, such that N/|V | = ρ. In
this limit, it is possible to show that ρ(x) satisfies the following equation,
known as the first BBGKY equation,
Z
1
∇x U1,2 (x − y)ρ(y | x) dy = 0,
(9.6)
∇ρ(x) + ρ(x)
kT
where ρ(y | x) describes the conditional density of particles at y, given the
presence of a particle at x. By definition,
ρ(y | x) =
ρ(x, y)
ρ(x)
where ρ(x, y) is the joint physical concentration of a pair of particles.
For homogeneous infinite systems ρ(x) is uniform, due to symmetry, and
thus equal to the bulk concentration ρ. The microscopic structure of the
solution is described by the pair correlation function g2 (x, y), which is the
non-dimensional version of ρ(x, y), given by
g2 (x1 , x2 ) =
ρ(x1 , x2 )
ρ2
(9.7)
Z
=
lim N (N − 1)
N,V →∞
Z
···
p(x1 , x2 , ..., xN )
V N −2
N
Y
dxi .
i=3
The pair correlation function is important for the determination of many
thermodynamic properties of an equilibrium system, such as the free energy,
pressure, chemical potential and compressibility of electrolytes, to name just
a few [12], [158].
By differentiating (9.7) with respect to x1 , it is possible to show that g2
satisfies the second BBGKY equation, which depends on the higher order
triplet correlation function g3 ,
Z
kB T ∇x1 g2 + g2 ∇x1 U1,2 + ρ ∇x1 U1,3 |x1 − x3 | g3 (x1 , x2 , x3 ) dx3 = 0.
(9.8)
174
From Langevin Equations to Partial Differential Equations
Equations (9.6) and (9.8) correspond to a system with only one species
of interacting particles. In the case of two species of interacting particles
—say positive and negative ions — there are two singlet density functions,
g1p and g1n , for the positive and negative species, respectively. There are
α,β
also four pair correlation functions,
g2 , where α, β are one of the four
possible combinations (α, β) ∈ (p, p), (p, n), (n, p), (n, n) , though obviously g2p,n (x, y) = g2n,p (y, x). All of these quantities satisfy first and second
BBGKY equations similar to (9.6) and (9.8).
The BBGKY equations constitute an infinite hierarchy of forwardly coupled integro-differential equations. Further approximations are needed to
solve the problem. For example, approximate solutions for the single and
pair densities are often computed using closure relations, such as HNC or
MSA, relating higher order correlation functions to lower order correlation
functions, and the relevant thermodynamical quantities are calculated from
the resulting densities [12], [158].
9.3.1
Equilibrium vs. Non-equilibrium Statistical Mechanics
In contrast to the simple equilibrium system described in the previous section,
the biophysical systems we consider are in steady state, but not equilibrium.
Our system is connected to an energy source, either a natural one, as in
living organisms, or an artificial one, that mimics the natural one, but also
allows experimental control of parameters. Usually the nanopore carries a
steady net flux of particles (e.g., either into or out of a biological cell). A
steady state system carrying a constant flow cannot be described by the
equilibrium theory, since the Boltzmann distribution assumes a symmetrical
velocity distribution with zero mean and thus zero net flux. We construct
below a non-equilibrium analogue of the Boltzmann distribution and use it
to calculate fluxes.
It is instructive to note that the first BBGKY equation (9.6) can be
rewritten as
J (x) = 0,
where J (x) is the average flux at location x,
f̄ (x)
,
J (x) = −D ∇ρ(x) − ρ(x)
kB T
(9.9)
9.4 Non-Equilibrium Statistical Mechanics
with D the diffusion coefficient and f̄ (x) the average force on a particle at
x, given by
Z
f̄ (x) = − ∇x U1,2 (x − y)ρ(y | x) dy.
Upon comparison of equation (9.9) with equations (9.1) and (9.2) for the
PB and PNP fluxes, it is tempting to write the following equation for the
non-equilibrium case,
∇ · J (x) = 0.
This result, however plausible, cannot be derived from the assumptions of
equilibrium statistical mechanics, because those assumptions imply J (x) =
0. We present below a derivation of this non-equilibrium equation from the
dynamics of diffusing ions.
9.4
9.4.1
Non-Equilibrium Statistical Mechanics
A Trajectory Based Approach
Our point of departure for the statistical description of non-equilibrium systems of diffusing particles is a probability measure defined on their trajectories in phase space, rather than on the statistics of points in configuration
space, which is the starting point of traditional equilibrium statistical mechanics.
First, we introduce some notation. We consider a finite domain Ω, that
consists of the two macroscopic reservoirs and the connecting rigid nanopore
(see Figure 2.1). Its boundary ∂Ω is composed of reflecting boundaries ∂ΩR
and the feedback boundaries ∂ΩF . We assume that there are N h ions of
species
h (h =Ca++ , Na+ ,Cl− , . . . ) in Ω, which are numbered at time t = 0,
P
h
h N = N , and we follow their trajectories at all times t > 0. The coordinates of a point are x = (x, y, z), while the location and velocity coordinates
of the j-th ion of species h at time t are xhj (t) and v hj (t), respectively. According to our assumptions, an ion that reaches ∂ΩF is instantly re-injected
by the feedback mechanism at one or another part of the boundary, so that
its individual identity is preserved, and consequently the total number of
ions inside Ω is fixed at all times. The feedback mechanism serves as the
energy source for the system, keeping it in stationary non-equilibrium by
keeping average concentrations cL and cR in the left and right reservoirs and
a constant applied voltage V across the system. In experimental situations,
175
176
From Langevin Equations to Partial Differential Equations
electronic devices and chemical apparatus provide the energy and feedback.
In biological systems, metabolic machinery and the active transport systems
it fuels provide the energy and feedback.
The evolution of the joint probability density function (pdf) of all the ions
and water molecules in this system can be described by the Liouville equation.
However, at the level of the primitive model, we follow the evolution of the
pdf of only the ions in the system, which is a lower dimensional projection
of the former.
Since the motion of ions in solution is strongly overdamped, on time scales
larger than the relaxation time of the solution, memory effects due to the
thermal motion of the solvent can be neglected [7], and the joint motion of
only the ions can be described as diffusion with interactions. Therefore, our
starting point is a memoryless system of coupled Langevin equations for the
different ion species h = Ca++ , N a+ , Cl− , etc., j = 1, . . . , N h ,
ẍhj
+γ
h
xhj
ẋhj
q
f hj
h γ h xh ẇ h ,
+
2ε
=
j
j
Mh
(9.10)
where a dot on a variable indicates differentiation with respect to time,
γ h (xh ) is the location dependent friction coefficient per unit mass, M h is
the effective mass of an ion of species h and εh = kB T /M h . The force on
the j-th ion of species h is f hj and includes all ion-ion interactions. It thus
depends on the locations of all ions. The functions ẇhj are, by assumption,
independent standard Gaussian white noises. Thus, the effects of the solvent are modeled as a source of noise and friction for ionic motion and as
an averaged dielectric coefficient for the ion-ion interactions. We discuss the
limitations of these approximations in Section 9.6.
9.4.2
The Fokker-Planck Equation
We define by pN (x1 , . . . , xN , v 1 , . . . , v N ) the stationary probability density
function (pdf) of the system of all N ions. Since the coupled motion of all ions
is governed by the Langevin system (9.10) with independent noise terms, the
stationary pdf pN satisfies the multi-dimensional stationary Fokker-Planck
equation (FPE) [166]
h
0 =
N
XX
h
j=1
Lhj pN ,
(9.11)
9.5 The C-PNP System
177
where Lhj is the Fokker-Planck operator acting on the phase space coordinates
of the j-th ion of species h. It is given by
Lhj pN
= ∇v hj ·
γ
h
(xhj )v hj
f hj
− h
M
!
pN +
∆v hj εh γ h (xhj )pN − v hj · ∇xhj pN ,
where the operators ∇v and ∆v denote the gradient and the Laplacian with
respect to the variable v, respectively. Equation (9.11) is defined in the 3N
dimensional region (x1 , . . . , xN ) ∈ ΩN and (v 1 , . . . , v N ) ∈ R3N .
The solution pN of the stationary FPE (9.11) is the non-equilibrium analog of the Boltzmann distribution. It is the stationary transition probability
density function of the 6N -dimensional trajectory of the system in phase
space and thus it is also the probability density function of the particle configurations in phase space. Obviously, since the FPE is defined in a finite
region, a unique solution is determined only after specification of appropriate boundary conditions. If no-flux boundary conditions are given, the FPE
(9.11) can be solved explicitly and the solution is the Boltzmann distribution.
Thus, our formulation of a non-equilibrium system reduces to the equilibrium
theory in this special case. It is thus clear that the boundary conditions are
what drive the system out of equilibrium. The boundary conditions for the
FPE (9.11) need to be determined from the action of the feedback mechanism
at the boundaries and will be described in Chapter 12.
Finally, we note that a time dependent FPE, similar to (9.11), was suggested [7], [40] as the starting point for the study of the transport characteristics of bulk electrolytes, by an analysis of the decay into equilibrium of
transient infinite non-equilibrium electrolyte systems.
9.5
The C-PNP System
With the interpretation of the stationary joint transition pdf of the phasespace trajectories, pN (x1 , . . . , v 1 , . . .), as the probability density of configurations of all particles in phase space, we can follow the steps taken in the
theory of equilibrium statistical mechanics, and study the single and pair
densities. An equation similar to (9.5), for ch (x), the time-averaged steady
178
From Langevin Equations to Partial Differential Equations
state physical concentration of ions of species h at location xh1 , is given by
Z
Y
Y
0
h
h
h0
h
h
h
p
dx
dv hi .
c (x1 ) = N
N (x1 , . . . , v 1 , . . .)
i
N −1
3N
Ω
×R
i,h0
(i,h0 )6=(1,h)
(9.12)
In the equilibrium case, equation (9.4) gives an explicit expression for
pN , so that some of the integrations in (9.12) can be performed explicitly.
Moreover, in deriving the BBGKY equation, the limit N, |V | → ∞ is taken.
In the non-equilibrium case, the system remains finite and pN is not known
explicitly. What is known is only that pN satisfies the FPE equation (9.11)
in a finite domain. However, by integration of this equation over all particle coordinates but one, the following Nernst-Planck type equation for the
concentration ch (x) can be derived [167],
0 = −∇x · J h (x),
where J h (x) is the flux density of type h ions, given by
#
"
h
f̄
(x)
ch (x) ,
J h (x) = −Dh (x) ∇ch (x) −
kB T
(9.13)
(9.14)
where Dh (x) = kB T /M h γ h (x) is the local diffusion coefficient of species h.
h
The quantity f̄ (x) in (9.14) is the average force on a single ion of type h.
It is given by
Z
h
(9.15)
f̄ (x) =
f hp
(x̃h | xh = x) dx̃h1 ,
ΩN −1 1 N −1 1 1
where x̃h1 is the vector of all N − 1 particle coordinates except xh1 and
pN −1 (x̃h1 | xh1 = x) is the conditional probability density of the N − 1 remaining ions given that the first ion of species h is located at x.
In the case of charged ions in solution, the ion-ion interaction forces are
pair-wise additive, and thus the force on the first ion of species h can be
written as
X
0
0
(9.16)
f h1 = f hed (xh1 ) +
f h,h (xhi , xh1 ),
(i,h0 )6=(1,h)
0
where f h,h is the ion-ion interaction force that an ion of type h0 acts on
an ion of type h. It includes both Coulombic interactions as well as short
9.5 The C-PNP System
179
range interactions, such as excluded volume or Lennard-Jones forces. The
force f hed contains both the effects of an applied external field as well as
the dielectric self-force near dielectric boundaries [167, 133, 4]. Interactions
between charges in the system and boundary charges, including both fixed
and induced charges, may determine the entire behavior of the system, e.g.,
to stop the flow of ions through an open pore. These interactions change the
energy of the system also in the equilibrium case.
As shown in [167], with the specific form (9.16) for the force on a single
ion, equation (9.15) for the average force simplifies to
h
h
h
h
h
f̄ (x) = f ed (x) + f̄ SR (x) − z e∇y φ̄ (y|x)
y =x
where
h
f̄ SR (x)
=
XZ
h0
Ω
0
0
h |h
f h,h
(y|x) dy,
SR (y, x)c
(9.17)
is the average short range force on a type h ion, z h is the valence of type h
ions and φ̄h (y|x) is the conditional electrostatic potential at y given a type
h ion at x. It satisfies the (conditional) Poisson equation
X 0 0
∇y · ε(y)∇y φ̄h (y|x) = −e
z h ch |h (y|x),
(9.18)
h0
where ε(y) is the dielectric coefficient at y. In both eq. (9.17) and (9.18),
0
ch |h (y, x) is the conditional concentration of h0 ions at y given an h-type ion
at x. In terms of unconditional quantities, it is given by
0
c
h0 |h
ch,h (x, y)
(y|x) =
.
ch (x)
To summarize, the density ch (x) satisfies a Nernst-Planck type equation
h
(9.13), with an average force f̄ that is the sum of a dielectric self force, an
averaged short range force (9.17) and an averaged electrostatic force. The
latter is a solution of a conditional Poisson equation that depends on conditional densities, in contrast to the unconditional densities in the standard
PNP formulation, equation (9.3). We call this resulting system of conditional
Poisson and Nernst-Planck equations C-PNP.
The NP equation (9.13) is defined in the finite domain Ω. Therefore, in
h
addition to the determination of the averaged force f̄ , boundary conditions
180
From Langevin Equations to Partial Differential Equations
on ∂Ω must be specified in order to determine the unique solution for ch (x).
Obviously, on ∂ΩR , ch (x) satisfies no flux boundary conditions,
h
J (x) · ν = 0.
x ∈ ∂ΩR
In addition, on ∂ΩF , according to our assumptions, the average concentrations ch (x) are maintained at fixed known values chF (x) by the feedback
mechanism. Therefore, regardless of the exact method by which the feedback
mechanism maintains these average concentrations,
ch (x) = chF (x),
for x ∈ ∂ΩF .
In case the domain consists of the entire space, separated by an impermeable
membrane with a small hole, the reflecting boundary ∂ΩR are the two sides
of the membrane, and the conditions ate infinity are
ch (x) → chR
for |x| → ∞,
z>0
(9.19)
h
c (x) →
9.5.1
chL
for |x| → ∞,
z < 0.
The C-PNP Hierarchy
An important difference between the PNP and the C-PNP systems is that
the C-PNP system is not closed, because, as seen from (9.17) and (9.18), the
averaged force in the Nernst-Planck equation (9.13) depends on conditional
higher order concentrations. Specifically, we consider the equation for the
pair concentrations. Employing similar methods to those of [167], we obtain
0
that ch,h (x, y) satisfies the six-dimensional Nernst-Planck equation
0
h,h0
0 · J
∇xh · J h,h
x,
y
+
∇
x,
y
= 0,
(9.20)
h
0
h
y
x
yh
where
"
h,h0
J xh (x, y) = −Dh (x) ∇x c
h,h0
h,h0
(x, y) −
f̄
h,h0
#
(x, y) h,h0
c (x, y)
kB T
and f̄ (x, y) is the average force on an ion of species h located at x,
0
given an ion of species h0 located at y. The second flux J h,h
0 is given by a
h
y
9.5 The C-PNP System
181
similar expression. For the case of pairwise additive forces, this force can be
simplified to
f̄
h,h0
0
h,h0
(x, y) = f hed (x) + f h,h (x, y) + f̄ SR (x, y)
(9.21)
h
h,h0
−ez ∇z φ̄ (z|x, y)
h,h0
0
z =x
where f̄ SR and φ̄h,h are the higher order analogs of (9.17) and (9.18), which
00
0
depend on the third order conditional concentrations ch |h,h . Equation (9.20)
is the non-equilibrium analog of the second BBGKY equation (9.8), and as
h,h0
in equilibrium, determination of the forces f̄
(x, y) requires knowledge of
0 00
the triplet densities ch,h ,h (x, y, z).
Similarly, it is possible to write an equation for the triplet density, whose
average forces depend on fourth order conditional densities. We arrive this
way at an infinite hierarchy of conditional Poisson and Nernst-Planck type
equations, all defined in finite domains. The resulting equations are very
similar to those used in the study of macroscopic bulk dynamical properties of
electrolytes [7], where the time dependence of similar equations is considered
in infinite domains. In these studies, closure relations between the triplet
and pair densities, similar to those of equilibrium statistical mechanics, are
employed in order to compute the average forces [38]. Closure relations
require further physics or approximations, or both.
In our case, however, since we are concerned with a finite system in nonequilibrium, a closure relation between the triplet and the pair densities is
not enough to close the system. Specifically, the Smoluchowski type equation
(9.20) is defined in the finite domain (x, y) ∈ Ω × Ω. Therefore, to uniquely
determine its solution, boundary conditions have to be prescribed on the
domain boundaries, e.g. for (x, y) ∈ ∂Ω × Ω and (x, y) ∈ Ω × ∂Ω. Only after
0
these boundary conditions are specified, the pair concentration ch,h (x, y) is
0
h,h
completely determined, provided the forces f̄
(x, y) are known. As in the
case of the single ion densities, the boundary conditions for the pair densities
should also be determined by the action of the feedback mechanism. The
derivation of boundary conditions for the pair concentrations, as well as for
higher order densities, requires a more detailed description of the feedback
mechanism, developed in Chapter 12.
182
From Langevin Equations to Partial Differential Equations
9.5.2
PNP revisited
The simplest possible closure is
0
0
ch |h (y|x) = ch (y),
(9.22)
which assumes independence of ions and thus neglects ion-ion finite size effects. Therefore, it is also necessary to neglect all short range forces in this
approximation, if one wishes to be consistent. This closure recovers the
(unconditional) PNP system, but with an additional force term, f ed , the dielectric boundary force on a single ion near dielectric interfaces. This term
represents the forces on a single ion by surface charges induced by the ion
itself at dielectric interfaces [133, 4]. This force term has a crucial importance in the determination of the permeation characteristics of the gramicidin
channel, see [133].
The above analysis of this closure clarifies the assumptions and validity
of PB and PNP. These two theories are obtained as approximations of the
exact BBGKY and C-PNP systems by use of the simple closure (9.22) that
neglects both the discreteness of charge and the finite size of ions. Therefore,
in any system where discreteness of charge and the finite size of particles
are important, such as near dielectric interfaces or in confined regions, the
validity of these theories is questionable.
9.6
Summary and Discussion
The function of biological systems such as ion channels typically involves
the atomic control of macroscopic flows. These channels are nanoscale nonequilibrium systems connecting large electrolyte reservoirs, so their description involves many different length and time scales. Yet, the majority of
published work on channels typically considers only a single level of modeling, which is valid or computable on limited time and length scales. In
order to study the function of such systems, connections between theories
at different levels are essential. In this chapter we presented a connection
between the level of Brownian dynamics (BD) and of continuum theories.
This connection is shown graphically as the diagonal red arrow in figure 9.1.
Our analysis shows an equivalence between BD simulations and the CPNP hierarchy. This equivalence can be used to validate specific closure
relations as well as BD computer codes by comparing the results of the two
9.6 Summary and Discussion
Figure 9.1: Typical time scales of various processes for permeation through
a protein channel
183
184
From Langevin Equations to Partial Differential Equations
computations. The resulting C-PNP hierarchy is not closed as it involves
conditional and unconditional charge densities. We find that it is the boundary conditions that drive the system out of equilibrium. When equilibrium
boundary conditions are imposed the equilibrium BBGKY hierarchy is recovered from the C-PNP equations. While the single ion densities satisfy simple
concentration and no flux boundary conditions, the corresponding conditions
for the higher order densities are not obvious and require further analysis (see
Chapter 12).
The two main force terms that appear in the C-PNP system and are
absent in standard PNP are the dielectric boundary force and the effects due
to the finite size of ions. Both of these terms have been shown to be important
in the context of narrow protein channels [58, 133, 136, 30, 119, 59]. However,
to the best of our knowledge, the validity of various closure relations has not
yet been considered in confined and highly non-homogeneous systems.
The derivation presented in this chapter is at the level of the primitive
model and thus relies on a few assumptions concerning the properties of the
implicit solvent. One of these assumptions is that the noise terms of different
ions are independent. This assumption may not hold in very narrow regions
occupied by one or more ions and only a few water molecules, where ions
and water may move in a highly coordinated fashion. The analysis of this
configuration requires separate analysis. Another assumption concerns the
representation of the solvent as an effective dielectric constant. Given the
substantial frequency dependence [8], and possible location dependence of
induced charge, higher resolution methods must be used, along with experimentation, to evaluate this representation and its parameters. Specifically,
the description of induced charge by a dielectric coefficient (with its inherent assumption of a linear invariant relation between induced charge and
local electrical field) must be validated, and the frequency/time and location
dependence of this dielectric coefficient evaluated. Such high resolution calculations are not trivial because they themselves must be shown to represent
the electric field accurately over the relevant length and time scales. We
note that in the context of proteins, the dielectric coefficient for dielectric
boundary forces may not be the same as the one coefficient for charge-charge
interactions [169].
Finally, we note that in this chapter we assumed that the nanopore is
rigid. If this is not the case, then the dynamics of the fluctuations of the
nanopore structure from its average configuration should be coupled to the
Langevin equations for the motion of all the ions as described in [167]. This
9.6 Summary and Discussion
would result in a conditional Poisson equation with a conditional averaged
structure of the nanopore, that depends on the location of the mobile ion
inside it. It seems likely that gating can be described by such a coupled
system of equations.
185
Chapter 10
Maximum Entropy Formulation
of the Kirkwood Superposition
Approximation
The contents of this chapter were published in [174]
Using a variational formulation, we derive the Kirkwood superposition
approximation for systems at equilibrium in the thermodynamic limit. We
define the entropy of the triplet correlation function and show that the Kirkwood closure brings the entropy to its maximal value. This approach leads to
a different interpretation for the Kirkwood closure relation, usually explained
by probabilistic considerations of dependence and independence of particles.
The Kirkwood closure is generalized to finite volume systems at equilibrium
by computing the pair correlation function in finite domains. Closure relations for high order correlation functions are also found using a variational
approach. In particular, maximizing the entropy of quadruplets leads to
the high order closure g1234 = g123 g124 g134 g234 /[g12 g13 g14 g23 g24 g34 ] used in the
BGY2 (Born-Green-Yvon 2) equations which are a pair of integral equations
for the triplet and pair correlation functions.
10.1
Introduction
The pair correlation function is one of the cornerstones in the theory of simple
liquids [12, 34, 7, 72, 67, 158]. Many thermodynamic properties of the fluid
10.1 Introduction
can be derived from the pair function. There are two main approaches to
finding the pair function. The first approach is based on the Ornstein-Zernike
integral equation and a closure relation for the direct and indirect correlation
functions. Many closure relations fall into this category, such as the PercusYevick approximation (PY), the hypernetted chain approximation (HNC)
and the mean spherical approximation (MSA). The second approach relies
on the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy, which
relates the n-th correlation function with the n + 1-th correlation function,
and assumes a closure relation that connects them. The Kirkwood superposition, that approximates the triplet correlation function by the product of
the three pair correlation functions (n = 2)
g3 (x1 , x2 , x3 ) = g2 (x1 , x2 )g2 (x1 , x3 )g2 (x2 , x3 ),
(see also eq. (10.42)), was the first suggested closure of this kind [99, 100].
