Download Łukasz Kuśmierz Lévy flights in non

Jagiellonian University
Faculty of Physics, Astronomy and Applied Computer Science
Łukasz Kuśmierz
Lévy flights in non-equilibrium
thermodynamics and search strategies
A thesis submitted for the degree of
Doctor of Philosophy
supervised by prof. dr hab. Ewa Gudowska-Nowak
Kraków 2016
Wydział Fizyki Astronomii i Informatyki Stosowanej
Uniwersytet Jagielloński
Oświadczenie
Ja niżej podpisany Łukasz Kuśmierz (nr indeksu:
XXX) doktorant Wydziału
Fizyki, Astronomii i Informatyki Stosowanej Uniwersytetu Jagiellońskiego oświadczam, że
przedłożona przeze mnie rozprawa doktorska pt. “Lévy flights in non-equilibrium thermodynamics and search strategies” jest oryginalna i przedstawia wyniki badań wykonanych
przeze mnie osobiście, pod kierunkiem prof. dr hab. Ewy Gudowskiej-Nowak. Pracę
napisałem samodzielnie. Oświadczam, że moja rozprawa doktorska została opracowana
zgodnie z Ustawą o prawie autorskim i prawach pokrewnych z dnia 4 lutego 1994 r. (Dziennik Ustaw 1994 nr 24 poz. 83 wraz z późniejszymi zmianami). Jestem świadomy, że
niezgodność niniejszego oświadczenia z prawdą ujawniona w dowolnym czasie, niezależnie
od skutków prawnych wynikających z ww. ustawy, może spowodować unieważnienie stopnia nabytego na podstawie tej rozprawy.
Kraków, dnia .....................
..........................
podpis doktoranta
Streszczenie
Niniejsza praca doktorska dotyczy zastosowania lotów Lévy’ego w modelowaniu zjawisk
naturalnych. Loty Lévy’ego razem z procesem Wienera współtworzą klasę procesów αstabilnych, czyli procesów stochastycznymi o stacjonarnych i niezależnych przyrostach
losowanych z rozkładów stabilnych. Choć procesy α-stabilne stanowią naturalne uogólnienie procesu Wienera, loty Lévy’ego posiadają wiele odmiennych od procesu Wienera
własności, np. ciężkie ogony i nieskończoną wariancję przyrostów oraz nieciągłe trajektorie. Loty Lévy’ego charakteryzują się samopodobieństwem i skalują się z czasem szybciej
niż procesy dyfuzyjne, stąd są one standardowym elementem modeli anomalnego transportu (w szczególności superdyfuzji).
Przedstawione wyniki opierają się na czterech opublikowanych artykułach naukowych.
Dwa z nich rozważają jednowymiarowy model Langevina, w którym fluktuacje Lévy’ego
są włączone jako albo zewnętrzna siła losowa, albo fragment nierównowagowego rezerwuaru. Oba scenariusze odnoszą się do układu ściśle nierównowagowego. Mimo, że momenty rozkładów siły i transferu energii w takich układach nie istnieją, wykazano że
możliwe jest zastosowanie teorii liniowej odpowiedzi dla właściwie zdefiniowanych zmiennych sprzężonych. Wprowadzono różne definicje stochastycznych odpowiedników ciepła i
pracy, w zależności od fizycznej interpretacji szumu Lévy’ego występującego w równaniu.
Wielkości te analizowano dla cząstki w harmonicznej studni potencjału przesuwanej ze
stałą szybkością wskazując na anomalny charakter relacji flukutacyjnych.
W dwóch pozostałych artykułach loty Lévy’ego rozważane są w kontekście losowych
strategii poszukiwawczych.
Wprowadzono tam modele superdyfuzji z powrotami do
położenia początkowego (tzw. resetowanie); jeden z czasem ciągłym, drugi z czasem
dyskretnym. W obu przypadkach otrzymano całkowe wzory na średnie czasy dochodzenia
do pojedynczego celu oraz przeprowadzono optymalizację tych czasów w przestrzeni dwóch
parametrów: intensywności resetowania i indeksu stabilności lotów Lévy’ego. Pokazano,
że w modelu z czasem dyskretnym optymalne parametry jako funkcje odległości od celu
są nieciągłe. Zbadano także wpływ rozkładu prawdopobieństwa odległości od celu na
wynik optymalizacji czasu poszukiwania. Wykazano, że informacja dotycząca tylko pierwszego momentu tego rozkładu jest niewystarczająca przy planowaniu optymalnej strategii poszukiwawczej.
3
Abstract
This doctoral thesis deals with applications of Lévy flights in the modeling of natural
phenomena. Lévy flights together with the Wiener process comprise the class of α-stable
processes, i.e. stochastic processes with stationary and independent increments drawn
from stable distributions. Although α-stable processes provide a natural generalization
of the Wiener process, Lévy flights possess some features very distinct from the Wiener
process, e.g. fat tails and infinite variance of the increments and discontinuous trajectories.
Lévy flights are self-similar and they scale in time faster than diffusion processes, therefore
they serve as a standard building block of models of anomalous transport (in particular
superdiffusion).
Results presented in the thesis are based on four published articles. Two of them treat
a one-dimensional Langevin model in which Lévy fluctuations are incorporated as either
a random external forcing, or a non-thermal bath. These scenarios correspond to strictly
non-equilibrium systems. Even though moments of force and energy transfer distributions
in these systems do not exist, it is shown that it is possible to apply the linear response
theory for properly defined conjugate variables. Different definitions of stochastic heat
and work are given, depending on the physical interpretation of the Lévy noise in the
equation. For a particle in a harmonic potential well moving with a constant velocity,
these quantities are analyzed pointing to the anomalous character of fluctuation relations.
In the other two articles Lévy flights are considered in the context of random search
strategies. Discrete- and continuous-time models of superdiffusion with resetting are
introduced. In both cases integral expressions for mean search times of a single target are
obtained and optimizations of these times are performed in the space of two parameters:
resetting intensity and stability index of the Lévy flight. It is shown that in the discretetime case the optimal parameters as functions of a distance to the target are discontinuous.
An influence of a distribution of a randomized distance to the target on the optimization
is also investigated. It is demonstrated that the information regarding solely the first
moment of this distribution is insufficient for the efficient search strategy planning.