Approximating the quadruplet correlation function by the triplet correlation
function (n = 3) using the Fisher-Kopeliovich closure [53] eq. (10.43) turned
out to yield a much better approximation for the pair correlation function
than the Kirkwood superposition approximation (SA), as noted by Ree et
al. [113, 151]. However, truncating the BBGKY hierarchy at a higher level
(n ≥ 4) is computationally impractical at the moment.
Both approaches and the underlying closure relations have their own advantages and disadvantages. Their success is often evaluated in comparisons
with either molecular dynamics (MD) or Monte-Carlo (MC) simulations. All
closures succeed in the limit of (very) low density. However, when the density
is increased they eventually fail, sooner or later. Choosing the “best” closure
relation is an art in itself. Obviously, each choice of closure relation results in
a different approximate solution for the pair function. The BBGKY equation
(with or without the SA) has a great advantage over any other approach. For
dp
= 0, and hence it predicts solidhard spheres it predicts a point where
dρ
ification. Neither the PY or HNC theories can do this and so one expects
the BBGKY approach to be superior at high densities, which is particularly
important in applications to protein and channel biology [44].
Rice et al. [152, 153] improved the Kirkwood SA for systems of hardspheres and Lennard-Jones 6-12 potential. Meeron [130] and Salpeter [160]
proposed a formal expression for the triplet correlation function in the form
g3 (x1 , x2 , x3 ) = g2 (x1 , x2 )g2 (x1 , x3 )g2 (x2 , x3 ) exp[τ (x1 , x2 , x3 , ρ)], (10.1)
187
188 Maximum Entropy Formulation of the Kirkwood Superposition Approximation
P
n
where τ (x1 , x2 , x3 , ρ) ≡ ∞
n=1 ρ δn+3 (x1 , x2 , x3 ). The coefficients δn+3 (x1 , x2 , x3 )
consist of simple 123-irreducible diagrams. Rowlinson [159] evaluated analytically the term δ4 for a system of hard spheres. Rice [152, 153] evaluated the
ρδ4
term δ5 numerically and showed that the Padé approximation τ ≈
1 − ρδ5
gives a better fit to the computer simulation of the hard-sphere system than
the Kirkwood SA. However, Ree proved that this improvement to the Kirkwood SA is inferior to the high order closure relation of Refs [113, 151],
because the latter takes into account all the diagrams of Refs [152, 153] and
more. Hu et al. [81] showed that the integral closure relations (PY, HNC and
MSA) can be obtained from the Kirkwood SA using another approximation.
In this chapter we derive the Kirkwood SA from a variational (EulerLagrange) formulation. We define an entropy functional for the triplet correlation function, and show that it produces the Kirkwood closure in the
thermodynamic limit of number of particles and volume taken to infinity,
while keeping the number density constant, i.e. N, V → ∞, N/V = ρ.
The triplet entropy functional is different from the physical entropy of the
full N-particle system. Maximizing the triplet entropy functional, given the
constraint that the pair correlation function is the marginal of the triplet
correlation function, yields the SA.
This maximum entropy formulation leads to a different closure for a finite
volume system which we write as an integral equation. In the thermodynamic
limit, the solution of this integral equation reduces to the Kirkwood SA. We
describe an iterative procedure for solving the BBGKY equation with this
generalized entropy closure.
Using this entropy closure instead of the Kirkwood SA produces different
pair correlation functions. The entropy closure can be used to calculate the
pair function in confined geometries. Specifically, we can deduce the resulting
pair function near the domain walls (boundaries), and compare it to MC or
MD simulations, and to the results of the original Kirkwood closure.
The present attempt to find the pair correlation function using a variational formulation is not new. Richardson [154] used a maximum entropy
(minimum Helmholtz free energy) argument to obtain an equation for the
pair correlation function. That variational approach consists in maximizing
the entropy while assuming an approximate relation (closure) between the
pair and triplet function, and resulted in an integral equation which differs
from the BGY equation. On the other hand, the variational approach applied in this chapter assumes that the BGY equation is exact, and maximizes
10.2 Maximum Entropy
189
the triplet entropy under the constraint that the pair correlation function is
the marginal of the triplet function. This approach leads to a closure for the
BGY equation. The resulting closure coincides with the Kirkwood SA for the
radial function. Mathematically, the approach of Richardson [154] consists in
solving a variational problem without imposing constraints, while the current
proposed variational formulation includes constraints and is therefore solved
by the method of Lagrange multipliers.
The definition of the n-particle entropy (3 ≤ n ≤ N ) is similar to the
definition of the triplet entropy functional. The N -particle entropy coincides with the physical entropy and its maximization yields the Boltzmann
distribution, as expected. High order closure relations are obtained by maximizing the n-particle entropy. In particular, maximizing the quadruplets
entropy leads to the Fisher-Kopeliovich closure [53] and the corresponding
BGY2 equation [113, 151], that has been shown to give a very good fit to
experimental data.
10.2
Maximum Entropy
We consider a finite domain Ω ⊂ R3 . Suppose p2 (x1 , x2 ) is a known symmetric probability distribution function (pdf)
Z
p2 (x1 , x2 ) dx1 dx2 = 1
Ω×Ω
p2 (x1 , x2 ) = p2 (x2 , x1 )
(10.2)
p2 (x1 , x2 ) ≥ 0,
where p2 (x1 , x2 ) represents the joint pdf of finding two particles at locations
x1 and x2 , as usually defined in the statistical mechanics of fluids. This
means a relationship between p2 (x1 , x2 ) and p3 (x1 , x2 , x3 ) can be expressed
190 Maximum Entropy Formulation of the Kirkwood Superposition Approximation
in the form of the constraints
Z
p3 (x1 , x2 , x3 ) dx1 − p2 (x2 , x3 ) = 0,
φ1 (p3 ) =
Ω
Z
p3 (x1 , x2 , x3 ) dx2 − p2 (x1 , x3 ) = 0,
φ2 (p3 ) =
(10.3)
Ω
Z
p3 (x1 , x2 , x3 ) dx3 − p2 (x1 , x2 ) = 0.
φ3 (p3 ) =
Ω
An approximate closure relation can be found by solving the optimization
problem of maximizing the triplet entropy functional
Z
H(p3 ) = −
p3 (x1 , x2 , x3 ) ln p3 (x1 , x2 , x3 ) dx1 dx2 dx3 ,
(10.4)
Ω×Ω×Ω
under the constraints (10.3). Note that H(p3 ) differs from the physical entropy of the full N -particle system
Z
pN (x1 , x2 , . . . , xN ) ln pN (x1 , x2 , . . . , xN ) dx1 dx2 · · · dxN ,
H ≡ H(pN ) = −
Ω
where pN (x1 , x2 , . . . , xN ) is the Boltzmann distribution. The motivation
for maximizing the triplet entropy in finding a closure relation, is that the
Boltzmann distribution brings the physical entropy to a maximum. Thus,
maximizing H(p3 ) instead of H introduces errors and is expected to be an
approximation to the closure problem. This issue will be further discussed
in Section 10.4. The notation ”triplet entropy” for H(p3 ) of equation (10.4)
is in agreement with the definition of the Shannon entropy of three random
variables in information theory (see eg. Ref [31]).
This variational problem can be solved by the method of Lagrange multipliers. Thus, we define Lagrange multipliers λ1 (x2 , x3 ), λ2 (x1 , x3 ), λ3 (x1 , x2 )
10.2 Maximum Entropy
191
and the functional
F (p3 , λ1 , λ2 , λ3 ) = H + λ1 φ1 + λ2 φ2 + λ3 φ3
Z
= −
p3 (x1 , x2 , x3 ) ln p3 (x1 , x2 , x3 ) dx1 dx2 dx3
Ω×Ω×Ω
Z
+λ1 (x2 , x3 )
p3 (x1 , x2 , x3 ) dx1 − p2 (x2 , x3 )
Ω
Z
+λ2 (x1 , x3 )
p3 (x1 , x2 , x3 ) dx2 − p2 (x1 , x3 )
Ω
Z
+λ3 (x1 , x2 )
p3 (x1 , x2 , x3 ) dx3 − p2 (x1 , x2 ) .
Ω
The Euler-Lagrange equation is
− ln p3 (x1 , x2 , x3 ) − 1 + λ1 (x2 , x3 ) + λ2 (x1 , x3 ) + λ3 (x1 , x2 ) = 0,
(10.5)
or equivalently
p3 (x1 , x2 , x3 ) = γ1 (x2 , x3 )γ2 (x1 , x3 )γ3 (x1 , x2 ),
(10.6)
where
γ1 (x2 , x3 ) = eλ1 (x2 ,x3 )−1/3 ,
γ2 (x1 , x3 ) = eλ2 (x1 ,x3 )−1/3 ,
γ3 (x1 , x2 ) = eλ3 (x1 ,x2 )−1/3 .
Clearly, γi ≥ 0 (i = 1, 2, 3), and therefore p3 ≥ 0. Moreover, γ1 = γ2 = γ3
because p2 is symmetric. Setting γ = γ1 = γ2 = γ3 we find that
p3 (x1 , x2 , x3 ) = γ(x1 , x2 )γ(x2 , x3 )γ(x1 , x3 ).
(10.7)
The constraint that p2 is the marginal of p3 gives an equation for γ,
Z
p2 (x1 , x2 ) = γ(x1 , x2 ) γ(x1 , x3 )γ(x2 , x3 ) dx3 .
(10.8)
Ω
192 Maximum Entropy Formulation of the Kirkwood Superposition Approximation
The symmetry of p2 (eq.(10.2)) implies that of γ,
γ(x1 , x2 ) = γ(x2 , x1 ).
If equation (10.8) has a unique solution, then p2 determines γ. However,
the pdf p2 is unknown. We know that p2 satisfies the BBGKY equation
0 = f ex (x1 )p2 (x1 , x2 ) + f (x2 , x1 )p2 (x1 , x2 ) − kB T ∇x1 p2 (x1 , x2 )
Z
+(N − 2)
f (x3 , x1 )p3 (x1 , x2 , x3 ) dx3 ,
Ω
where f (x2 , x1 ) is the force exerted on a particle located at x1 by another
particle located at x2 ; and f ex (x1 ) is an external force field acting on a
particle located at x1 . Substituting the maximum entropy closure (10.7)
into the BBGKY equation (10.9), together with the integral equation (10.8),
produces a system of integral equations for p2 and γ,
0 = f ex (x1 )p2 (x1 , x2 ) + f (x2 , x1 )p2 (x1 , x2 ) − kB T ∇x1 p2 (x1 , x2 )
Z
+(N − 2)
f (x3 , x1 )γ(x1 , x2 )γ(x2 , x3 )γ(x1 , x3 ) dx3
Ω
Z
γ(x1 , x3 )γ(x2 , x3 ) dx3 .
p2 (x1 , x2 ) = γ(x1 , x2 )
(10.9)
Ω
To determine the pair correlation function p2 , we need to solve the system
(10.9).
10.3
Two Examples
The system (10.9) can be solved or simplified in the case that the particles
interact only with the external field and in the thermodynamic limit.
10.3 Two Examples
10.3.1
193
Non-interacting particles in an external field
Non-interacting particles in an external field are described by f ≡ 0 and the
system (10.9) is reduced to
0 = f ex (x1 )p2 (x1 , x2 ) − kB T ∇x1 p2 (x1 , x2 )
(10.10)
Z
p2 (x1 , x2 ) = γ(x1 , x2 )
γ(x1 , x3 )γ(x2 , x3 ) dx3
(10.11)
Ω
Equation (10.10) can be integrated to yield
p2 (x1 , x2 ) = e−Uex (x1 )/kB T h(x2 ),
(10.12)
where f ex (x1 ) = −∇x1 Uex (x1 ), and h(x2 ) is an arbitrary function (the
integration constant). The symmetry condition (10.2) gives
p2 (x1 , x2 ) = C −1 e−[Uex (x1 )+Uex (x2 )]/kB T ,
Z
−Uex (x)/kB T
e
where C =
(10.13)
2
dx
is a normalization constant. As expected,
Ω
we find that x1 , x2 are independent random variables, p2 (x1 , x2 ) = p1 (x1 )p1 (x2 ).
In this case, the solution to (10.11) is given by
p
p
p
(10.14)
γ(x1 , x2 ) = p2 (x1 , x2 ) = p1 (x1 ) p1 (x2 ).
Indeed,
Z
γ(x1 , x2 )
γ(x1 , x3 )γ(x2 , x3 ) dx3 =
Ω
=
p
p1 (x1 )p1 (x2 )
Z p
p
p1 (x1 )p1 (x3 ) p1 (x2 )p1 (x3 ) dx3
Ω
Z
= p1 (x1 )p1 (x2 )
p1 (x3 ) dx3 = p2 (x1 , x2 ).
Ω
10.3.2
The Thermodynamic Limit
The thermodynamic limit is described by the following limiting process in
which the domain is gradually enlarged to be all space, Ω → R3 , keeping
194 Maximum Entropy Formulation of the Kirkwood Superposition Approximation
the ratio between the number of particles, N , and the volume of the domain,
V = |Ω|, constant. The ratio N/V = ρ is the number density. In this limiting
process, all the pdfs tend to zero, because the volume of the domain tends
to infinity, and the pdfs are normalized with respect to that volume. It is
therefore more convenient to work with number densities, such as ρ, which
are normalized with respect to a fixed volume, e.g., 1 cm3 , and do not vanish
in the limiting process.
First, we consider a bounded domain Ω ⊂ R3 . The previous example of
non-interacting particles suggests the definition
δ(x1 , x1 ) = p
γ(x1 , x2 )
p1 (x1 )p1 (x2 )
,
(10.15)
which transforms equation (10.11) into
Z
p2 (x1 , x2 )
= δ(x1 , x2 ) p1 (x3 )δ(x1 , x3 )δ(x2 , x3 ) dx3 .
p1 (x1 )p1 (x2 )
Ω
(10.16)
We rewrite equation (10.16) as
(Ω)
p2 (x1 , x2 )
=
(Ω)
(Ω)
p1 (x1 )p1 (x2 )
Z
(Ω)
(Ω)
δ (x1 , x2 ) p1 (x3 )δ (Ω) (x1 , x3 )δ (Ω) (x2 , x3 ) dx3 ,
(10.17)
Ω
(Ω)
(Ω)
where p2 (x1 , x2 ) = p2 (x1 , x2 ), δ (Ω) (x1 , x2 ) = δ(x1 , x2 ), p1 (x1 ) = p1 (x1 ),
to emphasize their dependency on the specific domain Ω. We set
(Ω)
g2 (x1 , x2 ) = lim3
Ω→R
p2 (x1 , x2 )
(Ω)
(Ω)
.
(10.18)
p1 (x1 )p1 (x2 )
For example, if the two particles become independent when they are separated,
(Ω)
(Ω)
(Ω)
p2 (x1 , x2 ) = p1 (x1 )p1 (x2 ) (1 + o(1)) , for |x1 − x2 | 1,
then lim|x2 |→∞ g2 (x1 , x2 ) = 1.
Next, we show that under the assumption (10.19)
lim δ (Ω) (x1 , x2 ) = g2 (x1 , x2 ).
Ω→R3
(10.19)
10.4 Minimum Helmholtz Free Energy
195
Indeed,
Z
Z
p2 (x1 , x3 ) p2 (x2 , x3 )
dx3 =
p1 (x3 ) (1 + o(1)) dx3
p1 (x3 )
p1 (x1 )p1 (x3 ) p1 (x2 )p1 (x3 )
Ω
Ω
= 1 + o(1).
(10.20)
Taking the limit Ω → R3 , the o(1) term vanishes, and equation (10.17) gives
δ(x1 , x2 ) = g2 (x1 , x2 ),
(10.21)
as asserted.
We interpret equation (10.21) as the Kirkwood SA. Equations (10.7) and
(10.15) imply that the triplet pdf satisfies
(Ω)
p3 (x1 , x2 , x3 )
(Ω)
(Ω)
(Ω)
p1 (x1 )p1 (x2 )p1 (x3 )
3
= δ (Ω) (x1 , x2 )δ (Ω) (x1 , x3 )δ (Ω) (x2 , x3 ).
(10.22)
Taking the limit Ω → R and using equation (10.21), we obtain
(Ω)
g3 (x1 , x2 , x3 ) =
lim
Ω→R3
p3 (x1 , x2 , x3 )
(Ω)
(Ω)
(Ω)
p1 (x1 )p1 (x2 )p1 (x3 )
= g2 (x1 , x2 )g2 (x1 , x3 )g2 (x2 , x3 ),
(10.23)
which is the Kirkwood closure relation for the triplet correlation function.
Usually, the motivation for using the Kirkwood SA is a probabilistic consideration of dependency and independency of particles [99] (see also Section
10.5.) Here we find another interpretation for the Kirkwood closure.
The Kirkwood closure is the (only) closure relation that brings the entropy
of triplets of particles to its maximum value. The principle of maximum
entropy is a well known principle in statistical mechanics, in testing statistical
hypotheses [114, 115] and beyond [155, 18]. The maximum entropy principle
is also applicable to systems out of equilibrium [35]; therefore Kirkwood’s SA
can be generalized to systems out of equilibrium. In the next section we give
further motivation for its use.
10.4
Minimum Helmholtz Free Energy
Elementary textbooks in statistical mechanics mention that the Boltzmann
distribution
1 −U (x1 ,...,xN )/kB T
e
,
(10.24)
pN (x1 , . . . , xN ) =
ZN
196 Maximum Entropy Formulation of the Kirkwood Superposition Approximation
brings the Helmholtz free energy
F (p) = U (p) − kB T H(p)
(10.25)
Z
=
U (x1 , . . . , xN )p(x1 , . . . , xN ) dx1 · · · dxN
ΩN
Z
+kB T
p(x1 , . . . , xN ) ln p(x1 , . . . , xN ) dx1 · · · dxN ,
ΩN
to its minimum under the normalization constraint
Z
p(x1 , . . . , xN ) dx1 · · · dxN = 1.
(10.26)
ΩN
For a pairwise additive potential together with an external field force
U (x1 , . . . , xN ) =
X
U (xi , xj ) +
N
X
Uex (xj ),
(10.27)
j=1
1≤i<j≤N
the potential energy term U (p) of the Helmholtz free energy (10.25) takes
the simple form
!
Z
N
X
X
U (p) =
U (xi , xj ) +
Uex (xj ) p(x1 , . . . , xN ) dx1 · · · dxN
ΩN
j=1
1≤i<j≤N
N (N − 1)
=
2
Z
Z
U (x1 , x2 )p2 (x1 , x2 ) dx1 dx2 + N
Ω2
Uex (x1 )p1 (x1 ) dx1 ,
Ω
where
Z
pN (x1 , . . . , xN ) dx1 · · · dxN ,
p2 (x1 , x2 ) =
N −2
ZΩ
p1 (x1 ) =
p2 (x1 , x2 ) dx2 ,
Ω
are the marginal densities. If the pdf p2 (x1 , x2 ) is assumed to be known,
as in Section 10.2, then the energy term of the Helmholtz free energy U (p)
is also known. Therefore, minimizing the Helmholtz free energy, under the
assumption that the pdf p2 (x1 , x2 ) is known, is equivalent to maximizing the
entropy, since U (p) is constant during the minimizing process.
10.5 Probabilistic Interpretation of the Kirkwood Closure
10.5
197
Probabilistic Interpretation of the Kirkwood Closure
The Kirkwood SA (10.23) was the first closure relation to be suggested [99]
and tested [100] in the theory of simple liquids. This fact might be explained
by its simplicity and its intuitive origin. In this section we give a probabilistic
interpretation of the Kirkwood SA, and find its generalization for closure
relations of higher orders of the BBGKY hierarchy. The problem at level n
(n ≥ 2) is to find an approximation for the n + 1-particle pdf in terms of the
n-particle pdf. For example, the Kirkwood SA (10.23) closes the hierarchy
at level n = 2. High order closures are much more accurate, fitting better
the experimental and simulated (MC or MD) data [113, 151]. However, the
computational complexity increases drastically with n, making the truncation
at level n ≥ 4 impractical at the moment.
First, consider the case n = 2. We assume that particles become independent as they are separated (10.19), although we are certainly aware that long
range forces such as the electric field can produce strong correlations over all
of even a very large domain, as already noted by Kirkwood in his original paper [99]. In order to make the exact equality (10.19) into an approximation,
we assume that there exists a distance d > 0 such that
p2 (x1 , x2 ) = p1 (x1 )p1 (x2 ),
(10.28)
for |x1 − x2 | > d. Three interchangeable particles can be in four different
configurations with respect to the distance d, depending on the number of
intersections (see Figure 10.1). In all configurations but configuration (d),
where all three particles intersect, there are at least two particles that do
not intersect. Since the particles are interchangeable we may assume that
|x1 − x3 | > d. Applying Bayes’ law we have
p3 (x1 , x2 , x3 ) = p3 (x3 |x1 , x2 )p2 (x1 , x2 ).
(10.29)
By the independence assumption (10.28) we have
p3 (x3 |x1 , x2 ) = p2 (x3 |x2 ).
(10.30)
Therefore,
p3 (x1 , x2 , x3 ) = p2 (x3 |x2 )p2 (x1 , x2 ) =
p2 (x2 , x3 )
p2 (x1 , x2 ).
p1 (x2 )
(10.31)
198 Maximum Entropy Formulation of the Kirkwood Superposition Approximation
Figure 10.1: Four configurations of three particles. (a) no intersections (b)
one intersection (c) two intersections (d) three intersections.
10.5 Probabilistic Interpretation of the Kirkwood Closure
Multiplying by 1 =
199
p2 (x1 , x3 )
we obtain
p1 (x1 )p1 (x3 )
p3 (x1 , x2 , x3 ) =
p2 (x1 , x2 )p2 (x2 , x3 )p2 (x1 , x3 )
,
p1 (x1 )p1 (x2 )p1 (x3 )
(10.32)
which is the Kirkwood SA. We see that the Kirkwood closure is a good
approximation when at least two particles are sufficiently far apart and independent. However, it fails when all three particles are close to each other.
Next, we find the n-level Kirkwood closure relation for the n + 1-particle
pdf in terms of the n-particle pdf, using probabilistic considerations.
Proposition: The n-level Kirkwood closure relation is given by
pn (x1 , x2 , . . . , xn ) =
n−1
Y
Y
pk (xi1 , xi2 , . . . , xik )(−1)
n−1−k
.
k=1 1≤i1 <i2 <...<ik ≤n
(10.33)
Proof: We have already seen that the approximation holds for n = 2, 3.
Assuming, by induction, that at least two particles are sufficiently far apart
and independent, we can assume without loss of generality, that particles 1
and n are far apart, and independent, |x1 − xn | > d. Using Bayes’ law, we
find that
pn (x1 , x2 , . . . , xn ) = pn−1 (x1 , x2 , . . . , xn−1 )pn (xn |x1 , x2 , . . . , xn−1 ).
Because particles 1 and n are sufficiently far apart it follows that
pn (xn |x1 , x2 , . . . , xn−1 ) = pn−1 (xn |x2 , x3 , . . . , xn−1 ) =
pn−1 (x2 , x3 , . . . , xn )
.
pn−2 (x2 , x3 , . . . , xn−1 )
Hence,
pn (x1 , x2 , . . . , xn ) =
pn−1 (x1 , x2 , . . . , xn−1 )pn−1 (x2 , x3 , . . . , xn )
.
pn−2 (x2 , x3 , . . . , xn−1 )
(10.34)
It follows from the induction assumption that for every j = 2, 3, . . . , n − 1,
with particles 1 and n far apart, we have
1 = pn−1 (x1 , . . . , xj−1 , xj+1 , . . . , xn )
(10.35)
n−2
Y
Y
n−1−k
×
pk (xi1 , xi2 , . . . , xik )(−1)
.
k=1 1≤i1 <i2 <...<ik ≤n, il 6=j
200 Maximum Entropy Formulation of the Kirkwood Superposition Approximation
Multiplying eqs.(10.34) and (10.35) (for all j = 2, 3, . . . , n−1) ends the proof.
Corollary: For n = 4 Kirkwood’s formula becomes
p4 (x1 , x2 , x3 , x4 ) = p1 (x1 )p1 (x2 )p1 (x3 )p1 (x4 ) ×
(10.36)
p3 (x1 , x2 , x3 )p3 (x1 , x2 , x4 )p3 (x1 , x3 , x4 )p3 (x2 , x3 , x4 )
.
p2 (x1 , x2 )p2 (x1 , x3 )p2 (x1 , x4 )p2 (x2 , x3 )p2 (x2 , x4 )p2 (x3 , x4 )
In the case Ω = R3 , we define
gn (x1 , x2 , . . . , xn ) = lim3
Ω→R
Dividing equation (10.33) by
Qn
j=1
n−1
pn (x1 , x2 , . . . , xn ) Y
Qn
=
j=1 p1 (xj )
k=1 1≤i
pn (x1 , x2 , . . . , xn )
Qn
.
j=1 p1 (xj )
(10.37)
p1 (xj ) gives
Y
1 <i2 <...<ik ≤n
pk (xi1 , xi2 , . . . , xik )
Qk
j=1 p1 (xij )
!(−1)n−1−k
,
(10.38)
where we used the combinatorial identity
n−1 X
n−1
k−1
k=1
(−1)n−1−k = 1.
(10.39)
Note that the k = 1 terms in the product of equation (10.38) cancel out, so
the product may begin from k = 2
n−1
pn (x1 , x2 , . . . , xn ) Y
Qn
=
j=1 p1 (xj )
k=2 1≤i
Y
1 <i2 <...<ik ≤n
pk (xi1 , xi2 , . . . , xik )
Qk
j=1 p1 (xij )
!(−1)n−1−k
.
(10.40)
Taking the limit Ω → R3 we obtain the n-level Kirkwood closure relation
gn (x1 , x2 , . . . , xn ) =
n−1
Y
Y
k=2 1≤i1 <i2 <...<ik ≤n
Examples:
gk (xi1 , xi2 , . . . , xik )(−1)
n−1−k
. (10.41)
10.6 High Level Entropy Closure
201
• n = 3 (Kirkwood SA)
g3 (x1 , x2 , x3 ) = g2 (x1 , x2 )g2 (x1 , x3 )g2 (x2 , x3 ).
(10.42)
• n = 4 (Fisher-Kopeliovich [53])
g4 (x1 , x2 , x3 , x4 ) =
(10.43)
g3 (x1 , x2 , x3 )g3 (x1 , x2 , x4 )g3 (x1 , x3 , x4 )g3 (x2 , x3 , x4 )
.
g2 (x1 , x2 )g2 (x1 , x3 )g2 (x1 , x4 )g2 (x2 , x3 )g2 (x2 , x4 )g2 (x3 , x4 )
10.6
High Level Entropy Closure
In this section we use the maximum entropy principle to derive the n-level
closure relation, and compare the resulting closure relation with the n-level
probabilistic Kirkwood closure of Section 10.5. The problem at level n (n ≥
2) is to find an approximation for the n + 1-particle pdf in terms of the nparticle pdf. For example, the Kirkwood SA (10.23) closes the hierarchy at
level n = 2. We use the principle of maximum entropy to obtain the closure
relation. As in the derivations of Section 10.2, we assume that the n-particle
pdf pn (x1 , x2 , . . . , xn ) is known, and we search for the n + 1-particle pdf
pn+1 (x1 , x2 , . . . , xn+1 ) that maximizes the entropy
H=
Z
−
pn+1 (x1 , x2 , . . . , xn+1 ) ln pn+1 (x1 , x2 , . . . , xn+1 ) dx1 dx2 · · · dxn+1 ,
Ωn+1
with the n + 1 constraints that the pn are the marginals of pn+1
pn (x1 , x2 , . . . , xn ) =
Z
pn+1 (x1 , . . . , xj , . . . , xn+1 ) dxj ,
j = 1, 2, . . . , n + 1.
Ω
Since p2 is the marginal of pn (n ≥ 2), it follows that p2 is also known.
Therefore, for a pairwise additive potential, maximizing the Helmholtz free
energy of the n + 1-particle system is equivalent to minimizing the entropy of the n + 1-particle system. Introducing the Lagrange multipliers
202 Maximum Entropy Formulation of the Kirkwood Superposition Approximation
λj (x1 , . . . , xj−1 , xj+1 , . . . , xn+1 ), j = 1, 2, . . . , n+1, the Euler-Lagrange equation gives
n+1
X
− ln pn+1 − 1 +
λj = 0.
(10.44)
j=1
Since the n particles are interchangeable,
pn+1 (x1 , . . . , xn+1 ) =
n+1
Y
γ(x1 , . . . , xj−1 , xj+1 , . . . , xn+1 ).
(10.45)
j=1
Integration with respect to xn+1 yields
pn (x1 , . . . , xn ) =
γ(x1 , . . . , xn )
Z Y
n
(10.46)
γ(x1 , . . . , xj−1 , xj+1 , . . . , xn+1 ) dxn+1 .
Ω j=1
Solving the non-linear integral equation (10.46) for γ and substituting in
equation (10.45) gives the n-level closure relation of the n-level BBGKY
hierarchy equation for pn
0 = f ex (x1 )pn (x1 , x2 , . . . , xn ) +
n
X
f (xj , x1 )pn (x1 , x2 , . . . , xn )
j=2
−kB T ∇x1 pn (x1 , x2 , . . . , xn )
Z
+(N − n) f (xn+1 , x1 )pn+1 (x1 , x2 , . . . , xn+1 ) dxn+1 .
(10.47)
Ω
10.6.1
The thermodynamic limit
We have seen in Section 10.3 that the maximum entropy principle yields the
Kirkwood SA (n = 2) in the thermodynamic limit. In this section we show
that in the thermodynamic limit, the maximum entropy principle results in
the probabilistic Kirkwood closure (10.33) for all levels n ≥ 2.
First, we consider a bounded domain Ω ⊂ R3 . We set
(Ω)
h(Ω)
n (x1 , x2 , . . . , xn ) = Qn−1 Q
k=1
pn (x1 , x2 , . . . , xn )
,
(Ω)
(−1)n−1−k
(x
,
x
,
.
.
.
,
x
)
p
i
i
i
1
2
k
1≤i1 <i2 <...<ik ≤n k
(10.48)
10.6 High Level Entropy Closure
203
and
γ (Ω) (x1 , x2 , . . . , xn )
δ (Ω) (x1 , x2 , . . . , xn ) = Qn−1 Q
n−k .
(Ω)
(−1)n−1−k n−k+1
p
(x
,
x
,
.
.
.
,
x
)
i
i
i
1
2
k
k=1
1≤i1 <i2 <...<ik ≤n k
(10.49)
Qn−1 Q
n−1−k
(Ω)
Dividing equation (10.46) by k=1 1≤i1 <i2 <...<ik ≤n pk (xi1 , xi2 , . . . , xik )(−1)
we obtain
(Ω)
h(Ω)
(x1 , x2 , . . . , xn ) ×
n (x1 , x2 , . . . , xn ) = δ
Z
F (x1 , x2 , . . . , xn , xn+1 )
n
Y
Ω
(10.50)
δ (Ω) (x1 , . . . , xj−1 , xj+1 , . . . , xn+1 ) dxn+1 ,
j=1
where
F (x1 , x2 , . . . , xn , xn+1 )
n−2
Y
Y
n−1−k (n−k−1)
(Ω)
=
pk (xi1 , xi2 , . . . , xik )(−1)
k=1 1≤i1 <i2 <...<ik ≤n
(10.51)
×
n−1
Y
Y
(Ω)
pk (xi1 , xi2 , . . . , xik−1 , xn+1 )(−1)
n−1−k (n−k)
.
k=1 1≤i1 <i2 <...<ik−1 ≤n
For min1≤j≤n |xn+1 − xj | 1, the n + 1-th particle becomes independent of
particles 1, 2, . . . , n, and we have
(Ω)
pk (xi1 , xi2 , . . . , xik−1 , xn+1 ) =
(Ω)
(Ω)
p1 (xn+1 )pk−1 (xi1 , xi2 , . . . , xik−1 ) (1 + o(1)) ,
for all k = 1, 2, . . . , n and all sets of indices 1 ≤ i1 < i2 < . . . < ik−1 ≤ n.
204 Maximum Entropy Formulation of the Kirkwood Superposition Approximation
Therefore, to leading order in (min1≤j≤n |xn+1 − xj |)−1
F (x1 , x2 , . . . , xn , xn+1 ) =
n−2
Y
Y
(Ω)
pk (xi1 , xi2 , . . . , xik )(−1)
n−1−k (n−k−1)
×
k=1 1≤i1 <i2 <...<ik ≤n
n−1
Y
Y
(Ω)
pk−1 (xi1 , xi2 , . . . , xik−1 )(−1)
n−1−k (n−k)
×
k=1 1≤i1 <i2 <...<ik−1 ≤n
n−1
Y
Y
(Ω)
n−1−k (n−k)
(Ω)
n−1−k (n−k)
p1 (xn+1 )(−1)
k=1 1≤i1 <i2 <...<ik−1 ≤n
=
n−1
Y
Y
p1 (xn+1 )(−1)
k=1 1≤i1 <i2 <...<ik−1 ≤n