4
Contents
List of publications
8
1 Introduction and motivation
9
1.1
Lévy flights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
Non-equilibrium thermodynamics . . . . . . . . . . . . . . . . . . . . . . . 10
1.3
Search strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Basic concepts and notions
2.1
2.2
14
Stochastic calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.1
Stochastic differential equations and noise-generating processes . . . 14
2.1.2
Stochastic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.3
Colored noise and Wong-Zakai theorem . . . . . . . . . . . . . . . . 19
Lévy α-stable distributions and Lévy α-stable processes . . . . . . . . . . . 21
2.2.1
Stable distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.2
Anomalous diffusion and Lévy flights . . . . . . . . . . . . . . . . . 23
2.2.3
Generating α-stable random numbers and processes on a computer
3 Summary of the papers
3.1
3.2
3.3
3.4
9
28
32
Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1.2
Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1.3
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.2
Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.3
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3.2
Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3.3
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6
CONTENTS
3.4.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4.2
Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4.3
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4 Conclusion
49
Acknowledgments
50
Appendices
51
A
B
Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
A.1
Paper I
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
A.2
Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
A.3
Paper III
A.4
Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Marcus canonical equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
B.1
Recipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
B.2
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
C
Lévy flights and detailed balance . . . . . . . . . . . . . . . . . . . . . . . 130
D
First arrival and first passage times . . . . . . . . . . . . . . . . . . . . . . 132
E
Continuous versus discrete jump distribution . . . . . . . . . . . . . . . . . 133
F
Discussion of α ≤ 1 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Bibliography
138
7
List of publications
Papers included in this thesis
I. L. Kusmierz, W. Ebeling, I. Sokolov, and E. Gudowska-Nowak. “Onsagers Fluctuation Theory and New Developments Including Non-equilibrium Lévy Fluctuations.”
In: Acta Physica Polonica B 44 (2013), p. 859.
II. L. Kusmierz, J. M. Rubi, and E. Gudowska-Nowak. “Heat and work distributions
for mixed Gauss-Cauchy process.” In: Journal of Statistical Mechanics: Theory and
Experiment 9 (2014), p. 002.
III. L. Kusmierz, S. N. Majumdar, S. Sabhapandit, and G. Schehr. “First order transition for the optimal search time of Lévy flights with resetting.” In: Physical Review
Letters 113.22 (2014), p. 220602.
IV. Ł. Kuśmierz and E. Gudowska-Nowak. “Optimal first-arrival times in Lévy flights
with resetting.” In: Physical Review E 92.5 (2015), p. 052127.
Coauthored papers not included in this thesis
V. K. Trojanowski, D. Allender, L. Longa, and Ł. Kuśmierz. “Theory of Phase Transitions of a Biaxial Nematogen in an External Field.” In: Molecular Crystals and
Liquid Crystals 540.1 (2011), pp. 59–68.
VI. H. Goh, Ł. Kuśmierz, J.-H. Lim, N. Thome, and M. Cord. “Learning invariant
color features with sparse topographic restricted Boltzmann machines.” In: Image Processing (ICIP), 2011 18th IEEE International Conference on. IEEE. 2011,
pp. 1241–1244.
VII. B. Lisowski, L. Kusmierz, M. Zabicki, and M. Bier. “"Cargo-mooring" as an operating principle for molecular motors.” In: Journal of Theoretical Biology 374 (2015),
pp. 13–25.
8
Chapter 1
Introduction and motivation
This thesis is organized as follows. The present Chapter briefly reviews and states the
relations between the key concepts that are used throughout the thesis. Chapter 2 introduces definitions, notation, and some basic features related to stochastic processes, with
a particular emphasis on Lévy flights. In Chapter 3 the attached articles are described
in, pointing to the motivation for the research, methods applied, and the most important
results obtained in the study. Finally, Chapter 4 concludes the main body of the thesis
with a succinct list of possible extensions of the analyzed models.
The attached articles, already published in peer-reviewed journals, constitute an integral part of the thesis. Appendices complement them with some additional results and
comments.
1.1
Lévy flights
More than 100 years ago, famous works by Bachelier [Bac00], Einstein [Ein05], Smoluchowski [Smo06], and Langevin [Lan08] started a completely new branch of mathematics,
dealing with randomness changing in time. Bachelier studied price fluctuations in stock
markets, whereas the others were trying to explain random movements of organic and inorganic particles suspended in water, which had been observed in 1827 by Brown [Bro28].
These different phenomena turned out to be well described by the same mathematical
object: a stochastic process, today known as the Brownian motion (also, the Wiener
process). Moreover, the equation governing the time-evolution of the probability density
function (PDF) of the Brownian motion belongs to the same class as phenomenological
Fick’s law of diffusion [Fic55] and the heat equation [Fou78].
This remarkable universality of the Wiener process is related to the central limit
theorem: an infinite sum of statistically independent, infinitesimal random numbers with
finite variance is distributed according the Gaussian law. Imagine a discrete-time random
walker, with each step (a displacement) drawn from a given distribution (with finite
9
1.2. NON-EQUILIBRIUM THERMODYNAMICS
variance) independently from the past. If the number of steps per unit of time becomes
infinite, the effective displacement in any finite time becomes Gaussian. Thus, increments
of the (continuous-time) Brownian motion have to be Gaussian.
The existence of a finite variance is essential in the above assertion. After waiving this
assumption, a whole family of possible distributions is identified in the generalized central
limit theorem [GK54] and named Lévy α-stable distributions. Similarly, stochastic processes generated from independent increments distributed according to α-stable laws are
called α-stable processes. Non-Gaussian α-stable processes are also called Lévy flights1 .
Lévy flights have already been applied as models of turbulence [SWK87], financial
markets [M+95], quantum systems coupled to chaotic subsystems [KBD99], and foraging
animals (see Section 1.3), thus they seem to be rather ubiquitous. This is not surprising, since they are grounded on the generalized central limit theorem. Additionally, the
detailed balance condition does not hold for Lévy flights in a harmonic potential (cf.
Appendix C), i.e. their presence has to be associated with a non-equilibrium situation.
We may therefore expect that they should serve as a natural model in non-equilibrium
thermodynamics.
1.2
Non-equilibrium thermodynamics
Establishing the connection between classical thermodynamics and probabilistic description of many microscopic degrees of freedom has been a great success of statistical mechanics. From its very beginnings, much effort in statistical mechanics has been devoted to constructing a practical theory of systems out of equilibrium, starting from the
Maxwell-Stefan equations [Max67; Ste71; Bot11] and the celebrated Boltzmann transport
equation, devised by Ludwig Boltzmann in 1872 [Kre10; KTH12]. Further work has been
concentrated around the linear response of an initially equilibrated system to an external perturbation. One of the most important contributions has been made by Onsager
[Ons31a; Ons31b], who, assuming the principle of microscopic reversibility, derived reciprocal relations occurring between pairs of generalized forces and fluxes. Another classical
result is embodied in the Green-Kubo relations [Gre54; Kub57]. They link transport
coefficients with integrals of equilibrium time-correlation functions [Zwa65] and can be
seen as a generalization of the Einstein-Smoluchowski relation, which connects the diffu1
The name “Lévy flight” was coined by Mandelbrot [Man83].