n−1


X




n
n−1−k
(−1)
(n
−
k)
h
i


k−1 


(Ω)
= p1 (xn+1 ) k=1
(Ω)
= p1 (xn+1 ),
(10.52)
where we have used the combinatorial identity
n−1
X
n−1−k
(−1)
k=1
n
(n − k)
= 1.
k−1
(10.53)
We conclude that
h(Ω)
n (x1 , x2 , . . . , xn )
= δ
(Ω)
Z
(Ω)
p1 (xn+1 ) (1 + o(1))
(x1 , x2 , . . . , xn )
Ω
×
n
Y
δ (Ω) (x1 , . . . , xj−1 , xj+1 , . . . , xn+1 ) dxn+1 .
j=1
Therefore,
lim δ (Ω) (x1 , x2 , . . . , xn ) = lim3 h(Ω)
n (x1 , x2 , . . . , xn )
Ω→R3
Ω→R
(10.54)
10.6 High Level Entropy Closure
205
Substituting eq. (10.49) in eq. (10.45), using the relations (10.48) and
(10.54), and the definition (10.37), we find that
gn+1 (x1 , . . . , xn+1 ) =
n
Y
Y
gk (xi1 , xi2 , . . . , xik )(−1)
n−k
.
k=2 1≤i1 <i2 <...<ik ≤n+1
(10.55)
We observe that for systems in the entire space, the probabilistic Kirkwood
closure (10.41) agrees with the maximum entropy closure (10.55) for all orders
n.
10.6.2
Closure at the highest level n = N − 1
Although the probabilistic Kirkwood closure and the maximum entropy closure agree in the case of systems in the entire space, they differ in the case
of confined systems with a finite number of particles N . It appears that the
maximum entropy closure (10.45) is exact in a confined system when applied
at the highest level n = N − 1, whereas the probabilistic Kirkwood closure
(10.33) is not exact. In other words, the maximum entropy closure relation
yields the Boltzmann distribution (10.24). What appeared at first to be an
approximation, turns out to be the exact result. Indeed, setting
1/N
1
×
(10.56)
γ(x1 , x2 , . . . , xN −1 ) =
ZN
( "
#
)
N
−1
X
X
1
1
exp −
U (xi , xj ) +
Uex (xj ) /kB T ,
N − 2 1≤i<j≤N −1
N − 1 j=1
we see that clearly,
N
Y
j=1
γ(x1 , . . . , xj−1 , xj , . . . , xN ) =
1 −U (x1 ,...,xN )/kB T
e
,
ZN
(10.57)
which is the Boltzmann distribution. The Boltzmann distribution obviously
satisfies the BBGKY equation (10.47). Therefore, we have found a solution to
BBGKY equation which satisfies the closure relation (10.45). This solution
coincides with the Boltzmann distribution, and so we conclude that it is the
exact solution. For the N − 1-particle pdf pN −1 we have
Z
1
pN −1 (x1 , . . . , xN −1 ) =
e−U (x1 ,...,xN )/kB T dxN ,
(10.58)
ZN Ω
206 Maximum Entropy Formulation of the Kirkwood Superposition Approximation
which is both the exact and “approximate” result.
The probabilistic Kirkwood approximation (10.33) gives however a different result. If it were to be exact, then the resulting N − 1-particle pdf
would have been given by equation (10.58). Therefore, all lower level pdf’s
pn (n ≤ N − 1) would be (multiple) integrals of the Boltzmann distribution.
Therefore, by the closure (10.33), pN should be a product of integrals of the
Boltzmann distribution, which is a contradiction to the known form of the
Boltzmann distribution (10.24).
Although the maximum entropy closure is exact at the highest order
n = N −1, it is not exact at lower orders. The observation that the maximum
entropy closure is exact at the highest level, while the probabilistic Kirkwood
closure is not, may indicate that the maximum entropy closure may fit the
experimental data better than the probabilistic Kirkwood closure does in
confined systems, even when used at lower levels (n = 2, 3).
10.7
Confined Systems
Systems in bounded domains are particularly important, because only bounded
domains can include the spatially nonuniform boundary conditions needed
to describe devices, with spatially distinct inputs, outputs, and (sometimes)
power supplies. A large fraction of electrochemistry involves such devices
as batteries or concentration cells. A large fraction of molecular biology involves such devices as transport proteins that move ions across otherwise
impermeable membranes [45, 46].
In the general case, where Ω ⊂ R3 is a bounded domain, there is no known
analytic solution to the system (10.9). We propose to solve this system by
the following iterative scheme:
1. Initial guess γ (0) (x1 , x2 ). Set i = 0.
(i)
2. Solve the non-homogeneous linear equation for p2 (x1 , x2 )
(i)
(i)
(i)
kB T ∇x1 p2 (x1 , x2 ) − f (x2 , x1 )p2 (x1 , x2 ) − f ex (x1 )p2 (x1 , x2 )
Z
= (N − 2)
Ω
f (x3 , x1 )γ (i) (x1 , x2 )γ (i) (x2 , x3 )γ (i) (x1 , x3 ) dx3 .
10.7 Confined Systems
207
3. Solve the non-linear system for γ (i+1)
Z
(i)
(i+1)
p2 (x1 , x2 ) = γ
(x1 , x2 ) γ (i+1) (x2 , x3 )γ (i+1) (x1 , x3 ) dx3 .
Ω
(10.59)
4. i ← i + 1. Return to step 2, until convergence is achieved.
The analysis of the preceding section indicates that a good initial guess might
be
p
γ (0) (x1 , x2 ) = p1 (x1 )p1 (x2 ) g2 (x1 , x2 ),
(10.60)
where g2 (x1 , x2 ) is the solution to the BBGKY equation with the Kirkwood
SA in the entire space R3 . This solution can be found rather easily using the
inherited symmetries of the problem. For example, it is well known that if
f ex = 0, then g2 (x1 , x2 ) = g2 (|x1 − x2 |), and the problem for g2 becomes
one dimensional.
Step 2 requires the solution of a linear partial differential equation in a
bounded region. This equation can be written in a gradient form as
∇x1 e(U (x1 ,x2 )+Uex (x1 ))/kB T p2 (x1 , x2 ) =
N − 2 (U (x1 ,x2 )+Uex (x1 ))/kB T
e
−
kB T
Z
∇x1 U (x1 , x3 )p3 (x1 , x2 , x3 ) dx3 .
Ω
The identity ∇x1 × ∇x1 u(x1 ) = 0, for all u, imposes a solvability condition
for γ. Indeed, taking the curl of the last equation, together with the closure
(10.7) results in
0 = ∇x1 ×
Z
(U (x1 ,x2 )+Uex (x1 ))/kB T
e
∇x1 U (x1 , x3 )γ(x1 , x2 )γ(x1 , x3 )γ(x2 , x3 ) dx3 .
Ω
In step 3 we solve a non-linear integral equation. We suggest solving
the non-linear equation (10.8) by a Newton-Raphson iterative scheme. Let
γ (n) (x1 , x2 ) be the n-th iteration. Define the operator Γ(n) : Ω2 → Ω2 as
follows
Z
(n)
Γ u(x, z) =
γ (n) (x, y)u(z, y) dy.
(10.61)
Ω
208 Maximum Entropy Formulation of the Kirkwood Superposition Approximation
Let the operator S : Ω2 → Ω2 be the symmetrization operator
Su(x, y) = u(y, x).
(10.62)
The Newton-Raphson iteration scheme suggests
γ (n+1) = γ (n) + ∆(x1 , x2 ),
(10.63)
where ∆(x1 , x2 ) satisfies the linear integral equation
(n)
p2 (x1 , x2 ) −
p (x1 , x2 )
= 2(n)
∆(x1 , x2 )
γ (x1 , x2 )
(n)
p2 (x1 , x2 )
+γ (n) (x1 , x2 )Γ(n) ∆(x1 , x2 )
+γ (n) (x1 , x2 )SΓ(n) ∆(x1 , x2 ),
where
(n)
p2 (x1 , x2 )
=γ
(n)
Z
(x1 , x2 )
γ (n) (x1 , x3 )γ (n) (x2 , x3 ) dx3 .
(10.64)
Ω
We may write the iteration equivalently as
(n)
γ
(n+1)
=γ
(n)
+
p2
+ γ (n) Γ(n) + γ (n) SΓ(n)
γ (n)
!−1
p2 −
(n)
p2
.
(10.65)
The steps in the algorithm are iterated until convergence is achieved.
We have yet to test our generalized Kirkwood closure in practice. The
resulting pair correlation function should be compared with MD or MC simulations of particles in a confined region. The observation of Subsection 10.6.2
and the generality of the maximum entropy principle (minimum Helmholtz
free energy) may indicate that it will outperform the regular Kirkwood SA
in bounded domains. The difference between the results of the two closure
methods should be seen near the boundary walls of the domain.
10.8
Mixtures
The maximum entropy principle can also be used to find closure relations
of mixtures, both in confined domains and in the entire space. Suppose a
10.8 Mixtures
209
mixture ofPS ≥ 2 species, with Nα (α = 1, 2, . . . , S) particles of each species.
Let N = Sα=1 Nα be the total number of particles of all specie. There are
S 2 2-particle pdfs,
pαβ
2 (x1 , x2 ) α, β ∈ {1, 2, . . . , S},
βα
that exhibit the symmetry pαβ
2 (x1 , x2 ) = p2 (x2 , x1 ). In this section we
briefly discuss how to find the closure relation in the mixture problem.
In the maximum entropy approach, one is searching for S 3 3-particle pdfs
αβγ
p3 (x1 , x2 , x3 ), α, β, γ ∈ {1, 2, . . . , S}, that bring the entropy
Z
S
X
Nα Nβ Nγ
αβγ
pαβγ
H=−
3 (x1 , x2 , x3 ) ln p3 (x1 , x2 , x3 ) dx1 dx2 dx3
N N N Ω
α,β,γ=0
(10.66)
3
to maximum, with the 3S marginal constraints
Z
αβ
pαβγ
p2 (x1 , x2 ) =
3 (x1 , x2 , x3 ) dx3 ,
ZΩ
pαγ
pαβγ
2 (x1 , x3 ) =
3 (x1 , x2 , x3 ) dx2 ,
Ω
Z
βγ
pαβγ
p2 (x2 , x3 ) =
3 (x1 , x2 , x3 ) dx1 .
(10.67)
Ω
This variational problem is solved using the Euler-Lagrange formulation similar to the derivation done in Section 10.2.
In the case of a system in the entire space, the methods of subsections
10.3.2 and 10.6.1 show that the mixture entropy closure coincides with the
probabilistic Kirkwood closure. Both subsections suggest that the triplets
correlation functions are related to the pair correlation function through
g3αβγ (x1 , x2 , x3 ) = g2αβ (x1 , x2 )g2αγ (x1 , x3 )g2βγ (x2 , x3 ),
(10.68)
for α, β, γ ∈ {1, 2, . . . , S}. Closures of higher orders can be obtained in a
similar manner.
In confined systems, the Euler-Lagrange formulation leads to integral
equations of the form (10.8). Note that since the entropy (10.66) depends
Nα
, we expect the resulting confined system pair
on the particle fraction
N
correlation will also depend on the particle fraction.
210 Maximum Entropy Formulation of the Kirkwood Superposition Approximation
10.9
Discussion and Summary
We have used the maximum entropy principle to derive a closure relation for
the BBGKY hierarchy. It is possible to consider functionals over distributions
other than the entropy that can yield different (known) closures. In fact, a
somewhat similar approach is used in density functional theory (DFT). In
the DFT problem setup, a functional of the pair correlation function (e.g.,
the Helmholtz free energy) is maximized. The function that brings the given
functional to its maximal value is the resulting pair correlation function.
Using this method, one can recover some of the Ornstein-Zernike integral
closures that relate the direct and indirect correlation functions, such as the
PY closure, for instance. Our approach differs from that of the DFT in
that we optimize over relations between the probability correlation (density)
functions of successive orders, rather than over relations between the direct
and indirect correlation functions.
In this chapter we have used the maximum entropy principle to derive
a closure relation for the BBGKY hierarchy. This approach to the closure
problem appears to be new. We proved that for systems in the entire space,
the maximum entropy closure relation coincides with the probabilistic Kirkwood SA for all orders of the hierarchy. In finite systems the maximum
entropy closure differs from Kirkwood’s SA. In particular, when applied to
the highest level of the hierarchy, the maximum entropy closure is exact,
whereas the probabilistic Kirkwood approximation is not. Besides the advantage of generality, the maximum entropy closures are expected to perform
better than the Kirkwood SA, even in low order approximations. We expect
the differences between the pair correlation functions predicted by the two
methods to be significant especially near the domain boundaries. The maximum entropy closure may be applicable to non-equilibrium systems, and
in particular to systems with spatially inhomogeneous boundary conditions.
That implementation of the maximum entropy closure will be the subject of
a separate paper.
Chapter 11
Attenuation of the Electric
Potential and Field in
Disordered Systems
The contents of this chapter were published in [176]
We study the electric potential and field produced by disordered distributions of charge to see why clumps of charge do not produce large potentials
or fields. The question is answered by evaluating the probability distribution of the electric potential and field in a totally disordered system that is
overall electroneutral. An infinite system of point charges is called totally
disordered if the locations of the points and the values of the charges are
random. It is called electroneutral if the mean charge is zero. In one dimension, we show that the electric field is always small, of the order of the field
of a single charge, and the spatial variations in potential are what can be
produced by a single charge. In two and three dimensions, the electric field
in similarly disordered electroneutral systems is usually small, with small
variations. Interestingly, in two and three dimensional systems, the electric
potential is usually very large, even though the electric field is not: large
amounts of energy are needed to put together a typical disordered configuration of charges in two and three dimensions, but not in one dimension. If
the system is locally electroneutral—as well as globally electroneutral—the
potential is usually small in all dimensions. The properties considered here
arise from the superposition of electric fields of quasi-static distributions of
charge, as in nonmetallic solids or ionic solutions. These properties are found
212
Attenuation of the Electric Potential and Field in Disordered Systems
in distributions of charge far from equilibrium.
11.1
Introduction
There is no danger of electric shock when handling a powder of salt or when
dipping a finger in a salt solution, although these systems have huge numbers
of positive and negative charges. It seems intuitively obvious that the alternating arrangement of charge in crystalline Na+ Cl− should produce electric
fields that add almost to zero; it also seems obvious that Na+ and Cl− ions
will move in solution to minimize their equilibrium free energy and produce
small electrical potentials. But what about random arrangements of charge
that occur in a random quasi-static arrangement of charge such as a snapshot
of the location of ions in a solution? Tiny imbalances in charge distribution
produce large potentials, so why doesn’t a random distribution of charge
produce large potentials, particularly if the distribution is not at thermodynamic equilibrium? Indeed, some arrangements of charge produce arbitrarily
large potentials, but as we shall see, these distributions occur rarely enough
that the mean and variance of stochastic distributions are usually finite and
small. More specifically, we determine the conditions under which stochastic
distributions of fixed charge produce small fields.
The quasi-static arrangements of charge can represent the fixed charge
in amorphous non-metallic solids or snapshots of charge arrangement of ions
in solution, due to their random (Brownian) motion. Our analysis does not
apply to quantum systems [17], and in particular it fails if electrons move
in delocalized orbitals, as in metals. Note that the random arrangements of
charge considered here do not necessarily minimize free energy.
We consider the field and potential in overall electroneutral random configurations of infinitely many point charges. An infinite system of point
charges is called totally disordered if the locations of the points and the
charges are random, and it is called overall electroneutral if the mean charge
is zero. The configurations of charge may be static or quasi-static, that
is, time dependent, but varying sufficiently slowly to avoid electromagnetic
phenomena: the electric potential is described by Coulomb’s law alone. In
one dimensional systems of this type, the potential is usually finite—even
though the system usually contains an infinite number of positive and negative charges. Even if the system is disordered and spatially random, charges
of the same sign do not clump together often enough to produce large fields
11.1 Introduction
213
or potentials, in one dimensional systems.
Our approach is stochastic. We ask how disordered can a random electroneutral system be, yet still have a small field or potential. We find the
answer by evaluating the probability distribution of the electric potential and
field of a disordered system of charges. We find that the electric field in a totally disordered one dimensional system is small whether the system is locally
electroneutral or not. The potential behaves differently; it can be arbitrarily
large in a one dimensional system, but it is usually small in electroneutral
systems.
In two or three dimensional disordered systems, the electric field is not
necessarily small. We show that in such systems that are also electroneutral
the field is usually small. The potential, however, is usually large, even if the
system is electroneutral. Both potential and field are small, if the system is
locally—as well as globally—electroneutral (see definition below) in one, two
and three dimensions.
We consider several types of random arrays of charges: (a) A lattice with
random distances between two nearest charges; (b) A lattice (of random
or periodic structure) with a random distribution of positive and negative
charges (charge ±1). Charges in the lattice need not alternate between positive and negative, nor need they be periodically distributed; (c) A lattice
(of random or periodic structure) with random charge strengths. Not all
charges are ±1, but they are chosen from a set q1 , q2 , . . . , qn with probabilities p1 , p2 , . . . , pn , respectively, such that
n
X
qi pi = 0.
(11.1)
i=1
Equation (11.1) is our definition of electroneutrality in an infinite system.
We use renewal theory [97], perturbation theory [10], and saddle point approximation [90] to calculate the electric potential of one dimensional systems
of charges and show that it is usually small. That is to say, the probability
is small that the potential takes on large values. Thus, randomly distributed
particles produce small potentials even in disordered systems in one dimension, if the system is electroneutral. The analysis of one dimensional systems
requires the calculation of the probability density function (pdf) of weighted
independent identically distributed (i.i.d.) sums of random variables. This
pdf looks like the normal distribution near its center, but the tail distribution has the double exponential decay of the log-Weibull distribution [156].
214
Attenuation of the Electric Potential and Field in Disordered Systems
We conclude that the electric potential of totally disordered electroneutral
one dimensional systems is necessarily small, comparable to that of a single
charge.
Later in the chapter, we define local electroneutrality precisely and show
that two and three dimensional systems with local electroneutrality usually
have small potentials, because the potential of a locally neutral system of
charges decays like the potential of a point dipole, as 1/r2 . We show that
the potential of typical totally disordered arrays of charges in two and three
dimensions is infinite even if the system is electroneutral.
Historically, little attention seems to have been paid to quasi-static random arrangements of charge, although much attention has been paid to the
equilibrium arrangements of mobile charge. In systems of mobile charges,
such as liquids and ionic solutions, the decay of the electric potential may
even be exponential, after the mobile charges assume their equilibrium distribution. The early theory of Debye-Hückel [7] shows a nearly exponential
decay (with distance from a given particle) of the average electric potential
at equilibrium, originally found by solving the linearized Poisson-Boltzmann
equation. In classical physics, perfect screening of multipoles (of all orders)
occurs in both homogeneous and inhomogeneous systems at equilibrium in
the thermodynamic limit, when boundary conditions at infinity are chosen
to have no effect [129] and there is no flux of any species. This type of
screening in electrolytic solutions is produced by the equilibrium configuration of the mobile charges [72, 125], which typically takes 100 psec to establish
(compared to the 10−16 time scale of most atomic motions) [8]. Many other
systems are screened by mobile charges after they assume their equilibrium
configuration of lowest free energy [23], such as ionic solutions, metals and
semiconductors.
The spatial decay of potential in ionic solutions determines many of the
properties of ionic solutions and is a striking example of screening or shielding. “Sum rules” of statistical mechanics [72, 125] describe these properties. These rules depend on the system assuming an equilibrium distribution,
which can only happen if the charges are mobile.
We consider finite and infinite systems of charges which may or may not
be mobile and which are not necessarily at equilibrium. We show that the
potential of a finite disordered locally electroneutral system is attenuated to
the potential of a single typical charge, whether the potential is evaluated
inside or outside a finite system or in an infinite system. We note that
the behavior of the electric potential and field outside the line or plane of
11.2 A One-Dimensional Ionic Lattice
the lattice can be analyzed in a straightforward manner by the methods
developed below.
11.2
A One-Dimensional Ionic Lattice
Consider a semi-infinite array of alternating electric charges ±q with a distance d between neighboring charges. The electric potential Φ at a point P ,
located at a distance R from and to the left of the first charge (see Fig. 11.1)
is given by
1
1
1
1
q
−
+
−
+ ···
Φ =
4πε0 R R + d R + 2d R + 3d
q
1
1
1
=
+
−
+ ···
1−
4πε0 R
1 + a 1 + 2a 1 + 3a
∞
q X (−1)n
,
(11.2)
=
4πε0 R n=0 1 + na
where a = d/R is a dimensionless parameter. The series (11.2) is conditionally convergent, so it can be summed to any value by changing the order of
summation [108]. The order of summation reflects the order of construction
of the system; different orders may lead to different potential energies of the
system. However, the infinite series that determines the electric field
∞
X
q
(−1)n
E=
4πε0 R2 n=0 (1 + na)2
is absolutely convergent, so the field does not depend on the order of summation of its defining series. Thus, all potentials differ from each other by
a constant, which presumably reflects the different ways the charge distribution could be constructed, while having the same electric field. From here
on, we consider the ordering in equation (11.2).
Setting R = d (a = 1) we find the potential at a vacant lattice point
(to avoid infinite potentials) due to charges located at both directions of the
infinite lattice is
∞
2Φ(R = d) = 2
q X (−1)n−1
q
=
· 2 log 2.
4πε0 d n=1
n
4πε0 d
215
216
Attenuation of the Electric Potential and Field in Disordered Systems
Figure 11.1: A semi infinite lattice of alternating charges with a distance d
between neighboring charges. The point P is located at a distance R from
and to the left of the first charge.
The constant 2 log 2 is known as the Madelung constant of a one dimensional
lattice [101].
Next we find the asymptotic behavior of the potential Φ away from the
semi-infinite lattice, that is for R d, or equivalently a 1. The following
analysis is independent of the order of summation of the series (11.2). Clearly,
the infinite sum in eq. (11.2) converges, because it is an alternating sum with
a decaying general term. We expand the potential for a 1 (away from the
lattice) in the asymptotic form
Φ=
q 1
V0 + aV1 + a2 V2 + · · · .
4πε0 R
(11.3)
The effect of the first charge can be separated from all the others,
Φ=
q 1
q
1
−
V0 + ãV1 + ã2 V2 + · · · ,
4πε0 R 4πε0 R + d
(11.4)
where
a
d
=
.
R+d
1+a
Comparing eqs.(11.3) and (11.4) we obtain
"
#
2
1
a
a
V0 + aV1 + a2 V2 + · · · = 1 −
V0 +
V1 +
V2 + . . . .
1+a
1+a
1+a
ã =
The coefficients V0 , V1 , . . . are found by equating the coefficients of like powers
of a. In particular, we find that V0 = 1/2, V1 = 1/4, V2 = 0, so the potential
11.2 A One-Dimensional Ionic Lattice
217
has the asymptotic form
q 1 1 1
3
.
+ a+O a
Φ=
4πε0 R 2 4
(11.5)
All coefficients Vn can easily be computed in a similar fashion. This result
also determines the rate at which the potential far away reaches its limiting
1 q
value,
. The divergent series for x = 1 has the value V0 = 12 if
2 4πε0 R
interpreted as a limit using the Abel sum [108]
1 − 1 + 1 − 1 + 1 − 1 + . . . = lim−
x→1
∞
X
(−1)n xn = lim−
n=0
x→1
1
1
= .
1+x
2
We note that the asymptotic expansion (11.5) can also be found directly from
the differential equation that the sum
∞
X
(−1)n n
y(x) =
x
1 + na
n=0
satisfies [131]
axy 0 + y =
1
,
1+x
(11.6)
with initial condition y(0) = 1. The asymptotic form of y(x) can easily be
found by standard methods [10]. In particular,
lim y(x) =
x→1−
∞
X
(−1)n
.
1
+
na
n=0
The physical interpretation of the asymptotic expansion (11.5) is that
the electric potential away from an infinite lattice of charged particles is
about the same as if half a single charge were located at the origin. The
spatial arrangement of the lattice attenuates the effect of its charge. The
potential near the lattice is determined by a few of the nearest charges and
the contribution of the remaining charges reduces to that of a half charge
placed at a distance R d. Obviously, as R → 0 the potential becomes
infinite, approaching the potential produced by just the nearest charge.
218
Attenuation of the Electric Potential and Field in Disordered Systems
11.3
One Dimensional Random Ionic Lattice
We turn now to solids in which the charges are distributed randomly in several different ways. First, consider a semi-infinite lattice of electric charges, in
which the sign of each charge is determined randomly by a flip of a fair coin.
That is, the charges that are located at the lattice points Xn (n = 0, 1, 2, . . .)
are independent Bernoulli random variables that take the values ±1 with
probability 1/2. The electric potential of this random lattice is given by
∞
q X Xn
.
Φ=
4πε0 R n=0 1 + na
(11.7)
Some discussion of the nature of convergence of the series (11.7) is needed at
this point. The convergence of the sum of variances means that the partial
sums converge in L2 with respect to the probability measure, so the sum
(11.7) exists as a random variable Φ ∈ L2 , whose variance is the sum of
the variances. Now, the Cauchy-Schwarz inequality implies that Φ ∈ L1 , so
hΦi = 0. Note that (11.7) also converges with probability 1 [15].
We use fair coin tossing to maintain the condition of global electroneutrality, though arbitrary long runs of positive or negative charges occur in this
distribution. Thus some realizations of the sequence Xn have runs (‘clumps’)
of substantial net charge and potential. The standard deviation of the net
charge in a region gives some feel for the size of the clumps. The √
standard
deviation in the net charge of a region containing N charges is q N . For
large values of N , substantial regions are not charge neutral. The condition of
local charge neutrality (defined later) is violated for many of the realizations
of charge in this distribution.
Note that a particular set of Xn can produce an infinite potential, despite
our general conclusions. If, for example, Xn = 1 for all n, the electric poten∞
X
1
= ∞. Nonetheless,
tial becomes infinite (see eq. (11.7)), because
1 + na
n=0
the L2 convergence of (11.7) implies that the probability that (11.7) is infinite is 0. In other words, even though the potential is infinite for a particular
set of Xn , the potential is finite with probability 1. This is a striking example of the attenuation of the electric field, even without mobile charge.
The attenuation of the potential produced by some ‘clumpy’ configurations
of charges occurs even though there is no correlation in position, and there
is no motion whatsoever.
11.3 One Dimensional Random Ionic Lattice
219
The electric field, given by
∞
X
q
Xn
E=−
,
2
4πε0 R n=0 (1 + na)2
remains finite for all realizations of Xn , because the sum
∞
X
1
q
S=
2
4πε0 R n=0 (1 + na)2
converges.
The electric field is bounded (above and below) by S and so there is
zero probability that the function is outside the interval (−S, S). The pdf of
the electric field is compactly supported, even when all charges are positive
(or negative). The electric field—unlike the potential—is attenuated even if
the net charge of the system is not zero, taken as a whole. The standard
(∞
)1/2
X
1
q
, which is of the order of
deviation of the field is
4πε0 R2 n=0 (1 + na)4
the field of a single charge at a distance R.
11.3.1
Moments
The expected value of Φ is hΦi = 0, as mentioned above. The variance of Φ
is given by
2 X
∞
q
1
V ar (Φ) =
.
(11.8)
4πε0 R n=0 (1 + na)2
A vacant lattice point in an infinite (not semi-infinite) lattice corresponds to
R = d for both the charges to the right and to the left. It follows that the
variance of the potential there is twice that given in (11.8) with a = 1, that
is,
2 X
2 2
∞
q
1
q
π
V ar (Φ) = 2
=2
,
(11.9)
2
4πε0 d n=1 n
4πε0 d
6
so that the standard deviation is
σΦ =
q
π
√ .
4πε0 d 3
(11.10)
220
Attenuation of the Electric Potential and Field in Disordered Systems
√
As expected, the constant π/ 3 is larger than the Madelung constant 2 log 2
of the periodic lattice, because the potential of the disordered system is larger
than that of the ordered one.
Away from the semi infinite lattice, i.e., for a 1, we can approximate
the variance (11.8) by the Euler-Maclaurin formula, which replaces the sum
by an integral,
2 Z ∞
1
1
q
dx + + O(a)
V ar (Φ) =
4πε0 R
(1 + ax)2
2
0
2 q
1 1
=
+ + O(a) ,
(11.11)
4πε0 R
a 2
so the standard deviation is
σφ |R =
q
√
4πε0 dR
(1 + O(a)) .
(11.12)
1
1
of a single
The decay law of √ is more gradual than the decay law
R
R
charge.
11.3.2
The electrical potential as a weighted i.i.d. sum
P
The potential (11.7) is a weighted sum of the form
an Xn , where Xn are
i.i.d. random variables. The distribution of P
potential is generally not normal.
∞
−n
Xn , where Xn are the
For example, consider the weighted sum
n=1 2
same Bernoulli random variables. This weighted sum represents the uniform
distribution in the interval [−1, 1]. It is, in fact equivalent to the binary
representation of real numbers in the interval. Not only does this distribution
not look like the Gaussian distribution for small deviations, it does not look
at all Gaussian for large deviations. In fact, this distribution has compact
support. It is zero outside a finite interval, without the tails of the better
endowed Gaussian. Other unusual limit distributions can be easily
P obtained
−n
from sums of the form (11.7). For example, the weighted sum ∞
Xn
n=1 3
is equivalent to the uniform distribution on the Cantor “middle thirds” set
[195] in [−1, 1], whose Lebesgue measure (length) is 0.
Note that the sum
∞
X
Xn
(1 + na)1+ε
n=0
11.3 One Dimensional Random Ionic Lattice
221
has compact support for every ε > 0, because the series
∞
X
n=0
1
(1 + na)1+ε
converges for every ε > 0. In our case ε = 0, so that the limit distribution
does not necessarily have compact support. Nonetheless, we expect that the
probability distribution function of the potential will have tails that decay
steeply, even steeper than those of the normal distribution.
11.3.3
Large and small potentials. The saddle point
approximation
The existence of the first moment of the sum (11.7) depends on its tail
distribution, which we calculate below by the saddle point method [90]. That
is, we calculate the chance of finding a pinch of (noncrystalline) salt with a
very large potential. For a potential Φ defined in equation (11.7), we denote
−1
q
Φ by f (x). The Fourier transform fˆ(k) of this pdf is
the pdf of
4πε0 R
given by the infinite product
fˆ(k) =
∞
Y
cos
n=0
k
1 + na
,
(11.13)
which is an entire function in the complex plane, because the general term
is 1 + O(n−2 ). The inverse Fourier transform recovers the pdf
Z ∞
1
fˆ(k)eikx dk,
(11.14)
f (x) =
2π −∞
which we want to evaluate asymptotically for large x. Setting
g(k, x) =
∞
X
log cos
n=0
we write
1
f (x) =
2π
Z
k
1 + na
+ ikx,
(11.15)
∞
exp{g(k, x)} dk.
−∞
(11.16)
222
Attenuation of the Electric Potential and Field in Disordered Systems
d
g(k, x) = 0. Differentiating
dk
equation (11.15) with respect to k, we find that
k
∞ tan
X
d
1 + na
g(k, x) = −
+ ix.
(11.17)
dk
1 + na
n=0
The saddle point is the point k for which
We look for a root of the derivative on the imaginary axis, and substitute
k = is. The vanishing derivative condition of the saddle point method is
then
s
∞ tanh
X
1 + na
.
(11.18)
x=
1 + na
n=0
The infinite sum on the right hand side represents a monotone increasing
function of s in the interval 0 < s < ∞, so equation (11.18) has exactly one
solution for every x. Near the saddle point k = is, we approximate g(k) by
its Taylor expansion up to the order
g(k) ≈ g(is) +
1 d2
g(is)(k − is)2 ,
2 dk 2
(11.19)
to find the leading order term of the full asymptotic expansion (derivatives
of higher order of the Taylor expansion can be used to find all terms of the
asymptotic expansion [21]). We use the Cauchy integral formula to calculate
our Fourier integral (11.16) on the line parallel to the real k axis through
k = is (see Fig. 11.2)
Z
1 00
1 g(is) ∞
2
e
exp
g (is)(k − is) dk
f (x) ≈
2π
2
−∞
Z
1 g(is) ∞
z2
eg(is)
00
=
e
exp g (is)
dz = p
. (11.20)
2π
2
−2πg 00 (is)
−∞
Equation (11.18) has no analytic solution, so we construct asymptotic approximations for large and small values of s separately.
11.3.4
Tail asymptotics
Throughout this subsection we assume that a is small and s is large and we
find the tail asymptotics of the pdf away from the system (for a 1). For
11.3 One Dimensional Random Ionic Lattice
Figure 11.2: The integration contour passes through the saddle point k = is
in the complex plane.
223
224
Attenuation of the Electric Potential and Field in Disordered Systems
s 1 the Euler-Maclaurin sum formula gives
s
Z ∞ tanh
1
1 + ax
x=
dx + tanh s + O(a).
(11.21)
1 + ax
2
0
s
, we obtain
Substituting z =
1 + ax
Z
1 s tanh z
1
x=
dz + tanh(s) + O(a).
(11.22)
a 0
z
2
Writing
Z s
Z 1
Z s
Z s
tanh z
tanh z
tanh z − 1
dz
dz =
dz +
dz +
z
z
z
0
0
1
1 z
Z 1
Z ∞
tanh z
tanh z − 1
= log s +
dz +
dz + O(e−2s ),
z
z
0
1
we obtain (11.22) in the form
ax = log s + C +
a
+ O(a2 , e−2s ),
2
where the constant C is given by
Z 1
Z ∞
tanh z
tanh z − 1
C=
dz +
dz.
z
z
0
1
(11.23)
(11.24)
Exponentiation of equation (11.23) gives the location of the saddle point
asymptotically for small a and large s as
s = eax−C−a/2+O(a
2 ,e−2s )
.
(11.25)
The saddle point approximation (11.20) requires the evaluation of g and its
second derivative at k = is. The Euler-Maclaurin sum formula gives
∞
X
s
− sx
g(is) =
log cosh
1 + na
n=0
Z
s s log cosh z
1
=
dz + log cosh s − sx + O(as)
2
a 0
z
2
Z 1
Z s
Z ∞
s
log cosh z
dz
log cosh z − z
1
=
dz +
dz + O
+
2
2
a
z
z
s
0
1 z
1
s log 2
+ −
− sx + O(as).
2
2
11.3 One Dimensional Random Ionic Lattice
225
Using equations (11.23) and (11.24), we find
s log 2
1
g(is) = C1 −
+ O a, , as ,
a
2
a
where
Z
C1 =
(11.26)
Z ∞
Z ∞
Z 1
log cosh z
tanh z − 1
log cosh z − z
tanh z
dz−
dz,
dz+
dz−
2
2
z
z
z
z
0
1
1
0
(11.27)
and integration by parts shows that C1 = −1. It follows that
s log 2
1
g(is) = − −
+ O a, , as .
(11.28)
a
2
a
1
The second derivative of g is evaluated in a similar fashion
s
2
∞ 1 − tanh
X
d2
1 + na
g(k)
= −
2
dk
(1 + na)2
k=is
n=0
Z s
1
1
= −
1 − tanh2 z dz −
1 − tanh2 s + O(ase−2s )
as 0
2
= −
tanh s
+ O(ase−2s , e−2s )
as
1
1
+ O(as, 1, )e−2s .
(11.29)
as
as
Substitution of (11.28), (11.29), and (11.25) into the saddle point approximation (11.20) gives
√
1 ax−C−a/2
a 1
,
(11.30)
f (x) ≈ √ e 2 (ax−C−a/2) e− a e
2 π
= −
where the constant C = .8187801402 · · · is given by equation (11.24). Therefore, the small a and large s approximation to the tail of the pdf of Φ is given
by
√
4πε0 R a
√ ×
fΦ (x) ∼
(11.31)
q 2 π
1 4πε0 d
1
4πε0 d
exp
x − C − a/2 − exp
x − C − a/2
, x → ∞.
2
q
a
q
226
Attenuation of the Electric Potential and Field in Disordered Systems
It follows from equation (11.31) that the pdf decays to zero as a double
exponential as x → ∞, which implies that all moments exist. This decay is
similar to the extreme value or the log-Weibull (Gumbel) distributions [156].
The compact support of the distributions of convergent series is replaced here
with a steep decay. Note also that the decay becomes steeper further away
from the system, as expected, because the pre-exponential factor of the inner
exponent is 1/a = R/d.
For small x the pdf can be approximated by a zero mean Gaussian with
variance V ar (Φ), which for small a is
(
r
2 )
Rd 4πε0 x
4πε0 Rd
exp −
, x → 0.
(11.32)
fΦ (x) ∼
q
2π
2
q
Near its center, the distribution
looks like a Gaussian with a standard devia√
tion that decays like 1/ R, in agreement with equation (11.12). We conclude
that the pdf looks normal near its center, but, far away from there, it decays
to zero much more steeply, rather like a cutoff. This conclusion is the answer
to the question posed in subsection 11.3.2 about the normality of weighted
sums of i.i.d. random variables. The non Gaussian tails of the distribution
are characteristic of large deviations [90].
11.4
Random Distances
Consider a one-dimensional system of alternating charges without the restriction of equal distance between successive charges. In particular, we assume
a renewal model, in which the distances between two neighboring charges
are non-negative i.i.d random variables with pdf f (l) and finite expectation
value
Z ∞
d=
lf (l) dl < ∞.
0
The potential of this random system is also a random variable.
We show below that away from the system the mean value of the potential
V̄ has the asymptotic form
1
q
+ O(a) ,
(11.33)
V̄ =
4πε0 R 2
where a = d/R. Equation (11.33) defines the attenuation produced by the
configuration of charges. The mean potential of the system is produced by
11.5 Dimensions Higher than One
227
(in effect) half a charge. We note that the value 1/2 is exactly the same for
both random and non-random systems of alternating charges (eq.(11.5)). We
first note that
Z ∞
f (l) Pr {V (R + l) = V ∗ − V } dl,
(11.34)
Pr{V (R) = V } =
0
q
where V ∗ =
. To find the mean value, we multiply (11.34) by V and
4πε0 R
integrate (note that 0 ≤ V ≤ V ∗ ), and then change the order of integration
Z ∞
Z V∗
V dV
f (l) Pr {V (R + l) = V ∗ − V } dl
V̄ (R) =
0
0
∞
Z
=
Z
f (l) dl
0
∗
V Pr {V (R + l) = V ∗ − V } dV
0
Z
∞
= V −
Z
Z
= V −
V∗
V Pr{V (R + l) = V } dV
f (l) dl
0
0
∗
V∗
∞
f (l)V̄ (R + l) dl.
(11.35)
0
We look for an asymptotic expansion of the form
q
V̄ (R) =
V̄0 + aV̄1 + a2 V̄2 + . . . .
4πε0 R
(11.36)
Substituting this asymptotic expansion into (11.35) gives V̄0 = 1/2 for the
O(1) term, because
Z ∞
R
1−a≤
f (l)
dl ≤ 1.
(11.37)
R+l
0
1
≥ 1 − x. Hence (11.33)
The first inequality is due to the inequality
1+x
follows.
11.5
Dimensions Higher than One
11.5.1
The condition of global electroneutrality
In dimensions higher than one, global electroneutrality is enough to dramatically attenuate the electric field, but it is not enough to produce a small
potential, as shown below.
228
Attenuation of the Electric Potential and Field in Disordered Systems
Consider the electric potential at a vacant site of random charges located
at the points of a 2D square lattice
X
X
√ nm .
Φ=
(11.38)
2 + m2
n
(n,m)6=(0,0)
The variance of Φ is
X
V ar(Φ) =
(n,m)6=(0,0)
n2
1
= ∞.
+ m2
(11.39)
The infinite value of the variance means that arbitrarily large potentials can
occur with high probability. That is, the electric potential is not attenuated.
The divergence of the variance of the potential of three dimensional systems
is even steeper. Therefore, attenuation of the potential of totally disordered
systems can occur in two or three dimensional systems only if some correlation is introduced into the distribution of the locations of the charges. If, for
example, the signs of all charges alternate, as in a real Na+ Cl− crystal, the
distribution of potential will be dramatically different, and greatly attenuated, compared to a two or three dimensional system in which many charges
of one sign are clumped together.
The condition of global electroneutrality is enough to ensure the dramatic
attenuation of the electric field. Indeed, consider a 3D cubic lattice of random
charges. The z-component of the electric field at a vacant lattice point is
n
Xnml cos √
X
n2 + m2 + l 2
.
(11.40)
Ez =
2
n + m2 + l 2
(n,m,l)6=(0,0,0)
The variance of Ez is finite,
V ar(Ez ) =
X
(n,m,l)6=(0,0,0)
n
cos √
n2 + m2 + l 2
2
(n + m2 + l2 )2
2
< ∞,
because convergence is determined by the integral
Z π
Z ∞
1 2
2
r dr < ∞.
2π
cos θ sin θ dθ
r4
0
d
The large potential means that much work has to be done to create the
given spatial configuration of the charges, however, the resulting field remains
usually small.
11.5 Dimensions Higher than One
11.5.2
229
The condition of local electroneutrality
Here we show that the condition of local electroneutrality implies the attenuation of the potential in two and three dimensions. For example, the
potential of a two or three dimensional lattice of extended dipoles is finite
with probability 1, if the orientation of dipoles is distributed independently,
identically, and uniformly on the unit sphere (see Fig. 11.3). Paraphrasing
[88, p.136], we say that a (net) charge distribution ρ(x) has local charge neutrality if the (net) charge inside a sphere of radius R falls with increasing R
faster than any power, that is, for any x
Z
n
lim R
ρ(y) dy = 0 for all n > 0.
(11.41)
R→∞
|x−y |<R
On a lattice, the number of charges that are assigned to each lattice point
can be larger than in our example of dipoles (Fig. 11.3), thus forming multipoles. The Debye-Hückel distribution also satisfies the local charge neutrality
condition.
The potential of a single lattice point can then be written as an expansion
in spherical harmonics, if the charges of each multipole are contained in a
single lattice box. It can be also expanded, if the charge density of each
multipole decays sufficiently fast, as [88]
∞
l
1
Ylm (θ, φ)
1 X X
qlm
,
Φ0,0,0 (x) =
4πε0 l=0 m=−l 2l + 1
rl+1
(11.42)
where qlm are the multipole moments. In particular, the zeroth order multipole moment is
Z
1
q00 = √
ρ(y) dy = 0,
(11.43)
4π
by the condition of local electroneutrality (11.41): the far potential due to a
single lattice point decays as 1/r2 (or steeper). The coefficients qlm assigned
to each lattice point are randomized as in the previous sections so their
mean value vanishes, meaning that there is no preferred orientation in space.
(Compare the example of dipoles which do not have a preferred orientation.)
The mean value of the potential of the entire lattice is then hΦi = 0. The
variance is given by
X
V ar(Φ) =
V ar(Φijk ),
(11.44)
ijk
230
Attenuation of the Electric Potential and Field in Disordered Systems
Figure 11.3: Two dimensional lattice of dipoles of randomly chosen orientations produce attenuation due to the condition of local electroneutrality.
11.5 Dimensions Higher than One
where Φijk is the potential of the charge at lattice point (i, j, k). The potential decays as 1/r2 (or steeper); therefore the variance decays as 1/r4 =
1/(i2 + j 2 + k 2 )2 (or steeper). The convergence of the infinite sum (11.44) is
determined by the convergence of the integral
Z
Z ∞
1
1
4π
< ∞.
(11.45)
dV = 4π
dr =
4
2
r
d
r>d r
d
Thus, the variance of the potential is finite and we have shown that local
electroneutrality produces a dramatic attenuation of potential. As above,
the potential away from a charge is usually of the order of the potential of a
single charge.
11.5.3
The liquid state
Screening in the liquid state involves a least three phenomena. (1) The movement of charge to a distribution of minimal free energy. (2) The properties
of a static charge distribution with minimal free energy. (3) The properties
of any charge distribution.
If the charge correlation function ρ(x) minimizes free energy, and is at
equilibrium, as in ionic solutions, the far field potential is strongly screened.
However, the relaxation into such a state takes time, typically psec to nsec
in an ionic solution under biological conditions (see measurements reported
in [8], and theory summarized in [107]). As long as local charge neutrality
exists during the relaxation period, the potential changes from attenuated
(as described above) to exponentially screened, as equilibrium is reached.
In fact, the spread of potential in ionic solutions has the curious property
that it is much less shielded at short times than at long times; potentials
on the (sub) femtosecond time scale of atomic dynamics spread macroscopic
distances while potentials on long time scales spread only atomic distances.
Specifically, potentials on a time scale greater than nano or microseconds
spread a few Debye lengths, only a nanometer or so under biological conditions, although potentials on a femtosecond time scale can spread arbitrarily
far depending on the configuration of dielectrics at boundaries that govern the violations of local electroneutrality. To make this verbal analysis of
fast phenomena rigorous, the potentials and fields should be computed from
Maxwell’s equations, not Coulomb’s law.
Non-equilibrium fluctuations may violate local charge neutrality, therefore field fluctuations can be large. For example, in systems which are not lo-
231
232
Attenuation of the Electric Potential and Field in Disordered Systems
cally electroneutral, potential can spread a long way, as in the telegraph [57],
Kelvin’s transatlantic cable, or the axons of nerve cells [87]. In such systems,
d.c. potential spreads arbitrarily far—kilometers in telegraphs; thousands of
kilometers in the transatlantic cable; centimeters in a squid nerve filled with
salt water—even if an abundance of ions (≈ 1023 ) are present. Local electroneutrality is violated in such systems (at the insulating boundary which
separates the inside and outside of the cable, e.g., the cell membrane) and
that violation allows large far field potentials.
11.6
Summary and Discussion
Global electroneutrality ensures the dramatic attenuation of the electric potential and field of a one dimensional system of charges. Even if local electroneutrality is violated, and the local net charge is not zero, the potential
remains finite in these one dimensional systems, even in a random lattice that
includes arbitrarily long strings of equal charges. We have shown that the
distribution of the weighted sum of i.i.d. random variables that define the
one-dimensional electric potential is almost normal near its center, but has
very steep double exponentially decaying tails. The distances between neighboring charges can also be random, without changing the attenuation effect.
In higher dimensions, global electroneutrality is sufficient to dramatically
attenuate the electric field, but not the potential. However, local electroneutrality ensures a small potential in two and three dimensions, so the electric
potential and field is short range in one, two, and three dimensions, if the
systems are locally electroneutral.
Chapter 12
Boundary Conditions and a
Closure Relation for the Pair
Correlation Function in
Non-Equilibrium Diffusion
12.1
Infinite system in Steady-State
Consider the following physical setup (Fig.12.1). The space is divided by an
infinite long impermeable membrane located at z = 0. Particles of several
species can traverse the membrane only through a narrow channel. The
interaction between particles is described by short ranged forces, such as the
Lennard-Jones 6-12 interaction, whereas long ranged interactions are ignored.
Far away from the membrane the concentrations are maintained fixed: chL is
the concentration of the h species at z → −∞ and chR is the concentration
of the h species at z → ∞. The total number of particles in this system is
infinite.
This system is carrying net flux of particles due to the concentration gradient. In the case of non-interacting point particles the concentration profile
and flux were calculated explicitly in [98, 50]. This is a mixed boundary
value problem that is similar to the well known electrified disk problem of
electrostatics [88]. Mixed boundary value problems occur in many field of
applications (see [184] for different applications and methods of solving such
problems). For example, in Chapter 5 we calculated the time it takes a non
Boundary Conditions and a Closure Relation for the Pair Correlation Function in
234
Non-Equilibrium Diffusion
Figure 12.1: The physical setup: the space is divided by an infinite membrane. Particles can permeate only through the channel. The concentrations
to the left and to the right of the membrane are different.
12.1 Infinite system in Steady-State