10
1.2. NON-EQUILIBRIUM THERMODYNAMICS
sion coefficient with the mobility [Ein05; Smo06]. These results are summarized in the
fluctuation-dissipation theorem, which, for quantities of interest, links the linear response
to external perturbations with equilibrium time correlations of their spontaneous fluctuations [Kub66; Mar+08]. A reformulation of non-equilibrium statistical mechanics in terms
of the projection operator formalism has been developed by Zwanzig [Zwa60; Zwa61] and
Mori [Mor65].
More recently, a series of new results have been proven for systems out of equilibrium.
They relate equilibrium quantities, such as temperature of heat bath and free energy, with
fluctuations of work, heat, and entropy production, obtained from irreversible processes
far from equilibrium, thus their name: fluctuation relations (also fluctuation theorems).
Importantly, these results should be still valid for “small” 2 systems, such as molecular
motors, quantum dots, and colloidal particles. There are many fluctuation relations,
relating different functionals of the corresponding stochastic trajectories and proven in
different setups. Most of them can be interpreted as a generalization of the second law of
thermodynamics and inequalities known from thermodynamics of irreversible processes
can be derived from them. Remarkably, these inequalities are superseded by equalities of
distributions or averages.
As an example, let us quote the Jarzynski equality [Jar97]. It relates free energy
differences (∆F ) between two equilibrium states at the same temperature T and work
(W ) performed on the system during a (possibly irreversible) process joining these two
states:
he−W/kT i = e−∆F/kT .
(1.1)
Here k is the Boltzmann constant and hi denotes the ensemble average. From (1.1)
combined with Jensen’s inequality [Nee93] we get
hW i ≥ ∆F,
(1.2)
which is known in thermodynamics. The Jarzynski equality can be derived from the
Crooks fluctuation theorem [Cro99], which relates the probability of a trajectory to the
probability of the time-reversal of this trajectory. Other fluctuation theorems describe
fluctuations of entropy production, which have been first observed in numerical inves2
By small we mean systems in which thermal fluctuations are already very important—energy scales
should be of the order of kB T . Typically, in experiments verifying fluctuation relations, length scales of
such systems range from 1 to 1000 nanometers [BLR05].
11
1.3. SEARCH STRATEGIES
tigations using a Gaussian thermostat [ECM93], e.g. the transient fluctuation theorem
by Evans and Searles [ES94] and the steady-state fluctuation theorem by Gallavotti and
Cohen [GC95a; GC95b]. A simpler proof of the latter theorem for systems undergoing
Langevin dynamics has been given by Kurchan [Kur98]. This approach has been further
extended by Hatano and Sasa [HS01] and Seifert [Sei05], who have precisely defined microscopic analogues of all relevant macroscopic quantities, including entropy. Sagawa and
Ueda [SU10] have further generalized the Jarzynski equality to non-equilibrium systems
under feedback control. Many of the theoretically derived fluctuation relations have been
verified experimentally, see e.g. [CL98; Wan+02; Wan+05; Col+05; Sch+05; Imp+07;
Bat+14; An+15].
The theory of fluctuation relations provides an exciting subject of research. It is an
actively developing field and its recent achievements have been reported in a series of
review articles [ES02; Tou09; Jar11; Sei12] and books [ME07; Sek10; Sch+13].
1.3
Search strategies
People seek something all the time: food, lovers, clothes, misplaced keys, television channels, etc. Answering our needs by performing a search is a recurrent theme in our lives,
and so search strategies are of a significant interest in both science and technology. In
the age of information technology the importance of search engines (e.g. web search engines) cannot be overestimated. Search strategies have also been intensively studied in the
context of animal foraging. Since in many cases sensing skills, knowledge, memory and
processing power of a searcher (e.g. foraging animal) are very limited, random processes
are often used to model its motion. However, patterns observed in animal movement do
not fit the prediction of the Brownian motion [BB88; Bla97; BF08].
Lévy flights (LFs), along with similar Lévy walks (LWs)[SK86], have been extensively
studied in the context of random search strategies. After the pioneering theoretical work
by Shlesinger, Klafter, and collaborators [SK86; SWK87; KSZ96], LFs and LWs have been
hypothesized to correctly model motion of foraging animals, e.g. albatrosses [Vis+96],
deer, bumblebees [Vis+99], jackals [Atk+02], spider monkeys [Ram+04], fruit flies [RF07],
honey bees [Rey+07], marine predators [Sim+08], and extinct sea urchins [Sim+14]. Both
LFs and LWs have even been claimed to be observed in human mobility patterns [BHG06;
Rhe+11]. Additionally, LFs have been proven to be optimal in some setups [Vis+99;
Bar+02; Rap+03; Bar+05].
12
1.3. SEARCH STRATEGIES
In another framework, proposed as an alternative to Lévy flight strategies [Bén+05;
Bén+06], a search process is divided into two distinct phases: a slow-motion phase, allowing the searcher to detect a target, and a fast-motion phase, during which the target cannot
be observed. Such models are known as the intermittent search strategies [Bén+11]. As
shown in [Lom+08], even in this framework LWs are still preferable, i.e. they optimize
the expected search time.
These findings, although questioned by some authors [Boy+06; Edw+07; Ben07], have
inspired many theoretical studies [Rey05; LAM05; Bén+06; Lom+08; PCM14; Zha+15]
and also have led to the construction of biologically inspired computer search algorithms
(i.e. stochastic optimization procedures), such as LFs-based evolutionary algorithms
[Gut01; Obu03], Cuckoo Search [YD09], Lévy-flight firefly algorithm [Yan10], Eagle strategy [YD10], Bat algorithm [YH13], Lévy-flight krill herd algorithm [Wan+13], and a blend
of LFs and simulated annealing [Pav07].
13
Chapter 2
Basic concepts and notions
In this part we review some basic definitions that are used in the remaining of the thesis.
The notation introduced in this section is consequently used throughout the main body
of the thesis and its appendices, but it varies across the attached papers.
Most of the definitions briefly reviewed in the forthcoming paragraphs refer to basic introductions to stochastic processes applied in natural sciences presented in [HL84;
Gar+85; Van92] and to a more formal treatment discussed in mathematical textbooks
[KT98; Pro04; Ros06; App09]. In particular, references [MK00; MK04; DSU08; ST94;
BP03; KS11] provide introductory texts on Lévy flights and other models of anomalous
diffusion. More information regarding numerical methods for simulating α-stable processes can be found in [JW93].
2.1
2.1.1
Stochastic calculus
Stochastic differential equations and noise-generating processes
A stochastic differential equation (SDE) is formally an integral equation of the following
form
Xt = X0 +
Zt
a(Xs , s)ds +
0
Zt
b(Xs , s)dZs ,
(2.1)
0
where a(·, ·) and b(·, ·) are given functions and (Xt )t≥0 (also written as (Xt , t ≥ 0) or
{Xt , t ≥ 0}) is an unknown stochastic process with a given initial condition X0 . Large
letters indicate that, at a given instance of time, Xt and Zt are random variables. The
last term on the RHS, often referred to as the noise term, is a stochastic integral with
respect to some given stochastic process (Zt )t≥0 . In the special case of b(Xt , t) = const.
the noise is called additive, otherwise it is called multiplicative. Note that in the latter
case the stochastic integral has to be carefully defined (see Section 2.1.2).
14
2.1. STOCHASTIC CALCULUS
Let us now recall stochastic processes which are typically used to model the noise. In
each case, we assume that the process is cádlág (right continuous with left limits) and that
the natural filtration has been chosen, so that the process is adapted (non-anticipating).
Additionally, we restrict our attention to real-valued processes.
Definition 1. A stochastic process (Wt )t≥0 is called a Wiener process (or one dimensional
standard Brownian motion) iff (if and only if ) the following conditions hold:
(1) W0 = 0,
(2) increments of the process are stationary and independent of the past,
(3) increments of the process are distributed according to the Gaussian probability distribution function (PDF):
x2
1
fWt+τ −Wt (x) = √
e 2τ ,
2πτ
(2.2)
(4) the function t → Wt is almost surely continuous.
Noise generated by the Wiener process is called white Gaussian noise.
Definition 2. A stochastic process (Nt )t≥0 is called a (counting) Poisson process with
intensity (or rate) λ > 0 iff the following conditions hold:
(1) N0 = 0,
(2) increments of the process are stationary and independent of the past,
(3) increments are distributed according to the Poisson distribution
P (Nt+τ − Nt = n) =
(λτ )n −λτ
e .
n!
(2.3)
Noise generated by the Poisson process is called shot noise. From the third condition
it is evident that the Poisson process attains values in natural numbers only. Since it
is a pure jump process and jumps are always equal to 1, the Poisson process is fully
characterized by its arrival times T1 , T2 , ..., i.e. times at which jumps occur:
Nt =
∞
X
1[Tk ,∞) (t),
k=1
15
(2.4)
2.1. STOCHASTIC CALCULUS
where 1A (t) = 1 iff t ∈ A. Waiting times are exponentially distributed
fTk+1 −Tk (t) = λe−λt ,
(2.5)
which can be used in the definition of the Poisson process instead of the third condition.
A compensated Poisson process (Mt )t≥0 is a centered version of the Poisson process Mt =
Nt − λt. Similarly to the Wiener process it is a martingale and it has a zero expected
value for all t ≥ 0.
Definition 3. Let (Nt )t≥0 be the Poisson process with arrival times T1 , T2 , ... and let
(Jk )k∈N+ be a sequence of i.i.d. (independent and identically distributed) random variables
independent from (Nt )t≥0 . Then a stochastic process given by
Pt =
Nt
X
k=1
Jk =
∞
X
Jk 1[Tk ,∞) (t)
(2.6)
k=1
is called a compound Poisson process.
Noise generated by a compound Poisson process consists of random kicks with intensities distributed according to some given PDF. Times between consecutive kicks are
exponentially distributed, which implies that at a given instance of time, a waiting time
for the next kick is independent from a time that has already passed till the last kick.
There are two other notations that are often used for the description of stochastic
processes. In the first one, SDE is written in a differential form, wherein the initial
condition and integrals are omitted:
dXt = a(Xt , t)dt + b(Xt , t)dZt .
(2.7)
In physics literature, SDEs are mostly written in a simplified form (referred to as a
Langevin equation [Van92; Gar+85])
ẋt ≡
dxt
= a(xt , t) + b(xt , t)ξt ,
dt
(2.8)
where ξt = Żt is a time derivative of the noise-generating process. If (Zt )t≥0 is the
Wiener process (Zt = Wt ), ξt is called white Gaussian noise. Trajectories of the Wiener
process are (almost surely) nowhere differentiable and have infinite length on any finite
time interval. Therefore, white Gaussian noise has to be understood in a generalized
16
2.1. STOCHASTIC CALCULUS
function (distribution) sens. This is related to the fact that infinitely many nonequivalent
definitions of a stochastic integral are possible—see Section 2.1.2 for details.
2.1.2
Stochastic integrals
In the following we are interested in calculating integrals of the form
R
f (x)dg(x). Two
basic definitions of an integral are in use in standard calculus: the Riemann-Stieltjes integral and the Lebesgue-Stieltjes integral. The latter is more general, nevertheless both lead
to the same results when the former exists. Among the assumptions used in definitions
of these integrals is the bounded variation of g(x). However, trajectories of the Brownian
motion are very irregular—they are nowhere differentiable and are of unbounded variation
almost surely. Other stochastic processes which we may want to integrate with respect to
are even more erratic. Therefore, a different definition of stochastic integration is needed.
Here we define three stochastic integrals that are most commonly used in practice. We
also include a few examples of simple integrals with respect to the processes introduced
in the previous section.
The Itô stochastic integral is defined as [Gar+85; Pro04; App09]
Zt
0
Xs dZs = lim
n→∞
n
X
k=1
Xtk−1 Ztk − Ztk−1 ,
(2.9)
where 0 = t0 < ... < tn = t and max0≤k≤n |tk − tk−1 | → 0 when n → ∞. We will assume
a deterministic, regular mesh, although one can take random partitions instead. To see
how it works in practice we calculate a simple example of the Itô integral with respect to
the Wiener process:
Zt
0
Ws dWs = lim
n→∞
n
X
k=1
W(k−1) nt Wk nt − W(k−1) nt =
n X
1
= lim
2W(k−1) nt + Wk nt − Wk nt
Wk nt − W(k−1) nt =
2 n→∞ k=1
n 2 W 2
X
W2 W2 1
t
= t − 0 − lim
Wk nt − W(k−1) nt = t −
(2.10)
2
2
2 n→∞ k=1
2
2
The term marked with blue color is called a quadratic variation and plays an important
17
2.1. STOCHASTIC CALCULUS
role in the theory of stochastic calculus. It is quite easy to show that the quadratic
variation of the Wiener process is equal to t [Pro04]. Let us calculate an example of the
Itô integral with respect to the Poisson jump process
Zt
0
Ns dNs =
X
s:∆Ns
1 X
1
1
1
Ns− ∆Ns = Nt2 −
(∆Ns )2 = Nt2 − Nt ,
2
2 s:∆N 6=0
2
2
6 0
=
(2.11)
s
where Ns− = limt→s− Nt . In contrast to the Wiener process, the quadratic variation of
the Poisson counting process is a random variable.