interacting diffusing particle to get to the pore. The predicted flux density
1
is singular at the edges of the pore, where it blows up like p
, where
a2 − ρ 2
a is the radius of the pore, and ρ is the distance from the center of the
pore. Though the flux density is singular, the total flux through the pore is
finite, as this is an integrable singularity. Away from the channel, the flux
density decreases to zero as does the density of the field lines of a dipole in
electrostatics. The concentrations tend to their values at infinity as fast as
(a/z)2 . This is the rate at which the system tends to equilibrium, which is
maintained by fixed concentrations there.
Non-interacting point particles can pass through each other, so important
phenomena are lost, such as blocking (especially when the LJ radius is of the
order of a) and selectivity [73]. These are expected to be effects of the pair
correlation function, which is the joint pdf of two particles (see Chapter
9). The pair function is the basis to the theory of simple liquids [12, 7, 34,
72, 67, 158] of equilibrium statistical mechanics, from which thermodynamic
properties of matter can be calculated. There are two main approaches to
finding the pair function. The first approach is based on the Ornstein-Zernike
integral equation and a closure relation for the direct and indirect correlation
functions. Many closure relations fall into this category, such as the PercusYevick approximation (PY), the hypernetted chain approximation (HNC),
and the mean spherical approximation (MSA). The second approach relies
on the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy, which
relates the n-th correlation function with the n + 1-th correlation function,
and assumes a closure relation that connects them. However, the starting
point of all approaches is the assumption that the system is in equilibrium.
Certainly, this is not the case at hand, where a non vanishing steady state
flux is flowing through the system.
In [167] an equation for the singlet function is derived in terms of the
pair correlation function for systems out of equilibrium, i.e. carrying steady
state flux. The motion of each particle was described by a separate Langevin
equation, all coupled by the mutual forces. Integrating the full Fokker-Planck
equation in the Smoluchowski overdamped limit resulted a partial differential equation (PDE) for the singlet function in terms of the pair correlation
function. Using similar integrations, we find in Chapter 9 a PDE for the
pair function in terms of the triplet function 9.20, and in general, an entire
hierarchy of PDEs. However, two ingredients, that are necessary for the wellposedness of the pair function problem, are still missing. First, the PDE is
235
Boundary Conditions and a Closure Relation for the Pair Correlation Function in
236
Non-Equilibrium Diffusion
not well posed, unless (correct) boundary conditions are specified. Second,
a closure relation that relates the pair function and the triplet function is
needed. In this chapter we supply these two missing ingredients.
The boundary conditions for the pair function, which satisfies the equations (9.20)-(9.21), are to be specified when one particle is somewhere inside
the domain, while the second particle is at the boundary, which can be either the reflecting membrane, or infinity. At the reflecting wall we impose a
no flux condition for the two particle flux density (Neumann type boundary
condition),
0
J h,h
(12.1)
xh (x, y) = 0, x ∈ ∂ΩR , y ∈ Ω.
At infinity, the two particles become independent, therefore the pair function
satisfies
0
0
ch,h (x, y) = chL ch (y),
|x| → ∞,
z < 0,
y ∈ Ω,
(12.2)
c
h,h0
(x, y) =
0
chR ch (y),
|x| → ∞,
z > 0,
h,h0
y ∈ Ω.
Thus the singlet and the pair densities ch (x), c (x, y) satisfy the nonlinearly coupled system (9.13)-(9.19), (9.20)-(9.21), (12.1)-(12.2).
We choose Kirkwood’s superposition approximation, because it is grounded
in solid probabilistic considerations, which are independent of the state of
equilibrium or non-equilibrium of the systems, as shown in Chapter 10.
Therefore, it is appropriate to use the Kirkwood closure in systems carrying
steady state flux. Other closure relations that have been successful in calculating pair function in equilibrium system are not guaranteed to give correct
h,h0
results for systems out of equilibrium. The force term f̄
(x, y) is now defined in terms of the pair function, thus rendering (9.20) a nonlinear partial
differential equation. We assume that the nonlinear problem (9.13)-(9.19),
(9.20)-(9.21), (12.1)-(12.2) is well posed.
Still, this leaves us with the computational problem of solving the above
problem, numerically or otherwise. The computational complexity of such
high dimensional problem seems to be impractical at present. The rotational
symmetry of the problem (the distance from the wall and the distance from
the center of the pore suffice to describe the position of a single particle)
reduces the dimensionality from six to five, which still requires heavy computational resources (even the two dimensional analogous problem requires
solving a four dimensional PDE, 2D for each particle.) This remains a numerical challenge that hopefully will be solved someday.
12.2 Quasi Steady State: The Neumann Problem in Finite Domains
12.2
Quasi Steady State: The Neumann Problem in Finite Domains
In this section we consider a finite system with all boundaries reflecting (see
Figure 2.1). In other words, there are no control mechanisms to maintain
fixed concentrations and voltages. In the case of all reflecting boundary
condition, no spurious boundary layers appear in the solution of the FokkerPlanck equation. Moreover, the Smoluchowski overdamped approximation is
valid in the entire domain. In this model, time evolution is governed by the
Nernst-Planck equation and no feedback mechanisms (of the kind considered
in Chapter 2) are needed. In this setup, which describes actual channel
current measurements made with polarizing (e.g., platinum) electrodes, there
is no flux in the steady state. However, the long transient is the quasi steady
state that Hodgkin, Huxley, and Katz actually observed on the (biological)
time scale of 1 − 10 msecs [76, 77].
The eigenfunction expansion of the solution of the Smoluchowski equation
[167]
N
h
XX
∂pN (x̃, t)
∇xhj · J xhj (x̃, t),
=−
∂t
j=1
h
(12.3)
where J xhj (x̃, t) is the 3-dimensional probability flux vector of the j-th particle of species h, given by
f hj (x̃)
kB T
J xhj (x̃, t) = h h h pN (x̃, t) − h h h ∇xhj pN (x̃, t),
M γ (xj )
M γ (xj )
(12.4)
results in
pN (x̃, t) =
∞
X
an e−λn t φn (x̃),
(12.5)
n=0
where λ0 < λ1 < . . . < λn < . . . are the eigenvalues of the linear elliptic
operator
Nh
XX
h
j=1
∇xhj · J nxh (x̃) = λn φn (x̃),
j
(12.6)
237
Boundary Conditions and a Closure Relation for the Pair Correlation Function in
238
Non-Equilibrium Diffusion
with Neumann (no-flux) boundary conditions
J nxh (x̃) · ν(x̃) = 0,
j
x̃ ∈ ∂Ω,
(12.7)
where
f hj (x̃)
kB T
J xh (x̃) = h h h φn (x̃) − h h h ∇xhj φn (x̃).
j
M γ (xj )
M γ (xj )
n
(12.8)
Clearly, the first eigenvalue is λ0 = 0, which corresponds to the equilibrium
solution J 0xh (x̃) = 0 for all x̃, j, h, or equivalently, φ0 (x̃) = e−U (x̃)/kB T
j
is the equilibrium Boltzmann distribution. This means, that the system
equilibrates after sufficiently long time (t 1/λ1 ). As mentioned above, the
long transient (1/λ2 t 1/λ1 ) describes measurements on the biological
time scale. In this scale, a quasi steady flux flows through the channel.
For such time scales, we can approximate pN (x̃, t) by only the first two
eigenfunctions, as the other eigenfunctions decay exponentially faster,
pN (x̃, t) ≈ a0 e−U (x̃)/kB T + a1 φ1 (x̃)e−λ1 t .
(12.9)
Indeed, for non-interacting particles, it can be shown ([62, 32] and Chapter 5)
that the second eigenvalue λ1 is much smaller than all other eigenvalues (λ1 D
aD
, while λ2 ∝
, where a is the radius of the channel
λ2 ), as λ1 ∝
|Ω|
|Ω|2/3
hole, D is the diffusion coefficient and |Ω| is the volume of the chamber.
We conjecture that λ1 λ2 also for the case of interacting particles. From
the calculation of the eigenfunction φ1 (x̃), one can find the pair correlation
function of the quasi steady state. The resulting pair correlation function
depends on the choice of the closure relation used in the calculation, as is
the case in equilibrium systems [34].
12.3
The Electrical Current and the RamoShockley Function
We consider the diffusion of interacting ions in an electrolytic solution in
a finite domain that consists of two large baths connected by one or more
narrow channels, and includes two or more metal electrodes maintained at
fixed voltages, imposing a controlled voltage difference across the domain.
12.3 The Electrical Current and the Ramo-Shockley Function
In addition, an external circuitry is connected to one of the electrodes and
measures the current flowing through it, similar to a typical experimental
voltage clamp experiment. The problem at hand is to determine the electric
current flowing through this electrode due to the diffusion currents of the
different ionic species in the system. Obviously, the total current through
the electrode is the sum of the contributions of the diffusion currents of all
ionic species.
Note that in this formulation, we neglect magnetic fields, radiation terms,
convection terms, water flow, and electrolysis. The Ramo-Shockley theorem
[146, 172] relates the microscopic motion of mobile charges in the finite domain to the electric current measured at any given electrode.
For a single moving charge q at location x with velocity v, the instantaneous current at the j-th electrode is given by
Ij = qv · ∇uj (x)
(12.10)
where uj is the solution of capacitor problem
∇ · [ε(x)∇uj ] = 0
with the boundary conditions
uj (12.11)
= 1
(12.12)
= 0 (i 6= j),
(12.13)
∂Ωj
uj ∂Ωi
where ∂Ωj is the boundary of the j-th electrode.
In addition, on the insulating parts of the boundary of the domain, the
conditions on uj are that
ε1
∂uj
∂uj
− ε2
= 0,
∂n
∂n
where derivatives are taken in the normal direction to the boundary, and
ε1 , ε2 are the dielectric coefficients on the two sides of the interface.
In the case of many particles, due to superposition, the total current
recorded at the j-th electrode is given by
X
Ij =
qi v i · uj (xi )
i
239
Boundary Conditions and a Closure Relation for the Pair Correlation Function in
240
Non-Equilibrium Diffusion
12.4
The Connection between the Diffusion
Current and the Electric Current
We now relate the diffusion current carried by the mobile ions inside the
domain to the electrical current measured by the Ampére meter connected to
the electrode. Recall that pN (x̃, ṽ, t) denotes the probability density function
of all particles of all species at time t. Using the Ramo-Shockley theorem
(12.10), the macroscopic current due to the motion of all N h particles of
species h through the j-th electrode is given by
Z
h
h h
Ij = N q
v h1 · ∇uj (xh1 )pN (x̃, ṽ, t) dx̃ dṽ,
(12.14)
where we compute the contribution to the current due only to the first particle of species h, and multiply by N h , because all particles of species h are
interchangeable. The total current measured at electrode j is the sum over
all species,
X
Ijh .
(12.15)
Ij =
h
Note that in equation (12.14), we can easily integrate over all coordinates,
except that of the first particle, which gives the marginal density
Z
h
h h
Ij = N q
v h1 · ∇uj (xh1 )p(xh1 , v h1 , t) dxh1 dv h1 .
(12.16)
As shown in [167], by integrating the full Fokker-Planck equation (9.11)
over all particle coordinates but those of the first particle of species h, we
obtain a reduced Fokker-Planck equation, that can be written in the conservation law form
∂
p(xh1 , v h1 , t) + ∇x · J hx (xh1 , v h1 , t) + ∇v · J hv (xh1 , v h1 , t) = 0,
∂t
(12.17)
with J hx and J hv given by
J hx = v h1 p(xh1 , v h1 , t)
(12.18)
h
J hv = −(γ h v h1 −
f̄ 1
)p(xh1 , v h1 , t) − ∇Dh p(xh1 , v h1 , t).
Mh
12.4 The Connection between the Diffusion Current and the Electric Current
Therefore, combining equations (12.18) and (12.16), and suppressing the
index j of the electrode, the electric current due to species h is
Z
h
h h
I =N q
∇u(x) · J hx (x, v, t) dx dv.
(12.19)
12.4.1
The Steady State
We first consider the current measured in the steady state, in which ∂p/∂t = 0
in equation (12.17). In this case we can write
Z
h
h h
I = N q
∇x u(x) · J hx + u∇x · J hx + u∇v J hv dxdv
Z
h h
= N q
∇x · uJ hx + u∇v J hv dxdv.
(12.20)
The second term in (12.20) vanishes, because integration with respect to v
extends over R3 . Integration with respect to v in the first term yields the
total probability flux J hx (x) at location x. The divergence theorem converts
the integral with respect to x into
Z
h
h h
I = N q
∇ · u(x)J hx (x) dx
Ω
= N hqh
Z
∂Ω
u(x)J hx (x) · n(x) dSx .
Using the boundary conditions (12.12) for the Ramo-Shockley function u(x)
and the no-flux boundary condition for the x component of the flux at the
insulating boundaries, give
Z
h
h h
I =N q
J hx (x) · n(x) dSx .
(12.21)
∂Ωj
If the electrode surface is a reflecting wall for trajectories, the total diffusive flux through ∂Ωj is zero, and by equation (12.21) the total electric
current also vanishes. Indeed, in this case, the steady state of the system
is equilibrium, which means that there is neither a diffusive nor an electric
flux.
If the electrodes also serve as a feedback mechanism that injects and
absorbs ions, the system can be in steady state with a non vanishing flux
241
Boundary Conditions and a Closure Relation for the Pair Correlation Function in
242
Non-Equilibrium Diffusion
[180]. In this case, the external circuitry will measure a steady state electrical
current, that according to equations (12.21) and (12.15), is equal to the total
diffusion current multiplied by the corresponding electrical charges,
Z
X
h h
Itotal =
q N
J h (x) · n(x)dSx .
(12.22)
h
∂Ωj
This formula shows that the current measured by the external circuitry is
exactly equal to the total ionic current flowing between the two baths. An
example for such a (hypothetical?) system is an Ag + Cl− solution with solid
AgCl electrodes maintained at 1V (higher voltage can cause electrolysis of
the water molecules).
12.4.2
Quasi Steady State
We now consider the case of reflecting electrodes, before the system equilibrates. In this case, the solution for the time dependent probability density
function, equation (9.11), can be expanded in eigenfunctions (similarly to
the expansion (12.9))
pN (x̃, ṽ, t) ≈ a0 p0 (x̃, ṽ) + a1 p1 (x̃, ṽ)e−λ1 t .
(12.23)
Treating the time-dependent problem as was done above for the steady state,
we obtain that
Z
∂p
h
h h
dx̃ dṽ
I = N q
∇u(x1 ) · J x + u(x1 ) ∇x J x + ∇v J v +
∂t
Z
∂p
h h
(12.24)
= N q
u(x1 ) dx̃ dṽ,
∂t
where all integrals involving J x and J v vanish, due to the no flux boundary
conditions on the domain boundaries. Inserting the approximation (12.23)
into (12.24) integrating over all particles, except the first one of species h,
we obtain that in the quasi steady state, the electrical current due to species
h is given by
Z
h
h h
−λ1 t
I = N q λ1 e
u(x)ph1 (x) dx,
(12.25)
Ω
ph1
where
is the marginal density of p1 (x̃, ṽ) corresponding to a particle of
species h.
12.4 The Connection between the Diffusion Current and the Electric Current
Note that p1 is the second eigenfunction, and being orthogonal to the
first eigenfunction p0 it is approximately a constant +1 in one bath and a
constant −1 in the other, with additional boundary layers near the channel
(up to a normalization factor). Therefore, the integral is a constant times the
volume of the baths. However, since λ1 is proportional to a/|Ω| (see Chapter
5), the current is independent of the total volume of the baths, but depends
on the small diameter of the channel.
243
Chapter 13
Open Problems and Future
Research
There are still many problems related to the topics of this dissertation that
remain open. In this Chapter I list several open problems, and try to give
a lead when possible. The main problem that arises from part I of this dissertation is how to simulate systems of interacting particles. That is, how to
inject particles from a bath into the simulation, where there are interactions
between the simulated particles and the bath particles, and between bath
particles and themselves. In part I we have reasoned that all interactions
can be replaced by a mean field approximation sufficiently far away from a
channel. Deriving an estimate of the error such an assumption introduces,
as well as the correction needed in the simulation algorithm, remain open
problems.
There are several unsolved problems related to Narrow Escape: (a) Does
the error estimate O(ε log ε) remains valid for all three dimensional domains?
A possible direction could be to use the maximum principle while bounding the given domain between two balls; (b) How do boundary singularities
of three dimensional domains affect the MFPT? (c) The Narrow Escape
problem is closely related to the homogenization problem of a domain with
many small holes, that is, to the approximation of the complicated absorbing boundary conditions by a simpler killing measure [142, 93, 148]. This
direction can possibly be even further developed.
Part II raises many more questions. The problem of finding a “good”
closure relation in equilibrium occupies many physical chemists even today,
so there is no wonder that finding a closure relation for systems out of equi-
245
librium is a tough nut to crack. The pair correlation function of a few one
dimensional systems can be determined analytically, without any closure relation [94, 95, 188, 70, 71] by using the Markov property of a related stochastic process. The resulting pair function has a WKB form for high densities.
However, in higher dimensions the Markov property is no longer preserved:
there is no meaning of “the next particle” as in one dimension (the closest
particle interpretation fails too). A possible direction would be to look for a
solution in a WKB form in high densities for the BBGKY equation, and to
match it to the regular low density cluster expansions.
Appendix A
Algebra Solution of the Albedo
Problem
A.1
The Stationary Albedo Problem
The stationary albedo problem was solved by Klosek [103] using the halfrange expansion technique developed by Hagan, Doering and Levermore [64,
65]. The albedo problem stands for finding the exit probabilities and the first
passage time of a Langevin trajectory that is initiated with a positive velocity
exactly at the location of the absorbing boundary. Our interest in the albedo
problem arises from the need to connect a discrete simulation to a continuum
bath in simulating ion permeation through protein channels [47, 180], though
there are many other physical applications [96, 198]. The non-characteristic
case means that the Langevin particle moves in a non-vanishing force field.
The major part of the solution presented by Klosek consists of analyzing
the following boundary layer problem
f (x̃/γ)
Qv − vQx̃ = 0
(A.1)
Qvv + −v +
γ
Q(x̃ = 0, v > 0) = 0
where f (z) is the force field near the boundary, and γ is a large parameter
that stands for the friction. The analysis [103] is accurate but quite involved,
so one wishes to find a simpler solution.
Here we present an alternative solution to the boundary layer problem
(A.1), which is perhaps simpler. We construct the non-characteristic case
A.2 The Algebra Solution
247
solution from the characteristic case solution of Hagan et al [64] by using the
algebra of the differential operator L = ∂v2 − v∂v .
A.2
The Algebra Solution
The operator L satisfies the commute relations
[L, ∂v ] = ∂v ,
[L, v] = 2∂v − v,
(A.2)
(A.3)
where [A, B] = AB − BA is the commutator. For example, the identity
L∂v Q = LQv = Qvvv − vQvv = ∂v (Qvv − vQv ) + Qv
= ∂v LQ + ∂v Q,
(A.4)
proves relation (A.2). Substituting the Taylor expansion of f (z)
f (x̃/γ) =
∞
X
f (n) (0) x̃n
n=0
γn
n!
.
(A.5)
into (A.1) results in
Qvv +
∞
X
f (n) (0) x̃n
−v +
n! γ n+1
n=0
!
Qv − vQx̃ = 0.
(A.6)
We look for a solution of the form
1
1
Q = Q0 + Q1 + 2 Q2 + . . .
γ
γ
(A.7)
Substituting (A.7) in (A.6) we find that Q0 should satisfy
Q0vv − vQ0v − vQ0x̃ = 0,
(A.8)
and the recursion formula
Qn+1
vv
−
vQn+1
v
−
vQn+1
x̃
n
X
f (n−m) (0) n−m m
=−
x̃
Qv .
(n − m)!
m=0
(A.9)
248
Algebra Solution of the Albedo Problem
Equations (A.8) and (A.9) can be rewritten as
(L − v∂x̃ )Q0 = 0,
and
(L − v∂x̃ )Qn+1 = −
n
X
f (n−m) (0) n−m m
x̃
Qv ,
(n
−
m)!
m=0
(A.10)
(A.11)
The solution to problem (A.10) was found by Hagan et al [64]
" #
√
∞
X
√
1
N ( n) −
√ Vn (v)e− nx̃ . (A.12)
Q0 (x̃, v) = E −ζ
− (v − x̃) −
(−1)n
2
2n! n
n=1
We now proceed to find Q1 . Setting n = 0 in equation (A.11) follows
(L − v∂x̃ )Q1 = −f (0)Q0v .
(A.13)
Using the algebra of L (eqs. (A.2)-(A.3)) we find that Q1 is given by
Q1 = −
f (0)
(v − x̃) Q0 + AQ0 ,
2
(A.14)
where A is an arbitrary constant that should be determined from the matching condition [103]. Indeed,
f (0)
(L − v∂x̃ ) (v − x̃) Q0 + A(L − v∂x̃ )Q0
2
f (0)
−
{(v − x̃) (L − v∂x̃ ) + [L − v∂x̃ , v − x̃]} Q0
2
f (0)
−
{[L, v] + v [∂x̃ , x̃]} Q0
2
f (0)
−
[2∂v − v + v] Q0
2
−f (0)Q0v .
(A.15)
(L − v∂x̃ )Q1 = −
=
=
=
=
Clearly, Q1 also satisfies the boundary condition
Q1 (x̃ = 0, v > 0) = 0,
(A.16)
because
1
Q (x̃ = 0, v > 0) =
f (0)
v Q0 (x̃ = 0, v > 0) = 0.
A−
2
(A.17)
A.2 The Algebra Solution
We see that finding Q1 which is the O (γ −1 ) term is an immediate consequence of the algebra that L generates and the vanishing force fundamental
solution Q0 that was found by Hagan et al [64]. The use of the algebra saved
us the complicated complex variable analysis of the half range expansion.
249
Appendix B
Appendix of Chapter 5
Estimate of kKk2
B.1
B.1.1
Estimate of the kernel
A rough estimate of the kernel, for 0 ≤ u, v ≤ ε, is obtained from equation
(5.58) as
2
5
1
1
sin (v + u)
K 2 (u, v) ≤
cos
(v
+
u)
log
2
2
4π 2
2
2
1
5
1
.
+
cos
(v
−
u)
log
2
sin
(v
−
u)
2
4π 2
2
Furthermore,
2
Z 2 sin 1 (v+ε)
1
2
1
cos (v + u) log 2 sin (v + u) du =
(log x)2 dx
1
2
2
0
2 sin 2 v
2
1 1 2
1
1
1
≤ 2 sin (v + ε) − sin v
log 2 sin v ≤ ε cos v log 2 sin v 2
2
2
2
2
Z
ε
Z
ε
and
0
2
Z 2 sin 1 ε
2
1
1
ε cos v log 2 sin v
dv = ε
(log x)2 dx ≤ 2ε2 log2 ε.
2
2
0
B.2 Elliptic Hole
251
Similarly,
2
1
1
cos (v − u) log 2 sin (v − u) dv =
2
2
0
Z 2 sin 1 u
Z 2 sin 1 (ε−u)
2
2
2
(log x) dx +
(log x)2 dx ≤
ε
Z
0
0
2u log2 u + 2(ε − u) log2 (ε − u).
It follows that
Z ε
2u log2 u + 2(ε − u) log2 (ε − u) du ≤ 4ε2 log2 ε,
0
because u log u is an increasing function in the interval 0 ≤ u ≤ e−2 . Altogether, we obtain
√
1
30
kKk2 ≤
ε log
for ε e−2 ,
(B.1)
2π
ε
which is (5.68).
B.2
Elliptic Hole
We present here, for completeness, Lure’s [116] solution to the integral equation (5.16) in the elliptic hole case. We define for y = (x, y)
L(y) = 1 −
x2 y 2
− 2
a2
b
(b ≤ a)
and introduce polar coordinates in the ellipse ∂Ωa
x = y + (ρ cos θ, ρ sin θ),
with origin at the point y. The integral in eq.(5.16) takes the form
Z
∂Ωa
g0 (x)
dSx =
|x − y|
Z
2π
Z
dθ
0
0
ρ0 (θ)
g̃ dρ
p0
,
L(x)
(B.2)
252
Appendix of Chapter 5
where ρ0 (θ) denotes the distance between y and the boundary of the ellipse
in the direction θ. Expanding L(x) in powers of ρ, we find that
L(x) = 1 −
(x + ρ cos θ)2 (y + ρ sin θ)2
−
= L(y) − 2φ1 ρ − φ2 ρ2 ,
2
2
a
b
(B.3)
x cos θ y sin θ
cos2 θ sin2 θ
+
and
φ
=
+ 2 . Solving the quadratic
2
a2
b2
a2
b
equation (B.3) for ρ, taking the positive root, we obtain
1/2 o
1 n
,
(B.4)
−φ1 + φ21 + φ2 (L(y) − L(x))
ρ(x) =
φ2
where φ1 =
therefore, for fixed y and θ,
dρ(x) = −
1
dL(x)
,
2 [φ21 + φ2 (L(y) − L(x))]1/2
(B.5)
and the integral takes the form
Z
∂Ωa
g0 (x)
dSx =
|x − y|
Z
2π
Z L(y)
1
dL(x)
g̃
p 0
dθ
1/2
2 [φ21 + φ2 (L(y) − L(x))]
L(x)
0
2π
Z L(y)
g̃0 dz
1
p
p
dθ
.
2
2 φ1 + φ2 z L(y) − z
0
0
Z
=
0
φ21
z
and setting ψ =
, we find that
L(y)
φ2 L(y)
Z 2π
Z 1
g̃0
g0 (x)
ds
√
dθ √
dSx =
=
√
|x − y|
2 φ2 0
ψ+s 1−s
0
r
1
g̃0
ψ + s dθ √ 2 arctan
=
1 − s 0
2 φ2
Substituting s =
Z
∂Ωa
Z
2π
0
Z
2π
0
Z
0
2π
p g̃0 √
π − 2 arctan ψ dθ =
2 φ2