Note that in the Itô integral a value of the process (Zt )t≥0 in the sum is taken from
the beginning of the interval [ti , ti+1 ]. In general, one could take any intermediate point
τi ∈ [ti , ti+1 ]. For example, the anti-Itô integral is defined as [Pes+13]
Zt
Xs · dZs = lim
n→∞
0
n
X
k=1
Xtk Ztk − Ztk−1 ,
(2.12)
where the limit is understood in the same way as in the Itô integral definition (2.9).
Calculations similar to these performed before (2.10) give
Zt
Ws · dWs =
Wt2
t
+ ,
2
2
(2.13)
0
which is clearly different from the Itô integral result (2.10). In fact, there are infinitely
many possible results of this simple integral, depending on the particular choice of the
intermediate point τi [HL84]. Thus, when integrating with respect to the Wiener process,
a specific τi has to be chosen. In contrast, if the Riemann-Stjelties integral exists, it is
independent of this choice. The integral that we choose is a part of a model itself and
not of a physical reality [Van92], although there is still some debate regarding this issue
[Vol+10; MM11; Vol+11].
The Stratonovich integral is defined as1 [Gar+85; App09; CP14]
Zt
0
n
X
Xtk−1 + Xtk
Xs ◦ dZs = lim
(Ztk − Ztk−1 ),
n→∞
2
k=1
1
(2.14)
Note that this is different from the Fisk-Stratonovich integral [Pro04]. These two definitions are
equivalent for the Wiener process only.
18
2.1. STOCHASTIC CALCULUS
where the limit is understood in the same way as in the Itô integral definition (2.9).
We can easily calculate an exemplary Stratonovich integral with respect to the Wiener
process:
Zt
Ws ◦ dWs = lim
0
n→∞
n
X
Wk nt + W(k−1) nt 2
k=1
Wk nt − W(k−1) nt
=
Wt2 W02
W2
−
= t . (2.15)
2
2
2
The Stratonovich integral is convenient in applications because it preserves standard rules
of ordinary differential calculus (chain rule, Leibniz’s rule etc.) when applied to integrals
with respect to the Wiener process [Pro04; App09]. In particular
Zt
Wsn
Wtn+1
◦ dWs =
.
n+1
(2.16)
0
This is, however, not true for jump processes, as can be seen from the following examples:
Zt
Nt2
,
2
(2.17)
Nt3 Nt
+
.
3
6
(2.18)
Ns ◦ dNs =
0
Zt
Ns2 ◦ dNs =
0
Typically, in applications one assumes that rules of ordinary calculus hold, which for the
Wiener process implies that Stratonovich integral should be used. Another reason why
physicists prefer to employ Stratonovich integral is discussed in Section 2.1.3.
2.1.3
Colored noise and Wong-Zakai theorem
Let us define a new process as a low-pass filtered Wiener process (known as integrated
Ornstein-Uhlenbeck process):
τ Ẇtτ = −Wtτ + Wt .
(2.19)
Increments of (Wtτ )t≥0 are not independent, thus its time derivative is called colored Gaussian noise. In contrast to the Wiener process, trajectories of (Wtτ )t≥0 are differentiable
and its quadratic variation disappears. Therefore, Itô and Stratonovich integrals with
19
2.1. STOCHASTIC CALCULUS
respect to colored Gaussian noise are equivalent and the standard chain rule holds. With
10
10
0.05<q<0.95
0.25<q<0.75
q=0.5
0.05<q<0.95
0.25<q<0.75
q=0.5
5
quantiles of Xt
quantiles of Xt
5
0
0
−5
−10
0
−5
5
10
−10
0
15
5
10
t
150
150
0.05<q<0.95
0.25<q<0.75
q=0.5
0.05<q<0.95
0.25<q<0.75
q=0.5
100
quantiles of Xt
quantiles of Xt
100
50
0
−50
−100
0
15
t
50
0
−50
5
10
−100
0
15
5
t
10
15
t
Figure 2.1: Coloring of the Wiener (top) and the Cauchy (bottom) processes. In both cases
processes with independent increments (left) are filtered according to Eq. (2.19) with τ = 0.2
(right). Along with ten sample paths, median (red) and quantile areas (gray and violet) have been
plotted.
colored noise we can define a new process as a solution of the following SDE
Xtτ
=x+
Zt
a(Xsτ )ds
0
+
Zt
b(Xsτ )Ẇsτ ds.
(2.20)
0
The Wong-Zakai theorem [WZ65; CP14] states that in the limit of infinitesimal correlation
time, the process process Xtτ converges to the solution of the Stratonovich SDE:
lim Xtτ = Xto .
τ →0
20
(2.21)
2.2. LÉVY α-STABLE DISTRIBUTIONS AND LÉVY α-STABLE PROCESSES
Colored jump noises also preserve standard rules of calculus. However, the limit of disappearing correlation time in general does not lead to the Stratonovich integral. In the case
of jump processes another type of stochastic integral, a so-called Marcus integral, arises
from that limit. For more information regarding the Marcus integral, see Appendix B.
2.2
Lévy α-stable distributions and Lévy α-stable processes
2.2.1
Stable distributions
Definition 4. Let X be a random variable. The cumulative distribution function (CDF)
of X is defined as follows:
FX (x) = P (X ≤ x),
(2.22)
i.e. it is a probability that the random variable X takes on a value not greater than x. An
inverse of CDF:
QX (q) = FX−1 (q)
(2.23)
is called the quantile function of X. The probability distribution function (PDF) of X is
defined as follows:
fX (x) =
d
FX (x).
dx
(2.24)
The characteristic function of X is given by the Fourier transform of its PDF:
ΦX (k) = E[e
ikX
]=
Z
eikx fX (x)dx.
(2.25)
R
Definition 5. We say that two random variables X and Y are equivalent iff they differ
only on a null set, i.e. all possible probabilities associated with X and Y are the same.
We denote the equivalence between two random variables by X ∼ Y .
Definition 6. Let X1 and X2 be i.i.d. random variables. The distribution they share is
said to be stable iff:
∀a, b > 0 ∃c, d ∈ R ∀x : FaX1 +bX2 (x) = FcX1 +d (x).
If d = 0 for every pair (a, b) then the distribution is said to be strictly stable.
21
(2.26)
2.2. LÉVY α-STABLE DISTRIBUTIONS AND LÉVY α-STABLE PROCESSES
The family of all distributions which satisfy the requirement of stability has been
constructed by Paul Lévy [Lév25; Man60]. These distributions are referred to as α-stable
distributions or Lévy α-stable distributions.
Definition 7. Let α ∈ (0, 2] and β ∈ [−1, 1]. A random variable S(α, β) is called α-stable
iff its characteristic function is given by the formula

−|k|α 1 − iβsgn(k) tan πα , α 6= 1
2
ln ΦS(α,β) (k) =
−|k| 1 + iβsgn(k) 2 log |k| , α = 1
π
(2.27)
where sgn(·) is the sign function. In the special case of β = 0 this formula simplifies to
ln ΦS(α,0) (k) = −|k|α
(2.28)
and the corresponding S(α, 0) is called a symmetric α-stable random variable.