x cos θ y sin θ
+


g̃0 dθ
b2
π − 2 arctan r a2
.
r


cos2 θ sin2 θ
cos2 θ sin2 θ
2
+
+
L(y)
a2
b2
a2
b2
B.3 A Pathological Example
253
The arctan term changes sign when θ is replaced by θ + π, therefore its
integral vanishes, and we remain with
Z
Z
g0 (x)
πg̃0 2π
dθ
r
dSx =
2 0
∂Ωa |x − y|
cos2 θ sin2 θ
+ 2
a2
b
Z π
dθ
= 2πbg̃0 2 r
a2 − b 2
0
1−
sin2 θ
b2
= 2πbg̃0 K(e),
(B.6)
where K(·) is the complete elliptic integral of the first kind, and e is the
eccentricity of the ellipse
r
b2
(B.7)
e = 1 − 2 , (a > b).
a
We note that the integral (B.6) is independent of y, so we conclude that
(5.17) is the solution of the integral equation (5.16).
B.3
A Pathological Example
We have derived an integral equation for the leading order terms of the flux
and the MFPT in the case where the MFPT increases indefinitely as the
relative area of the hole decreases to zero. However, the MFPT does not
necessarily increase to infinity as the relative area of the hole decreases to
zero. This is illustrated by the following example. Consider a cylinder of
length L and radius a. The boundary of the cylinder is reflecting, except for
one of its bases (at z = 0, say), which is absorbing. The MFPT problem
becomes one dimensional and its solution is
z2
(B.8)
v(z) = Lz − .
2
Here there is neither a boundary layer nor a constant outer solution; the
MFPT grows gradually with z. The MFPT, averaged against a uniform iniL2
tial distribution in the cylinder, is Eτ =
and is independent of a, that is,
3
the assumption that the MFPT becomes infinite is violated.
Appendix C
Appendix of Chapter 6
C.1
Maximal Exit Time for the Circular Disk
Using equation (6.15) we find
∞
a0 X
+
an
2
n=1
Z π−ε
∞
X
1
a0
+√
[Pn (cos t) + Pn−1 (cos t)] dt.
=
h1 (t)
2
2 0
n=1
vmax = u(1, 0) =
(C.1)
Recall the generating function of the Legendre polynomials [1]
∞
X
1
√
=
Pn (x)tn ,
1 − 2tx + t2
n=0
(C.2)
from which it follows that
∞
X
n=0
Pn (cos t) = √
1
1
=
.
t
1 − 2 cos t + 1
2 sin
2
Together with equation (6.16), this gives


Z π−ε
Z π−ε
a0
1
1
h1 (t) dt
 1

vmax =
+√
h1 (t) 
− 1 dt = √
.
t
t
2
2 0
2 0
sin
sin
2
2
(C.3)
(C.4)
C.1 Maximal Exit Time for the Circular Disk
255
Combining with equation (6.21) and integrating by parts, we get
1
vmax = √
2
Z
π−ε
Z
t
u
du
2
√
dt
cos u − cos t
u sin
1 1 d
π sin t dt 0
0
2
Z t u sin u du π−ε
1
2
√
= √
t 0 cos u − cos t 0
2π sin
2
Z π−ε cos t
Z t u sin u du
1
2 dt
2
√
+ √
.
t
cos
u
−
cos
t
2
2 2π 0
0
sin
2
(C.5)
Equations (6.22) and (6.24) show that
√ Z t u sin u du
t
t
2
2
√
= −2 log cos + 2 log 1 + sin
+ k(t),
π 0 cos u − cos t
2
2
(C.6)
where

q
t
Z sin
arcsin
s2 + cos2

4
2

k(t) = −
 r
π 0
t
s2 + cos2
2

t
2

 ds.

(C.7)
Therefore,
√
lim
t→0
Z
2
π sin
t
2
0
t
t
t
u
−2 log cos + 2 log 1 + sin
+ k(t)
u sin du
2
2
2
√
= lim
t
t→0
cos u − cos t
sin
2
4
= 2 − arcsin(1) = 0.
(C.8)
π
256
Appendix of Chapter 6
Hence
vmax =
r

ε
Z cos
arcsin s2 + sin2

1 
ε
4
ε
2 r

ε 2 log 1 + cos 2 − 2 log sin 2 − π

ε
0
2 cos
s2 + sin2
2
2

1
+ √
2 2π
Z
0
π−ε
 
ε

2
 ds
 
t
Z t u sin u du
2 dt
2
√
.
cos
u
−
cos
t
2 t
0
sin
2
cos
For ε 1
t
Z t u sin u du
2 dt
2
√
vmax
+ O(ε).
(C.9)
t
cos u − cos t
0
0 sin2
2
Changing the order of integration, we get
t
Z π
Z π
cos dt
ε
1
u
2
vmax = − log + √
u sin du
+ O(ε).
√
t
2 2 2π 0
2
u sin2
cos u − cos t
2
(C.10)
Substituting
r
cos u − cos t
s=
(C.11)
2
in the inner integral results in
u
t
Z π
cos
cos dt
√
2
2.
= 2
u
√
t
2
2
u sin
sin
cos u − cos t
2
2
Therefore,
Z π
Z
ε
ε 2 π/2
1
u
vmax = − log +
du = − log −
log sin v dv
2 2π 0 tan u
2 π 0
2
ε
= − log + log 2.
(C.12)
2
ε
1
= − log + √
2 2 2π
Z
π
cos
C.2 Exit Times along the Ray
C.2
257
Exit Times along the Ray
Along the ray θ = π the MFPT is given by
∞
vray (r) ≡ v(r, θ = π) =
1 − r 2 a0 X
+
+
an (−r)n
4
2
n=1
1 − r 2 a0
1
=
+
+√
4
2
2
Z
π−ε
h1 (t)
0
∞
X
[Pn (cos t) + Pn−1 (cos t)](−r)n dt.
n=1
Using the generating function (C.2) of the Legendre polynomials to sum the
infinite series, we obtain
1 − r2 1 − r
vray (r) =
+ √
4
2
Z
π−ε
√
0
h1 (t) dt
.
1 + 2r cos t + r2
(C.13)
Combining with equation (6.21), integrating by parts, and hanging the order
of integration gives
1−r
1 − r2
+ √
a0
4
2 1 − 2r cos ε + r2
Z
Z π−ε
u
sin t dt
r(1 − r) π−ε
√
u sin du
.
− √
2
(1 + 2r cos t + r2 )3/2 cos u − cos t
2π
0
u
vray (r) =
The substitutions s =
Z
u
π−ε
√
cos u − cos t and x =
√
2r s lead to
√
sin t dt
2 cos u + cos ε
√
√
=
,
(1 + 2r cos t + r2 )3/2 cos u − cos t
(1 + 2r cos u + r2 ) 1 − 2r cos ε + r2
which implies that
vray (r) =
1 − r2
1−r
+ √
a0
(C.14)
4
2 1 − 2r cos ε + r2
√
√
Z π−ε
2 r(1 − r)
u cos u + cos ε
− √
u sin
du.
2 1 + 2r cos u + r2
π 1 − 2r cos ε + r2 0
258
Appendix of Chapter 6
The substitution (6.23) gives
√
Z π−ε
u cos u + cos ε
u sin
du =
2 1 + 2r cos u + r2
0
r
ε
ε
arccos s2 + sin2 s2 ds
√ Z cos 2
2
r
,
4 2
0
2
2 ε
2
2
(1 − 2r cos ε + r + 4rs ) sin + s
2
and we obtain the exact form of vray (r) as
1 − r2
1−r
+ √
a0
4
2 1 − 2r cos ε + r2
vray (r) =
8r(1 − r)
− √
π 1 − 2r cos ε + r2
For ε 1 and 1 − r (C.15)
r
ε
arccos s2 + sin2 s2 ds
2
r
.
2
2 ε
2
2
(1 − 2r cos ε + r + 4rs ) sin + s
2
Z cos ε
2
0
√
ε equation (C.15) becomes
Z
1 − r2
ε 8r 1 s arccos s ds
vray (r) =
− log −
+ O(ε).
4
2
π 0 (1 − r)2 + 4rs2
(C.16)
To evaluate the integral in (C.16), we write
arccos s =
π
− arcsin s,
2
(C.17)
and obtain
8r π
π 2
Z
0
1
s ds
= −2 log(1 − r) + log(1 + r2 ).
(1 − r)2 + 4rs2
The function q(r), defined by
8r
q(r) =
π
Z
0
1
arcsin s s ds
(1 − r)2 + 4rs2
(C.18)
in the interval 0 ≤ r ≤ 1, has the endpoint values
q(0) = 0,
q(1) = log 2.
(C.19)
C.3 Flux Profile
259
Therefore,
ε
1 − r2
+ 2 log(1 − r) +
− log(1 + r2 ) + q(r) + O(ε), (C.20)
2
4
√
is the MFPT for ε 1 and 1 − r ε. In particular,
vray (r) = − log
vcenter = vray (0) = − log
ε 1
+ + O(ε),
2 4
as asserted in (6.2).
C.3
Flux Profile
In this appendix we calculate the flux profile given by equation (6.36). Substituting equation (6.21) for h1 in equation (6.36) gives

u
1 d t u sin 2 du


Z π−ε
√


θ
d 
1
π dt 0 cos u − cos t 
√
dt .
f (θ) = − +
cos

2 dθ 
2 0
cos t − cos θ



Z
Integration by parts and changing the order of integration, we find that
θ
"
Z π−ε u sin u du
cos
1 1 d
2
2
p
√
f (θ) = − +
2 π dθ
cos u + cos ε
cos(π − ε) − cos θ 0
#
Z
Z π−ε
1
θ π−ε
u
sin t dt
− cos
u sin du
.
2
2 0
2
(cos t − cos θ)3/2 (cos u − cos t)1/2
u
We evaluate the inner integral by making the substitution x =
Z
u
π−ε
√
cos u − cos t,
√
sin t dt
2 cos u + cos ε
√
=
.
(cos t − cos θ)3/2 (cos u − cos t)1/2
(cos u − cos θ) − cos θ − cos ε
260
Appendix of Chapter 6
Therefore
θ
"
Z π−ε u sin u du
cos
1 1 d
2
2
√
√
f (θ) = − +
2 π dθ
cos u + cos ε
− cos ε − cos θ 0
√
θ
Z π−ε u sin u cos u + cos ε #
2
2
du
−√
cos
u − cos θ
− cos θ − cos ε 0