We give here a representation of standard α-stable distributions with two parameters
α (characteristic exponent or stability index) and β (skewness). The full family of αstable distributions is parameterized by four numbers, but two of them are responsible for
trivial affine transformations, i.e. shift and scale. Note also that there are many different
representations of the family of α-stable distributions. The one presented here and used
throughout the thesis follows Samoradnitsky and Taqqu [ST94] and is most commonly
used in practice. In this notation the standard normal variable Z is not equal to S(2, β)
√
(but Z ∼ S(2, β)/ 2). Other parametrizations with a discussion of their properties can
be found in [Wer96].
There are only three special cases for which a PDF of the α-stable random variable
can be expressed in terms of elementary functions. These are:
• Gaussian distribution (β is irrelevant)
x2
1
fS(2,β) = √ e− 4
4π
• Cauchy distribution
fS(1,0) =
22
1 1
π x2 + 1
(2.29)
(2.30)
2.2. LÉVY α-STABLE DISTRIBUTIONS AND LÉVY α-STABLE PROCESSES
• Lévy distribution
1
fS( 1 ,1)
2
1 e− 2x
=√
.
2π x3/2
(2.31)
One of the most striking features of α-stable distributions is a lack of finite moments, i.e.
second and higher moments are infinite or undefined for α < 2, whereas for α ≤ 1 the
distributions do not even have the first moment finite. This is related to the asymptotic
power-law behavior at ±∞, which is often referred to as fat or heavy tails.
The classical central limit theorem (CLT) states that distributions of normalized sums
of i.i.d. random variables with a finite variance converge (with a number of random
variables in the sum going to infinity) to a normal distribution. In a generalized central
limit theorem (GCLT) proven by Gnedenko and Kolmogorov [GK54], the finite variance
assumption is dropped and all possible attractors are identified with the family of α-stable
distributions.
Theorem 1 (Generalized central limit theorem). Let (Xi )i∈N be a sequence of i.i.d. ran-
dom variables. There exist constants an > 0, bn ∈ R and a nondegenerate random variable
Z for which the following convergence in distribution is true
an
n
X
i=1
d
Xi − bn = Sn −−−→ Z.
n→∞
(2.32)
Moreover, Z belongs to the family of α-stable random variables. The stability index α < 2
is achieved if the PDF of Xi has tails behaving as |x|−α−1 .
2.2.2
Anomalous diffusion and Lévy flights
One of the most famous equations associated with the Brownian motion, is the one describing the time-dependence of its variance
E[Wt2 ] = t.
(2.33)
A diffusion process is the solution of a SDE with white Gaussian noise:
dXt = a(Xt , t)dt + b(Xt , t)dWt .
23
(2.34)
2.2. LÉVY α-STABLE DISTRIBUTIONS AND LÉVY α-STABLE PROCESSES
0.3
α=2.0
α=1.8
α=1.6
−1
0.25
log10fα(x)
fα(x)
0.15
0.1
−2
log10fα(x)
−2
0.2
−1
−3
−4
−5
0.05
−5
0
x
5
−3
−4
−5
−6
0
5
10
x
15
20
−6
0
0.5
1
1.5
log10x
Figure 2.2: Symmetric α-stable distributions for α = (2; 1.8; 1.6). Differences between normal
distribution and others α-stable distributions are most pronounced in their tails and may be
overlooked at first sight (left). Tails are best visible in log (middle) or log-log (right) scales.
In all diffusion processes the linear time-dependence of variance is preserved in short time
scales. Therefore, Eq. (2.33) is sometimes used as a definition of diffusion.
In contrast, anomalous diffusion is expected to have a nonlinear scaling of variance
with time
E[Xt2 ] = tδ .
(2.35)
For δ < 1 the process is called subdiffusive, whereas for δ > 1 it is called superdiffusive.
In this thesis we restrict our attention to symmetric α-stable processes as models of
superdiffusive motion. Here we only mention that asymmetric, strictly increasing αstable processes can also be employed as so-called inverse-time subordinators to model
subdiffusion [PSW05; MWW07]. Another well-known example of an anomalous diffusion
model is the fractional Brownian motion [MV68], in which the increments are Gaussian
but not independent. It can serve as both super- and subdiffusion depending on the Hurst
exponent.
(α)
Definition 8 (Lévy flight). A stochastic process (Lt )t≥0 with α ∈ (0, 2] is called a
symmetric Lévy α-stable process (or Lévy flight if α < 2) iff the following conditions hold:
(α)
(1) L0 = 0,
(2) increments of the process are stationary and independent of the past,
(3) increments of the process are α-stable with β = 0:
α
ΦL(α) −L(α) (k) = e−τ |k| .
t+τ
t
24
(2.36)
2.2. LÉVY α-STABLE DISTRIBUTIONS AND LÉVY α-STABLE PROCESSES
The characteristic function of the Lévy α-stable process at a given instance of time is
given by a time-rescaled α-stable characteristic function
α
ΦL(α) (k) = e−t|k| .
(2.37)
t
In the special case of α = 1 the corresponding α-stable process is denoted by (Ct )t≥0 and
√
(2)
called a Cauchy process. It is easy to see that in our notation (Lt )t≥0 ∼ 2(Wt )t≥0 .
Although Lévy flights are recognized as models of superdiffusion, their variance is
infinite thus one cannot directly apply Eq. (2.35). LFs, however, have another useful
property—they are self-similar, i.e.
(α)
Lt
(α)
∼ t1/α L1 .
(2.38)
We can thus say that LFs (α < 2) scale faster with time than the Brownian motion
(α = 2) and call them superdiffusive. In general the quantile function can be used as a
measure of scaling with time. In order to account for a possible drift, the interquartile
range
?
g(t) = QXt (3/4) − QXt (1/4) ∝ tδ/2
(2.39)
can employed. If the proportionality is true, δ has the same interpretation as in Eq.
(2.35). The quantile function has been utilized to visualize processes in many figures in
this thesis, see for example Fig. 2.3 where a few examples of α-stable processes have been
compared.
Theorem 2. Sample paths of Lévy α-stable processes (i.e. realizations of the function
(α)
t → Lt ) are almost surely discontinuous for α < 2 and almost surely continuous for
α = 2.
Proof. We start by approximating the α-stable stochastic process as a sum of discrete(α)
time jumps (increments). At some fixed time t the random variable Lt
a finite sum
(α)
Lt
=
N
X
Ii (t/N ),
is thus given by
(2.40)
i=1
where Ii (τ ) ∼ τ 1/α S(α, 0), which follows from Eq. (2.36). The α-stable process is recovered in the limit N → ∞.