θ
cos
1 1 d 
π
2
√ a0 −
= − +
√
2 π dθ
− cos ε − cos θ 2
cos

θ
u√
Z
π−ε u sin
cos
cos u + cos ε

2
2
√
du .
cos u − cos θ
− cos θ − cos ε 0
The substitution (6.23) gives
u√
cos u + cos ε du
2
=
cos u − cos θ
0
r
ε 2
ε
2 + sin2
Z
arccos
s
s ds
cos
√
2
2
r
2 2
.
0
2 θ
2 ε
2 ε
2
2
s + sin − cos
s + sin
2
2
2
Z
u sin
π−ε
Therefore, the flux takes the form
θ
cos
1 d 
1
2
f (θ) = − + √
×
√
2
dθ
− cos θ − cos ε
2π

r


πa0 −

Z
0
cos(ε/2)

ε
s2 + sin2 s2 ds

2
 ,
r

ε
θ
ε
2
2
2
2
s2 + sin − cos
sin + s
2
2
2
4 arccos
C.3 Flux Profile
261
which is rewritten as
θ
1
1 d  cos 2
f (θ) = − +

2 2π dθ
b

√
Z
1−a2
πa0 − 4
0

!
arccos s2 + a2 s2 ds 
√
,
a2 + s2 (s2 + b2 )
√
(C.21)
ε
where a = sin and 2b2 = − cos θ − cos ε. Writing
2
√
∞
arccos s2 + a2 X
√
φ(a, s) =
=
φ2n (a) s2n ,
s 2 + a2
n=0
we find the Taylor coefficients
arccos a
φ0 (a) =
, φ2 (a) = −
a
arccos a
1
√
+
3
2
2a
2a 1 − a2
,
and so on. For all n ≥ 0 we find the asymptotic behavior
c2n
1
φ2n (a) ∼ 2n+1 + O
as a → 0.
a
a2n
To see this, consider the Taylor expansions
!
r
∞
s 2 2n+1
X
s2m
1+
=
cm
n 2m
a
a
m=0
!
r
∞
s 2
π X
αn a2n+1
=
+
arccos a 1 +
a
2 n=0
r
1+
s 2
!2n+1
a
∞
∞
X
π X
s2m
2n+1
=
+
αn a
cm
n 2m
2 n=0
a
m=0
∞
∞
π X s2m X m
=
+
cn αn a2n+1 ,
2m
2 m=0 a n=0
where αn and cm
n are (known) constants, and
a
∞
2n
1X
n (2n)! s
=
(−1)
.
s 2
a n=0
(2n n!)2 a2n
1+
a
1
r
(C.22)
262
Appendix of Chapter 6
Therefore
∞
(2n)! s2n
1X
φ(a, s) =
(−1)n n 2 2n
a n=0
(2 n!) a
∞
∞
π X s2m X m
+
c αn a2n+1
2 m=0 a2m n=0 n
!
,
(C.23)
from which it follows that
φ(a, s) =
∞ X
n=0
2n
(2n)!
s
(−1)
+ O(a)
.
n
2
2 (2 n!)
a2n+1
(C.24)
π
(2n)!
−2n
+
O
a
,
2 (2n n!)2 a2n+1
(C.25)
nπ
This shows that
φ2n (a) ∼ (−1)n
as asserted. The asymptotic behavior (C.25) of the coefficients φ2n (a) can
be used to estimate the integral in equation (C.21),
Z √1−a2 2n+2
Z √1−a2
∞
φ(a, s) s2 ds X
s
ds
φ
(a)
=
.
(C.26)
2n
2 + b2
s2 + b 2
s
0
0
n=0
To extract the asymptotic behavior of the integral as b → 0, we use the long
division
n
X
(−1)n+1 b2n+2
s2n+2
j 2j 2n−2j
=
(−1)
b
s
+
(C.27)
s2 + b 2
s2 + b 2
j=0
and integrate it to yield
Z √1−a2 2n+2
s
ds =
s2 + b 2
0
"
#
Z √1−a2
Z
n
X
j 2j
2n−2j
n+1 2n+2
(−1) b
s
ds + (−1) b
j=0
0
(C.28)
√
1−a2
0
ds
s2 + b 2
√
√
n 2 2n−2j+1
X
1 − a2
j 2j ( 1 − a )
n+1 2n+1
=
(−1) b
+ (−1) b
arctan
.
2n − 2j + 1
b
j=0
The Taylor expansion
√
∞
1 − a2
π X (−1)m+1
b2m+1
√
arctan
= +
b
2 m=0 2m + 1 ( 1 − a2 )2m+1
(C.29)
C.3 Flux Profile
263
gives the Taylor expansion of the integral (C.28) in powers of b as
√
Z √1−a2 2n+2
n
2 2n−2j+1
X
s
j 2j ( 1 − a )
ds
=
(−1)
b
s2 + b 2
2n − 2j + 1
0
j=0
∞
+(−1)n+1 b2n+1
=
n
X
π X (−1)m+1
b2m+1
√
+
2 m=0 2m + 1 ( 1 − a2 )2m+1
!
√
j(
(−1)
j=0
n+1 π
+(−1)
2
1 − a2 )2n−2j+1 2j
b
2n − 2j + 1
b
2n+1
∞
X
(−1)n+m
√
b2m+2n+2 .
+
2
2m+1
(2m + 1)( 1 − a )
m=0
Therefore, the Taylor expansion of the integral (C.26) is
√
X
Z √1−a2
n
∞
2 2n−2j+1
φ(a, s) s2 ds X
j( 1−a )
φ
(a)
(−1)
=
b2j
2n
2 + b2
s
2n
−
2j
+
1
0
n=0
j=0
n+1 π
+(−1)
2
b
2n+1
∞
X
(−1)n+m
2m+2n+2
√
+
b
.
2 )2m+1
(2m
+
1)(
1
−
a
m=0
Rearranging in powers of b, we find that
Z √1−a2
∞
φ(a, s) s2 ds X
=
βn (a) bn ,
2
2
s +b
0
n=0
where the first three coefficients are
√
Z √1−a2
∞
X
( 1 − a2 )2n+1
π
=
φ(a, s) ds = a0 ,
β0 (a) =
φ2n (a)
2n + 1
4
0
n=0
πφ0 (a)
π arccos a
=−
,
2
2a
√
∞
X
( 1 − a2 )2n−1
φ0 (a)
β2 (a) = −
φ2n (a)
+√
2n − 1
1 − a2
n=1
Z √1−a2
φ(a, s) − φ0 (a)
φ0 (a)
= −
ds + √
2
s
1 − a2
0
β1 (a) = −
(C.30)
264
Appendix of Chapter 6
and all other coefficients βn are recovered in a similar fashion,
π
π2
(2j)!
β2j+1 = (−1)j+1 φ2j (a) = −
+ O(a−2j ),
j
2
2j+1
2
4 (2 j!) a
√
∞
X
( 1 − a2 )2n−2j+1
j
β2j = (−1)
φ2n (a)
2n − 2j + 1
n=j
!
j−1
X
1
√
φ2n (a)
−
(2j − 2n − 1)( 1 − a2 )2j−2n−1
n=0
√
Z
j
1−a2
= (−1)
0
−
j−1
X
∞
1 X
φ2n (a)s2n ds
s2j n=j
1
√
φ2n (a)
!
(2j − 2n − 1)( 1 − a2 )2j−2n−1
P
Z √1−a2
2n
φ(a, s) − j−1
j
n=0 φ2n (a)s
= (−1)
ds
s2j
0
!
j−1
X
1
√
−
φ2n (a)
.
(2j − 2n − 1)( 1 − a2 )2j−2n−1
n=0
n=0
We see that extra effort should be put in finding the even coefficients β2n .
Expanding
∞
X (2n)! (s2 + a2 )n
π
1
√
φ(a, s) =
−
,
2 s2 + a2 n=0 (2n n!)2 2n + 1
(C.31)
and noting that the following infinite sum has a regular contribution
√
Z
lim
a→0
0
1−a2
∞
1 X (2n)! (s2 + a2 )n
ds = Cj ,
s2j n=j (2n n!)2 2n + 1
(C.32)
C.3 Flux Profile
265
where Cj are constants (also can be written in term of hypergeometric functions), we find an alternative representation for the even coefficients,
√
Z
j
1−a2
β2j = (−1)
φ(a, s) −
Pj−1
0
n=0
s2j
φ2n (a)s2n
j−1
X
φ2n (a)
√
−
(2j
−
2n
−
1)(
1 − a2 )2j−2n−1
n=0
j
= (−1) −Cj + O(a) +
ds
!
j−1
√
Z
1−a2
j−1
X (2n)! (s2 + a2 )n X
1
π
√
−
−
φ2n (a)s2n
2 s2 + a2 n=0 (2n n!)2 2n + 1
n=0
s2j
0
−
j−1
X
φ2n (a)
n=0
1
√
(2j − 2n − 1)( 1 − a2 )2j−2n−1
j−1
√
j
Z
1−a2
= (−1)
j−1
X (2n)! (s2 + a2 )n X
π
1
√
−
−
φ2n (a)s2n
2 s2 + a2 n=0 (2n n!)2 2n + 1
n=0
s2j
0
(2j − 2)!
1
+O
2 (2j−1 (j − 1)!)2 a2j−1
j−1 π
−(−1)
ds
1
a2j−2
.
The integrals are given in [145],
Z
n−j+1/2
j−1
1 X (−1)n
j−1
s2
ds
√
. (C.33)
= 2j
a n=0 2n − 2j + 1
n
s 2 + a2
s2j s2 + a2
The binomial expansion gives
Z
n
(s2 + a2 )n ds X
1
n 2k−2j+1 2n−2k
=
s
a
.
s2j
2k
−
2j
+
1
k
k=0
(C.34)
ds
266
Appendix of Chapter 6
Altogether, we find that the integral term in equation (C.33) is
j−1
j−1
√
Z
1−a2
X (2n)! (s2 + a2 )n X
π
1
√
φ2n (a)s2n
−
−
n
2
2
2
2 s +a
(2 n!) 2n + 1
n=0
n=0
s2j
0
ds =
j−1
j−1
1
π 1 X (−1)n
+O
.
=
2 a2j n=0 2n − 2j + 1
n
a2j−1
This sum has the closed form [145]
k
X
(−1)i k
(2k k!)2
=
,
2i + 1 i
(2k + 1)!
i=0
and we have obtained the asymptotic form of the even coefficients
π 1 (2j−1 (j − 1)!)2
1
β2j =
+O
.
2 a2j (2j − 1)!
a2j−1
(C.35)
(C.36)
We are now able to find the asymptotic expansion of the flux profile (C.21),


θ
!
√
√
Z
2
1−a
1
1 d  cos 2
arccos s2 + a2 s2 ds 
√
f (θ) = − +
πa0 − 4


2 2π dθ
b
a2 + s2 (s2 + b2 )
0
"
#
∞
1 2 d
θX
n
= − −
cos
βn+1 b .
2 π dθ
2 n=0
Setting εα = π − θ, we obtain after some manipulations that to leading order
in small ε the flux is given in the interval −1 < α < 1 by
"
#
∞
2
α2
1 X (2n+1 (n + 1)!) 2
(2n n!)2
f (α) = − √
−
α −
(1 − α2 )n+1/2
2
ε
(2n
+
2)!
(2n
+
1)!
ε 1−α
n=0
∞
π X (2n)!
(2n + 2)!(2n + 2) 2
−
−
α (1 − α2 )n + O(1). (C.37)
2ε n=0 (2n n!)2
(2n+1 (n + 1)!)2
Appendix D
Appendix of Chapter 7
D.1
Laplace Beltrami Operator on 2-Sphere
The Laplace Beltrami operator on a manifold is given by
X ∂ √
∂f
1
ij
g det G
,
∆M f = √
∂ξj
det G i,j ∂ξi
where
ti =
∂r
,
∂ξi
gij = hti , tj i,
G = (gij ),
g ij = gij−1 .
(D.1)
(D.2)
In spherical coordinates we have
gθθ = R2 ,
gφφ = R2 sin2 θ,
gθφ = gφθ = 0.
Therefore, for a function w = w(θ, φ)
2
∂f
∂ f
1 ∂2f
−2
∆M f = R
+ cot θ
+
.
∂θ2
∂θ sin2 θ ∂φ2
(D.3)
(D.4)
Bibliography
[1] Abramowitz M., Stegun I. A., Handbook of Mathematical Functions,
Dover Publications, NY (1972).
[2] Alberts B., Johnson A., Lewis J., Raff M., Roberts K., Walter P., Molecular Biology of the Cell, Garland; 4th edition 2002.
[3] Allen M. P. and Tildesley D. J., Computer simulation of liquids, Oxford
University Press, Oxford, 1991.
[4] Allen R., Hansen J-P and Melchionna S., “Electrostatic potential inside
ionic solutions confined by dielectrics: a variational approach”, Phys.
Chem. Chem. Phys. 3, pp.4177-4186 (2001).
[5] Andrews G.E., Askey R., Roy R., Special Functions, Cambridge University Press (2000).
[6] Aubin T., Some Nonlinear Problems in Riemannian Geometry,
Springer, NY (1998).
[7] Barthel J.M.G., Baumgartel A.H. (Editor), Krienke H., Physical Chemistry of Electrolyte Solutions: Modern Aspects, Steinkopf, Dietrich Pub.
1998.
[8] Barthel J., Münsterer M. and Buchner R., “Dielectric Properties of Water and Aqueous Electrolyte Solutions”, in Kreysa G. (ed.), Electrolyte
Data Collection (Part 2) , DECHEMA Chemistry Data Series, Vol. XII,
Frankfurt (1995).
[9] Beals R., Protopopescu V., “Half-Range Completeness for the FokkerPlanck Equation”, J. Stat Phys., Vol. 32, No. 3 (1983).
BIBLIOGRAPHY
[10] Bender C.M., Orszag S.A., Advanced Mathematical Methods for Scientists and Engineers, Springer, New York (1999).
[11] Berkowitz M. and McCammon J. A., “Molecular dynamics with stochastic boundary conditions,” Chem. Phys. Lett. 90, pp. 215–217 (1982).
[12] Berry R. S., Rice S., and Ross J., Physical chemistry, Oxford University
Press, NY, 2nd ed. (2000).
[13] Bockris J., Reddy A., Modern Electrochemistry, Vol. 1, Second Edition,
Plenum Press (1998).
[14] Borgdorff A.J., Choquet D., “Regulation of AMPA receptor lateral
movements”, Nature 417 (6889), pp.649-53 (2002).
[15] Breiman L., Probability (Classics in Applied Mathematics, No.7). SIAM
Publications (Reprint edition) (1992).
[16] Brooks C. L. III, and Karplus M., “Deformable stochastic boundaries in
molecular dynamics,” J. Chem. Phys. 79, p. 6312 (1983).
[17] Brydges D.C. and Martin Ph.A., “Coulomb systems at low density: a
review”, J. Stat. Phys. 96 (5/6), pp.1163-1330 (1999).
[18] Buck B., Macaulay V. A., Eds. Maximum Entropy in Action, New York,
Oxford (1995).
[19] Bunimovich L. A., Dettmann C. P., “Open Circular Billiards and the
Riemann Hypothesis”, Phys. Rev. Lett. 94 100201 (2005).
[20] Burschka M. A. and Titulaer U. M., “The kinetic boundary layer for
the Fokker-Planck equation with absorbing boundary,” J. Stat. Phys.
25, pp. 569–582 (1981).
[21] Carrier G. F., Krook M., Pearson C. E., Functions of a Complex Variable, McGraw-Hill, NY (1966).
[22] Chandrasekhar S., Rev. Mod. Phys. 15 (1) pp.1-89 (1943), also in N.
Wax, ed., Selected Papers on Noise and Stochastic Processes, Dover,
New York (1954).
269
270
BIBLIOGRAPHY
[23] Chazalviel J.N., Coulomb Screening by Mobile Charges, Birkhäuser,
Boston (1999).
[24] Chen D.P. and Eisenberg R.S., “Flux, coupling, and selectivity in ionic
channels of one conformation”, Biophys. J. 65, pp.727-746 (1993).
[25] Cohen J., Lewis R., “Ray Method for the asymptotic solution of the
diffusion equation”, J. Inst. Maths. Applics., 3, pp.266-290 (1967).
[26] Collins W. D., “On some dual series equations and their application to
electrostatic problems for spheroidal caps”, Proc. Cambridge Phil. Soc.
57, pp. 367-384 (1961).
[27] Collins W. D., “Note on an electrified circular disk situated inside an
earthed coaxial infinite hollow cylinder”, Proc. Cambridge Phil. Soc. 57,
pp. 623-627 (1961).
[28] Corry B., Kuyucak S., and Chung S. H., “Tests of Continuum Theories
as Models of Ion Channels. II. Poisson-Nernst-Planck Theory versus
Brownian Dynamics,” Biophys. J. 78, pp. 2364–2381 (2000).
[29] Corry B., Hoyles M., Allen T. W., Walker M., Kuyucak S., and Chung
S. H., “Reservoir Boundaries in Brownian Dynamics Simulations of Ion
Channels,” Biophys. J. 82, pp. 1975–1984 (2002).
[30] Corry B., Kuyucak S., Chung S., ”Dielectric Self-Energy in PoissonBoltzmann and Poisson-Nernst-Planck Models of on Channels”, Biophys. J. 84 (2003), pp.3594-3606.
[31] Cover T. M., Thomas J. A., Elements of Information Theory, Wiley,
New York (1991).
[32] Dagdug L., Berezhkovskii A. M., Shvartsman S. Y., Weiss G. H., “Equilibration in two chambers connected by a capillary”, J. Chem. Phys.
119 (23), pp.12473-12478 (2003).
[33] Dauge M., Elliptic Boundary Value Problems on Corner Domains:
Smoothness and Asymptotics of Solutions, Lecture Notes in Mathematics, 1341, Springer-Verlag, NY (1988).
BIBLIOGRAPHY
[34] Davis H. T., Statistical Mechanics of Phases, Interfaces, and Thin Films,
(Advances in Interfacial Engineering Series), John Wiley & Sons, NY
(1995).
[35] DeGroot S. R., Mazur P., Non-Equilibrium Thermodynamics, Amsterdam, North-Holland Publishing Co., 1962.
[36] Dembo A. and Zeitouni O., Large Deviations Techniques and Applications, Jones and Bartlett, Boston 1992.
[37] Duck P. W., Marshall T. W., Watson E. J., “First-passage times for the
Uhlenbeck-Ornstein process”, J. Phys. A: Math. Gen. 19 pp. 3545-3558
(1986).
[38] Dufrêche J.-F., Bernard O. and Turq P., “Transport equations for concentrated electrolyte solutions: Reference frame, mutual diffusion”, J.
Chem. Phys. 116 (5), pp.2085-2097 (2000).
[39] Dvoretzky A., Erdös P., and Kakutani S., “Nonincreasing everywhere of
the Brownian motion process”, Proc. Fourth Berkeley Symp. on Math.
Stat. and Probability II, pp.103-116. Univ. of Calif. Press (1961).
[40] Ebeling W., Feistel R., Kelbg G. and Sandig R., J. Non-Equilibrium
Thermodynamics 3, p.11 (1978).
[41] Einstein A., Ann. d. Physik 17, p.549 (1905), See also Investigations on
the Theory of the Brownian Movement, Dover (1956)
[42] Eisenberg R. S., “Ionic channels in biological membranes: electrostatic
analysis of a natural nano-tube,” Contemp. Phys. 39, pp. 447–466
(1998).
[43] Eisenberg R.S., “From structure to function in open ionic channels”. J.
Mem. Biol. 171 (1999), pp.1-24.
[44] Eisenberg R. S., “Proteins, Channels, and Crowded Ions.” Biophysical
Chemistry, 100, pp. 507-517 (2003).
[45] Eisenberg R.S., “Channels as enzymes.” J. Memb. Biol. 115, pp.112
(1990).
271
272
BIBLIOGRAPHY
[46] Eisenberg R.S., “Ion Channels as Devices”, Journal of Computational
Electronics 2, pp.245-249 (2003).
[47] Eisenberg R. S., Klosek M. M., and Schuss Z., “Diffusion as a chemical reaction: Stochastic trajectories between fixed concentrations,” J.
Chem. Phys. 102, pp.1767–1780 (1995).
[48] Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Tables of Integral Transforms, Volume 1, McGraw-Hill, NY (1954).
[49] Fabrikant V. I., Applications of Potential Theory in Mechanics, Kluwer
(1989).
[50] Fabrikant V. I., Mixed Boundary Value Problems of Potential Theory
and Their Applications in Engineering, Kluwer (1991).
[51] Feller W., An Introduction to Probability Theory and its Applications,
2nd ed, Wiley, New York (1957).
[52] Fick A., “On Liquid Diffusion,” Phil. Mag. and Jour. Sci. 10 pp. 31-39
(1855).
[53] Fisher I. Z., Kopeliovich B. I., Dokl. Akad. Nauk. USSR 133, 81 (1960);
(English transl.: Soviet Phys.–Doklady 5, 761, 1960-1961)
[54] Freidlin M., Functional Integration and Partial Differential Equations,
(Annuals of Mathematics Studies, 109), Princeton, NJ (1985).
[55] Garabedian P. R., Partial Differential Equations, Wiley, NY (1964).
[56] Gardiner C.W., Handbook of Stochastic Methods, 2-nd edition, Springer,
NY (1985).
[57] Ghausi M. S., Kelly J. J., Introduction to Distributed-Parameter Networks. New York, Holt Rinehart & Winston.331 (1968).
[58] Gillespie D., Nonner W. and Eisenberg R. S., ”Coupling Poisson-NernstPlanck and density functional theory to calculate ion flux”, J. Phys.
Condens. Matter 14 (46) (2002), pp.12129-12145.
[59] Gillespie D., Nonner W., Henderson W. and Eisenberg R. S., ”A physical
mechanism for large-ion selectivity of ion channels”, Phys. Chem. Chem.
Phys. 4, pp.4763-4769 (2002).
BIBLIOGRAPHY
[60] Goldman M., “On the first passage of the integrated Wiener process”,
Ann. Math. Stat. 42 (6), pp.2150-2155 (1971).
[61] Gordon J. S., A thread across the ocean: The heroic story of the trans
atlantic cable, p. 240, Walker, New York (2002).
[62] Grigoriev I. V., Makhnovskii Y. A., Berezhkovskii A. M., Zitserman V.
Y., “Kinetics of escape through a small hole”, J. Chem. Phys., 116 (22),
pp.9574-9577 (2002).
[63] Groeneboom P., Jongbloed G., Wellner J.A., “Integrated Brownian motion, conditioned to be positive”, Ann. Probab. 27 (3), pp.1283-1303
(1999).
[64] Hagan P. S., Doering C. R., and Levermore C. D., “Mean exit times for
particles driven by weakly colored noise,” SIAM J. Appl. Math. 49 (5),
pp. 1480–1513 (1989).
[65] Hagan P. S., Doering C. R., and Levermore C. D., “The distribution of
exit times for weakly colored noise,” J. Stat. Phys 54 (5/6), pp. 1321–
1352 (1989).
[66] Hänngi P., Talkner P., and Borkovec M., “50 year after Kramers”, Rev.
Mod. Phys. 62, p.251 (1990).
[67] Hansen J.-P., McDonald I. R., Theory of Simple Liquids, 2nd Ed., Academic Press, London (1990).
[68] Harris S., “Steady, one-dimensional Brownian motion with an absorbing
boundary”, J. Chem. Phys. 76, pp. 3103-3106 (1981).
[69] Helmholtz H.L.F. von, Crelle, Bd. 7 (1860).
[70] Hemmer P.C., Kac M., and Uhlenbeck G.E., “On the van der Waals
Theory of the Vapor-Liquid Equilibrium. III. Discussion of the Critical
Region,” J. Math. Phys. 5 60 (1964).
[71] Hemmer P.C., “On the van der Waals Theory of the Vapor-Liquid Equilibrium. IV. The Pair Correlation Function and Equation of State for
Long-Range Forces,” J. Math. Phys. 5 75 (1964).
273
274
BIBLIOGRAPHY
[72] Henderson D. (Editor), Fundamentals of Inhomogeneous Fluids, Marcel
Dekker, New York, (1992).
[73] Hille B., Ionic Channels of Excitable Membranes, pp. 1–814, Sinauer
Associates Inc., 3rd ed., Sunderland, Massachusetts (2001).
[74] Hille E., Analytic Function Theory, vol. 1, Chelsea Publishing Company,
New York (1976).
[75] Hodgkin A. L., The conduction of the nervous impulse, pp. 1–108, Liverpool University Press, Liverpool (1971).
[76] Hodgkin A.L., Huxley A.F., and Katz B., “Ionic currents underlying
activity in the giant axon of the squid,” Arch. Sci. physiol. 3, pp. 129–
150 (1949).
[77] Hodgkin A.L., Huxley A.F.,and Katz B., “Measurement of currentvoltage relations in the membrane of the giant axon of loligo,” J. Physiol.
(London) 116, pp. 424–448 (1952).
[78] Holcman D. and Schuss Z., “Diffusion through narrow openings: the
dynamics of AMPA receptors on a postsynaptic membrane”, J. Stat.
Phys. (in print).
[79] Holcman D. and Schuss Z., “Stochastic chemical reactions in microdomains”, (in print).
[80] Holcman D., Schuss Z., Korkotian E., “Calcium dynamics in denritic
spines and spine motility”, Biophysical Journal 87, pp.81-91 (2004)
[81] Hu Z. M., Gan H. H., Eu B. C., “On the Ornstein-Zernike relation, the
BBGKY hierarchy and closure”, Physica A 184, pp. 71–78 (1992).
[82] Im W., Seefeld S., and Roux B., “A grand canonical Monte CarloBrownian dynamics algorithm for simulating ion channels,” Biophys.
J. 79, pp. 788–801 (2000).
[83] Im W. and Roux B., “Ion permeation and selectivity of ompf porin: a
theoretical study based on molecular dynamics, brownian dynamics, and
continuum electrodiffusion theory,” J. Mol. Bio. 322 (4), pp. 851–869
(2002).
BIBLIOGRAPHY
[84] Im W. and Roux B., “Ions and counterions in a biological channel: a
molecular dynamics simulation of OmpF porin from Escherichia Coli in
an explicit membrane with 1 M KCl aqueous salt solution,” J. Mol. Bio.
319 (5), pp. 1177–1197, (2002).
[85] Isozaki Y., Watanabe S., “An asymptotic formula for the Kolmogorov
diffusion and refinement of Sinai’s estimates for the integral of Brownian
motion”, Proc. Japan Acad., Ser. A, 70, pp.271-276 (1994).
[86] Itô K. and McKean H.P. Jr., Diffusion Processes and their Sample Paths,
Springer, NY (1996).
[87] Jack J.J.B., Noble D., Tsien R.W., Electric Current Flow in Excitable
Cells. New York, Oxford, Clarendon Press (1975).
[88] Jackson J. D., Classical Electrodymnics, 2nd Ed., Wiley, NY (1975).
[89] Jacoboni C. and Lugli P., The Monte Carlo Method for Semiconductor
Device Simulation, pp. 1–356, Springer Verlag, New York (1989).
[90] Jensen J.L., Saddlepoint Approximations (Oxford Statistical Science Series, 16), Oxford University Press (1995).
[91] Joseph S., Mashl R.J., Jacobsson E. and Aluru N.R., ”Ion Channel
Based Biosensors: Ionic transport in carbon nanotubes”, Technical Proceedings of the 2003 Nanotechnology Conference 1 pp.158-161.
[92] Kac M., Probability and Related Topics in the Physical Sciences, Interscience, NY (1959).
[93] Kac M., “Probabilistic methods in some problems of scattering theory,”
Rocky Mountain J. Math. 4 pp. 511–538 (1974).
[94] Kac M., “On the Partition Function of a One-Dimensional Gas,” Phys.
Fluids 2 8 (1959), also in Mark Kac: Probability, Number Theory, and
Statistical Physics: Selected Papers, edited by K.Baclawski and M.D.
Donsker, Cambridge, Mass.: MIT (1979).
[95] Kac M., Uhlenbeck G.E., and Hemmer P.C., “On the van der Waals Theory of the Vapor-Liquid Equilibrium. I. Discussion of a One-Dimensional
Model,” J. Math. Phys 4 216 (1963).
275
276
BIBLIOGRAPHY
[96] Kainz A. J. and Titulaer U. M., “The analytic structure of the stationary
kinetic boundary layer for Brownian particles near an absorbing wall,”
J. Phys. A: Math. Gen. 24, pp. 4677–4695 (1991).
[97] Karlin S. and Taylor H.M., A First Course in Stochastic Processes, Academic Press, NY (1975).
[98] Kelman R. B., “Steady-State Diffusion Through a Finite Pore Into an
Infinite Reservoir: an Exact Solution”, Bulletin of Mathematical Biophysics 27, pp.57-65 (1965).
[99] Kirkwood J. G., “Statistical Mechanics of Fluid Mixtures.” J. Chem.
Phys. 3, pp. 300-313 (1935).
[100] Kirkwood J. G., Boggs E. M., “The Radial Distribution Function in
Liquids.” J. Chem. Phys. 10, pp. 394-402 (1942).
[101] Kittel C., Introducation to Solid State Physics, 7th ed., John Wiley
and Sons, New York (1996).
[102] Kleinert H., Path Integrals in Quantum Mechanics, Statistics, and
Polymer Physics, World Scientific, NY (1994).
[103] Klosek M. M., “Half-range expansion analysis for Langevin dynamics
in the high-friction limit with a singular absorbing boundary condition:
Noncharacteristic case,” J. Stat. Phys. 79 (1/2), pp. 313–345 (1995).
[104] Klosek M. M., Hagan P. S., “A non-characteristic level crossing problem for one-dimensional dynamics driven by wideband colored noise”,
unpublished.
[105] Kozlov V.A., Mazya V.G. and Rossmann J., Elliptic Boundary Value
Problems in Domains with Point Singularities, American Mathematical
Society, Mathematical Surveys and Monographs, vol. 52 (1997).
[106] Kozlov V.A., Rossmann J., Mazya V.G., Spectral Problems Associated
With Corner Singularities of Solutions of Elliptic Equations, Mathematical Surveys and Monographs, vol. 85, American Mathematical Society
2001.
BIBLIOGRAPHY
[107] Knödler D., Dieterich W. , Lonsky C., Nitzan A., “Nonlinear relaxation
and solvation dynamics in a Coulomb lattice gas”, J. Chem. Phys. 102
(1), pp. 465-470 (1995).
[108] Knopp K., Theory and Application of Infinite Series, Dover, NY (1990).
[109] Kramers H., “Brownian motion in a field of force,” Physica (Utrecht)
7, p. 284 (1940).
[110] Lachal A., “Sur le premier instant de passage de l’intégral du mouvement brownien,” Ann. Inst. Henri Poincaré 27 (3), pp. 385-405 (1991).
[111] Langevin P., C.R. Paris 146, p.530, (1908).
[112] Langouche F., Roekaerts D., and Tirapegui E., Functional Integration
and Semiclassical Expansions, (Mathematics & Its Applications), D.
Reidel, Dodrecht (1982).
[113] Lee Y.-T., Ree F. H., Ree T., “Distribution Function of Classical Fluids
of Hard Spheres. I.” J. Chem. Phys. 48 (8), pp. 3506-3516 (1968).
[114] Lehmann E. L., Testing Statistical Hypotheses, 2nd Ed., Springer Verlag (1997).
[115] Lehmann E. L., Casella G., Theory of Point Estimation, 2nd Ed.,
Springer Verlag (1998).
[116] Lur’e A. I., Three-Dimensional Problems of the Theory of Elasticity,
Interscience publishers, NY (1964).
[117] Magnus W., Oberhettinger F., Formulas and Theorems for the Special
Functions of Mathematical Physics, Chelsea Publishing Company, NY
(1949).
[118] Malenka R.C., Kauer J.A., Perkel D.J., Nicoll R.A., “The impact of
postsynaptic calcium on synaptic transmission–its role in long-term potentiation”, Trends Neurosci. 12 (11), pp.444-50 (1989).
[119] Mamonov A., Coalson R., Nitzan A., Kurnikova M., ”The role of the
dielectric barrier in narrow biological channels: a novel composite approach to modeling single channel currents”, Biophys. J. 84, pp.36463661 (2003).
277
278
BIBLIOGRAPHY
[120] Mandl P., Analytical treatment of one-dimensional Markov processes,
Springer Verlag, NY (1968).
[121] Marchewka A., Schuss Z., “Path integral approach to the Schrödinger
current”, Phys. Rev. A 61 052107 (2000).
[122] Markushevich A.I., Theory of Functions of a Complex Variable (3 Vols.
in 1), American Mathematical Society, 2nd edition (1985).
[123] Marshall T. W. and Watson E. J., “A drop of ink falls from my pen
... it comes to earth, I know not when,” J. Phys. A 18, pp. 3531–3559
(1985).
[124] Marshall T. W., Watson E. J., “The analytic solutions of some boundary layer problems in the theory of Brownian motion”, J. Phys. A: Math.
Gen., 20, pp. 1345-1354 (1987).
[125] Martin P.A., “Sum Rules in Charged Fluids,” Rev. Mod. Phys. 60, pp.
1076-1127 (1988).
[126] Matkowsky B. and Schuss Z., “The exit problem for randomly perturbed dynhamical systems”, SIAM J. Appl. Math. 33 (12), pp.365-382
(1977).
[127] McKean H.P. Jr., Stochastic Integrals, Academic Press, NY (1969).
[128] H.P. McKean, JR, “A winding problem for a resonator driven by a
white noise”, J. Math. Kyoto 2, pp.227-235 (1963).
[129] McQuarrie D.A., Statistical Mechanics, Harper and Row, NY (1976).
[130] Meeron E., J. Chem. Phys. 27, p. 1238 (1957).
[131] Mohazzabi P., Fournelle T.A., “Evaluation of Ill-Behaved Power Series”, The American Mathematical Monthly, 111, 4, pp. 308-321 (2004)
[132] Moy G., Corry B., Kuyucak S., and Chung S. H., “Tests of Continuum
Theories as Models of Ion Channels. I. Poisson-Boltzmann Theory versus
Brownian Dynamics,” Biophys. J. 78, pp. 2349–2363 (2000).
[133] Nadler B., Hollerbach U. and Eisenberg R.S., “Dielectric boundary
force and its crucial role in gramicidin”, Phys. Rev. E. 68 021905 (2003).
BIBLIOGRAPHY
[134] Nadler B., Naeh T., and Schuss Z., “The stationary arrival process of
independent diffusers from a continuum to an absorbing boundary is
Poissonian,” SIAM J. Appl. Math. 62, pp. 433–447 (2001).
[135] Nadler B., Naeh T., and Schuss Z., “Connecting a discrete ionic simulation to a continuum,” SIAM J. Appl. Math. 63 (3), pp. 850–873
(2003).
[136] Nadler B., Schuss Z., Hollerbach U., Eisenberg R.S., “Saturation of
conductance in single ion channels: The blocking effect of the near reaction field,” Physical Review E, 70 051912 (2004).
[137] Nadler B., Schuss Z., and Singer A., “Langevin trajectories between
fixed concentrations,” Phys. Rev. Lett. 94, 218101 (2005).
[138] Nadler B., Schuss Z., Singer A., Eisenberg R.S., “Ionic diffusion
through confined geometries: from Langevin equations to partial differential equations”, J. Phys.: Condens. Matter, 16, S2153-S2165 (2004).
[139] Naeh T., “Simulation of ionic solution,” Ph.D. dissertation, Department of Mathematics, Tel-Aviv University (2001).
[140] Nonner W. and Eisenberg R. S., “Ion permeation and glutamate
residues linked by Poisson-Nernst-Planck theory in l-type calcium channels,” Biophys. J. 75, pp. 1287–1305 (1998).
[141] Øksendal B., Stochastic Differential Equations, 5th ed., Springer,
Berlin Heidelberg (1998).
[142] Papanicolaou G.C., and Varadhan S.R.S, “Diffusion in Regions with
Many Small Holes”, in Proceedings of Vilnius Conference in Probability,
B. Grigelionis editor, Springer Lecture Notes in Control and Information
Sciences, vol. 25, pp. 190–206 (1980).
[143] Perrin J., C.R. Paris 146, p.967, (1908).
[144] Pinsky R. G., “Asymptotics of the principal eigenvalue and expected
hitting time for positive recurrent elliptic operators in a domain with
a small puncture”, Journal of Functional Analysis 200, 1, pp. 177-197
(2003).
279
280
BIBLIOGRAPHY
[145] Prudnikov A.P., Brychkov Y.A., Marichev O.I., Integrals and Series,
Vol. 1: Elementrary Functions, Gordon and Breach Science Publishers
(1986).
[146] Ramo S., “Currents induced by electron motion”, Proc. IRE. 27, pp.
584-585 (1939).
[147] Rayleigh J.W.S. Baron, The Theory of Sound, Vol. 2, 2nd Ed., Dover,
New York (1945).
[148] Rauch J., and Taylor M., “Potential and scattering theory on widely
perturbed domains,” J. Funct. Anal. 18 pp. 27–59 (1975).
[149] Razi Naqvi K., Mork K.J., Waldenstrøm K., “Steady-state one dimensional Fokker-Planck equation with an absorbing boundary: a half range
treatment”, Phys. Rev. A 40, pp. 3405-3047 (1989).
[150] Razi Naqvi K., Mork K.J., Waldenstrøm K., “Reduction of the FokkerPlanck Equation with an Absorbing or Reflecting Boundary to the Diffusion Equation and the Radiation Boundary Condition”, Phys. Rev.
Lett. 49, pp. 304-307 (1982).
[151] Ree F. H., Lee Y.-T., Ree T., “Distribution Function of Classical Fluids
of Hard Spheres. II.” J. Chem. Phys. 55 (1), pp. 234-245 (1971).
[152] Rice S. A., Lekner J., “On the Equation of State of the Rigid-Sphere
Fluid”, J. Chem. Phys. 42 (10), pp. 3559-3565 (1965).
[153] Rice S. A., Young D. A., “Equation of State of a Monatomic Fluid
with 6-12 Potential”, Dis. Faraday Soc. 43, pp. 16-25 (1967).
[154] Richardson J. M., “Variational Theory of the Radial Distribution Function”, J. Chem. Phys. 23 (12), pp. 2304-2308 (1955).
[155] Rosenkrantz R. D., Ed. Jaynes E.T., Papers on Probability, Statistics,
and Statistical Physics, Boston, Kluwer (1989).
[156] Rothschild V., Logothetis N. Probability Distributions, John Wiley,
New York (1985).
[157] Roux B., “Statistical mechanical equilibrium theory of selective ion
channels,” Biophysical Journal 77, pp. 139–153 (1999).
BIBLIOGRAPHY
[158] Rowley R. L., Statistical Mechanics for Thermophysical Property Calculations, Englewood Cliffs, NJ, PTR Prentice-Hall, (1994).
[159] Rowlinson J. S., Mol. Phys. 6, p. 517 (1963).
[160] Salpeter E. E., Ann. Phys. 5, p. 183 (1958).
[161] Saslow W. M., “Consider a spherical battery ...,” American Journal of
Physics 62, pp. 495–501 (1994).
[162] Saslow W. M., “What happens when you leave the car lights on
overnight: violation of local electroneutrality in slow, steady discharge
of a lead-acid cell,” Physical Review Letters 76, pp. 4849–4852 (1996).
[163] Saslow W. M., “Voltaic cells for physicists: two surface pumps and
an internal resistance,” American Journal of Physics 67, pp. 574–583
(1999).
[164] Schulman L.S., Techniques and Applications of Path Integrals, Wiley,
NY (1981).
[165] Schumaker M., “Boundary conditions and trajectories of diffusion processes”, J. Chem. Phys., 117, pp. 2469–2473, (2002).
[166] Schuss Z., Theory and Applications of Stochastic Differential Equations, Wiley, NY (1980).
[167] Schuss Z., Nadler B., and Eisenberg R. S., “Derivation of PNP equations in bath and channel from a molecular model,” Phys. Rev. E 64
(2-3) 036116 (2001).
[168] Schuss Z. and Spivak A., “On recovering the shape of a domain from
the trace of the heat kernel”, SIAM J. Appl. Math. (in press).
[169] Schutz C. and Warshel A., ”What are the dielectric constants of proteins and how to validate electrostatic models”, Proteins 44, pp.400-417
(2001).
[170] Selberherr S., Analysis and simulation of semiconductor devices, pp. 1–
293, Springer Verlag, New York (1984).
281
282
BIBLIOGRAPHY
[171] Selinger J. V. and Titulaer U. M., “The kinetic boundary layer for the
Klein-Kramers equation; a new numerical approach,” J. Stat. Phys. 36
(3/4), pp. 293–318 (1984).
[172] Shockley W., “Currents to conductors induced by moving point
charge”, J. Appl. Phys. 9, pp. 635-636 (1938).
[173] Sinai Y., “Distribution of some functionals of the integral of the Brownian motion”, Theoretical and Mathematical Physics (in Russian), 90,
pp.323-353 (1992).
[174] Singer A., “Maximum entropy formulation of the Kirkwood superposition approximation”, J. Chem. Phys., 121 (8), 3657-3666 (2004).
[175] Singer A. and Schuss Z., “Brownian simulations and uni-directional
flux in diffusion”, Phys. Rev. E 71 026115 (2005).
[176] Singer A., Schuss Z., Eisenberg R. S., “Attenuation of the electric potential and field in disordered systems”, J. Stat. Phys., (in press, available online).
[177] Singer A., Schuss Z., Holcman D., Eisenberg R.S., “Narrow Escape,
Part I: 3D”, (submitted for publication).
[178] Singer A., Schuss Z., Holcman D.,“Narrow Escape, part II: The circular
disk”, (submitted for publication).
[179] Singer A., Schuss Z., Holcman D., “Narrow Escape, Part III: Riemann
surfaces and non-smooth domains”, (submitted for publication).
[180] Singer A., Schuss Z., Nadler B., and Eisenberg R.S., “Memoryless control of boundary concentrations of diffusing particles”, Phys. Rev. E 70,
061106 (2004), also in Fluctuations and Noise in Biological, Biophysical,
and Biomedical Systems II, Eds. D. Abbot, S. M. Bezrukuv, A. Der, A.
Sanchez, SPIE Proc. 5467, pp. 345-358 (2004)
[181] Smoluchowski M.R. von Smolan, Rozprawy Kraków A46, p.257 (1906),
see also Ann. d. Physik 21, p.756 (1906).
[182] Smoluchowski M.R. von Smolan, Wien. Ber. 123, p.2381, (1914).
[183] Smoluchowski M.R. von Smolan, Phys. Zeits. 17, pp. 557 (1916).
BIBLIOGRAPHY
[184] Sneddon I. N., Mixed Boundary Value Problems in Potential Theory,
Wiley, NY (1966).
[185] Svedberg T., Z. Elektrochemie, 12, p.853 (1906).
[186] Svedberg T., Zeits. f. physik Chemie 77, p.147, (1911).
[187] Titulaer U. M., “A systematic solution procedure for the Fokker-Planck
equation of a Brownian particle in the high-friction case,” Physica A 91,
pp. 321–344 (1978).
[188] Uhlenbeck G.E., Hemmer P.C., and Kac M., “On the van der Waals
Theory of the Vapor-Liquid Equilibrium. II. Discussion of the Distribution Functions,” J. Math. Phys 4 229 (1963).
[189] van der Straaten T. A., Tang J. , Eisenberg R. S., Ravaioli U. , and
Aluru N. R., “Three-dimensional continuum simulations of ion transport
through biological ion channels: effects of charge distribution in the
constriction region of porin,” J. Computational Electronics 1, pp. 335–
340 (2002).
[190] van der Straaten T. A., Kathawala G., Kuang Z., Boda D., Chen D. P.,
Ravaioli U., Eisenberg R. S., and Henderson D., “Equilibrium structure of electrolyte calaculated using equilibrium monte carlo, molecular
dynamics, and transport monte carlo simulation,” Nanotech 3, p. 447
(2003).
[191] van der Straaten T. A., Kathawala G., and Ravaioli U., “Biomoca: A
transport monte carlo model for ion channels,” Journal of Computational Electronics 2, pp. 231–237 (2003).
[192] Vinogradov S. S., Smith P. D., Vinogradova E. D., Canonical Problems in Scattering and Potential Theory, Parts I and II, Chapman &
Hall/CRC (2002).
[193] Wang M. C. and Uhlenbeck G. E., “On the theory of the Brownian
motion II,” Rev. Mod. Phys. 17, pp. 323–342 (1945), also in N. Wax, ed.,
Selected Papers on Noise and Stochastic Processes, Dover, New York,
1954.
283
284
BIBLIOGRAPHY
[194] Weiss T. F., Cellular biophysics, p. 450, MIT Press, Cambridge, MA
(1996).
[195] Weisstein E.W. et al. ”Cantor Set.” From MathWorld–A Wolfram Web
Resource. http://mathworld.wolfram.com/CantorSet.html
[196] Westgren A., Arkiv for Matematik, Astronomi, och Fysik 11 (8,14)
(1916), 13, 14 (1918).
[197] Whittaker E. T., Watson G. N., A Course of Modern Analysis, Cambridge University Press, NY (1973).
[198] Widder M. E. and Titulaer U. M., “The moment method for boundary
layer problems in Brownian motion theory,” J. Stat. Phys. 56 (3/4),
pp. 471–498 (1989).
[199] Widder M. E. and Titulaer U. M., “The kinetic boundary layer around
an absorbing sphere and the growth of small droplets,” J. Stat. Phys.
55 (5/6), pp. 1109–1128 (1989).
[200] Wigger-Aboud S., Saraniti M., and Eisenberg R. S., “Self-consistent
particle based simulations of three dimensional ionic solutions,” Nanotech 3, p. 443 (2003).