Let l > 0 and t > 0 be fixed numbers and let Mt be a random variable equal to the
25
2.2. LÉVY α-STABLE DISTRIBUTIONS AND LÉVY α-STABLE PROCESSES
maximum absolute value of the jump of the process up to time t:
(α)
Mt = sup{|L(α)
τ − Lτ − | : 0 < τ ≤ t}
(2.41)
We want to calculate the probability that Mt exceeds l
P (Mt > l) = lim P (max |∆Ii (t/N )| > l).
N →∞
(2.42)
i
We proceed with the discretized version of the process and perform the limit at the end,
assuming in the calculations that N is already large, so that the asymptotic behavior of
the CDF can be used:
P (max |∆Ii (t/N )| ≤ l) = P (max |∆Ii (t/N )| ≤ l) =
i
i

N 
 1−
≈ = 1 − 2F t 1/α S(α,0) (−l)
(N )

 1−
N
Y
P (|∆Ii (t/N )| ≤ l) =
i=1
N
α
N →∞
2C(α)t
−−−→ e−C(α)t/l
N lα
N
√
2
N →∞
2 t − l4t N
√
e
−−−→ 1,
l πN
> 0, α < 2
α=2
where C(α) > 0 is some (known but not important for the proof) function. We arrive at

1 − e−C(α)t/lα > 0, α < 2
P (Mt > l) =
0,
α = 2.
(2.43)
We see that for α < 2 there is a nonzero probability of observing a discontinuity (i.e. a
jump) of magnitude at least l in a finite time, whereas for α = 2 this probability is equal
to 0.
According to the Lévy-Itô decomposition [App09], every Lévy process (i.e. stochastic
process with i.i.d. increments) can be represented by a linear combination of independent
components: a linear drift, the Wiener process, and some pure jump process. It turns out
that α-stable processes for α < 2 have no Brownian component and with probability 1 an
infinite number of discontinuities in a finite time is present [Pro04]. This is very different
from the Brownian motion, which has continuous trajectories, and from compound Poisson
processes, which have a finite number of discontinuities in a finite time.
Lévy flights may be used as noise generating processes. In this thesis we restrict our
attention to additive white α-stable noise, i.e. the corresponding stochastic process can
26
2.2. LÉVY α-STABLE DISTRIBUTIONS AND LÉVY α-STABLE PROCESSES
be written in a general form
(α)
dXt = a(Xt , t)dt + DdLt ,
(2.44)
where D is a generalized diffusion coefficient (note the unusual units [D] = [Xt ]α /[t]) and
a(x, t) is a drift term.
For our purposes Xt will denote a position of a particle, hence Eq.
(2.44) is a
stochastic equation of motion in the overdamped limit [Che+02; Str94] and a(Xt , t) =
−∂x U (x, t)|x=Xt is interpreted as force.
Since the solution of the SDE (2.44) is Markovian, all probability distributions related
to the process Xt can be in principle obtained from its propagator (conditional PDF)
f (x, t|y, s) and the initial distribution fX0 (x) [Gar+85]. The propagator corresponding
to (2.44) solves the following fractional Fokker-Planck equation (FFPE) [Zas94; Fog94;
JMF99]
∂
∂α
∂
f (x, t|x0 , t0 ) = D
f (x, t|x0 , t0 ) −
(a(x, t)f (x, t|x0 , t0 )) ,
α
∂t
∂|x|
∂x
(2.45)
where f (x, t0 |x, t0 ) = δ(x − x0 ) and ∂ α /∂|x|α stands for the Riesz fractional derivative
[Las00; Che+04; Her11], which is an integral operator defined as (0 < α < 2)
∂α
sin (πα/2)
g(x) = Γ(1 + α)
α
∂|x|
π
Z∞
g(x + z) − 2g(x) + g(x − z)
dz
z 1+α
(2.46)
0
and admits a simple form in the Fourier space (0 < α ≤ 2)
F
∂α
g (k) = −|k|α F{g}(k).
α
∂|x|
(2.47)
A nonlocal character of the integral operator [DGH06; Her11] makes it much harder to
deal with the FFPE (2.45) than with the standard diffusion equation (α = 2), especially
if the absorbing or reflecting boundaries have to be invoked. It leads to many nontrivial consequences, including multimodal stationary distributions in unimodal potentials
[Che+03a], nonuniform stationary distributions in an infinite, rectangular potential well
[DHH08], different distributions of first arrival and first passage times, and failure of the
method of images [Che+03b].
We have briefly reviewed here the theory of one-dimensional, symmetric Lévy flights.
27
2.2. LÉVY α-STABLE DISTRIBUTIONS AND LÉVY α-STABLE PROCESSES
The process and the corresponding FFPE can be generalized to include asymmetric αstable jumps in one dimension [Yan+00; DGS07] as well as to take place in multidimensional space [Vah+13; SD14a; SD14b; DS15]. Interestingly, components of a spherically
symmetric α-stable vector are not independent for α < 2, in contrast to the Gaussian
case.
Some authors refer to Eq. (2.45) as the space-FFPE, in contrast to another generalization of the Fokker-Planck equation involving a time- (Caputo or Riemann-Liouville)
fractional derivative, and called the time-FFPE. It can correspond to the process with
an inverse-time α-stable subordinator [MWW07] (or, equivalently, the CTRW scheme
[MK00]) describing subdiffusion. For the excellent review of different applications of
time-fractional derivatives, see [Mai97].
2.2.3
Generating α-stable random numbers and processes on a
computer
Most computer pseudo-random number generators (PRNMs) produce uniformly distributed sequences of numbers. In practice it is often desirable to generate pseudo-random
numbers distributed according to some cumulative distribution function FX . One way to
achieve this is to apply an inverse transform sampling, which is based on an observation that for a given random variable U uniform on the interval (0, 1), a transformed
random variable Y = QX (U ) = FX−1 (U ) has the same distribution as X. Thus, given a
sequence of uniformly distributed random numbers (u1 , u2 , ...), a transformed sequence
(QX (u1 ), QX (u2 ), ...) has the desirable distribution FX . Unfortunately, quite often the
quantile function QX (x) is not expressible in terms of elementary functions, which makes
implementations of the inverse transform sampling cumbersome or ineffective. This is
also the case with α-stable random variables, therefore another method based on a transformation involving two uniform random variables is presented here. This method was
introduced already in 1970s by Chambers et al. [CMS76], although a rigorous proof and
some minor corrections were given much later in [Wer96].
Let U, V be two mutually independent random variables, both uniform on (0, 1).
We introduce two auxiliary random variables: Φ = π U − 21 uniform on (− π2 , π2 ) and
W = − ln V exponentially distributed. Then the following transformation gives α-stable
28
2.2. LÉVY α-STABLE DISTRIBUTIONS AND LÉVY α-STABLE PROCESSES
5
8
6
4
t
0
X
Xt
2
0
−2
−4
−6
−8
−5
0
2
4
6
8
10
0
2
4
t
(a) α = 2
8
10
8
10
(b) α = 1.5
30
1000
20
0.10<q<0.90
0.25<q<0.75
q=0.5
500
10
t
0
X
Xt
6
t
−10
0
−500
−20
−1000
−30
0
2
4
6
8
10
0
2
4
t
6
t
(c) α = 1
(d) α = 0.5
Figure 2.3: Visualizations of 1-dimensional α-stable processes for α = (2; 1.5; 1, 0.5). Along with
ten exemplary sample trajectories, median (red) and quantile areas (gray and violet) have been
plotted. Quantiles have been obtained from 105 sample trajectories integrated with Euler scheme
(∆t = 0.01).
random variable

(1−α)/α
1

cos(Φ−α(Φ+Φ0 ))
0 ))
 1 + β 2 tan2 πα 2α sin (α(Φ+Φ
1/α
2
W
(cos Φ)
S(α, β) =

cos Φ
(1 + β 0 Φ) tan Φ − β 0 ln Wπ +βΦ
for α 6= 1
(2.48)
for α = 1,
2
where β 0 =
2
β
π
and Φ0 = arctan(β tan πα
)/α. In the symmetric case β = 0 and the
2
29
2.2. LÉVY α-STABLE DISTRIBUTIONS AND LÉVY α-STABLE PROCESSES
(a) α = 2
(b) α = 1.75
(d) α = 1.25
(e) α = 1
(c) α = 1.5
(f ) α = 0.75
Figure 2.4: Single realizations of 2-dimensional, axially symmetric α-stable processes for different
values of stability parameter α. Long straight lines represent instantaneous jumps, which are
becoming longer for lower values of α. For α = 2 trajectories are continuous2 .
transformation simplifies to
S(α, 0) =


sin αΦ
(cos Φ)1/α
tan Φ
cos((1−α)Φ)
W
(1−α)/α
for α 6= 1
(2.49)
for α = 1.
Two uniform random numbers are needed to generate one α-stable random number, with
a notable exception of Cauchy random variable S(1, 0), for which a simple transformation
of only one uniform random variable is required.
The most common way to simulate symmetric α-stable processes on a computer is to
2
Note that, for the sake of visualization, lines of a non-zero width have to be depicted. The resulting
object for the Brownian motion is known as the Wiener sausage [DV75].
30
2.2. LÉVY α-STABLE DISTRIBUTIONS AND LÉVY α-STABLE PROCESSES
use the Euler scheme. Its update rule is given by the following formula
Xt+∆t = Xt + ∆t1/α S(α, 0),
3
(2.50)
where ∆t is the integration step size. It can be generalize to the SDE with a drift term
(2.44)
Xt+∆t = Xt + a(Xt , t)∆t + S(α, 0)D∆t1/α .
(2.51)
Unfortunately, for nonlinear a(x, t) this algorithm may become numerically unstable. One
of the simplest methods to overcome that issue is to introduce a variable step size, e.g.:
∆t = min ∆tmax ,
Xt
η ,
a(Xt , t)
(2.52)
with η 1. This ensures that the drift term does not produce too large (relative to Xt )
steps.
3
An alternative method of simulating Lévy flights which does not need α-stable random numbers has
been proposed in [Man94].
31
Chapter 3
Summary of the papers
3.1
3.1.1
Paper I
Motivation
Onsager’s theory and Green-Kubo relations are grounded on the expansions around equilibrium states, which restricts their applicability to systems close to equilibrium. The aim
of the discussed paper is to extend this approach to systems far from equilibrium and
subjected to a non-equilibrated heat bath. We model interactions of the particle with
the reservoir by assuming that there are two independent sources of noise affecting the
particle: white Gaussian noise and white Cauchy noise. The latter introduces large kicks,
leading effectively to infinite moments of the position of the particle and infinite average
energy. Accordingly, the statistical temperature of the system cannot be properly defined,
i.e. the system is out of equilibrium even in its stationary state.
We further assume that the system is initiated in its stationary state and perturbed
with some known external time-dependent force. Can we still predict the response of
the system by means of the linear response theory? How should we identify conjugate
variables in that case? These questions have been addressed in the paper.
3.1.2
Methodology
In the framework of stochastic thermodynamics [Sek10; Van+13] a state of the system is
identified with the PDF of some random variable Xt . It may for instance correspond to
the position or momentum of a particle experiencing both deterministic and fluctuating
forces. We focus our attention on the solution of the following SDE
dXt = −λXt dt + f (t)dt +
32
√
2σdWt + γdCt ,
(3.1)
3.1. PAPER I
where (Wt )t≥0 and (Ct )t≥0 denote the Wiener and the symmetric Cauchy processes, respectively, and the external perturbation f (t) = sin t/10+t/100 becomes stronger in time.
Since the equation is linear, the linear response theory gives exact results in the Gaussian
(γ = 0) case. Note that E[Xt ] (expected value of the position at a given time) does not
exist for γ 6= 0.
For a constant perturbation f (t) = f the system attains a non-equilibrium steady state
(NESS) ps (x, f ) = limt→∞ pXt (x). By analogy to Gibbs equilibrium states we introduce
a non-equilibrium pseudo-potential
φ(x, f ) = − ln ps (x, f )
(3.2)
and use it to define a conjugate variable1 [DPG12]:
∂φ(Xt , f ) 1 ∂φ(x, 0) Zt = Z(Xt ) =
=−
,
∂f
λ
∂x x=Xt
f =0
(3.3)
where the second equality follows from the linearity of Eq. (3.1). The conjugate variable
defines a new stochastic process related to Xt by a nonlinear transform Z(x). In general,
the transform can be expressed in terms of the Faddeeva function [PW90] and simplifies
to a rational function for σ = 0 and to a linear function for γ = 0.
The expected value of the conjugate variable is finite due to the nonlinearity of the
transform. We can therefore proceed and calculate a linear response function (generalized
susceptibility)
χ(t) =
d
E[Zt Z0 ]0 ,
dt
(3.4)
where the averaging is performed in the stationary state of the unperturbed (f = 0)
system. The expected response of the conjugate variable to the perturbation f (t) can be
then determined exactly from
E[Zt ] =
Z∞
pXt (x)Z(x)dx,
(3.5)
−∞
where pXt (x) is calculated including f (t) and under stationary initial conditions. Other1
The conjugate variable is denoted by X in the paper. Note that the factor
there.
33
1
λ
is erroneously missing
3.1. PAPER I
wise, E[Zt ] can be estimated directly from the linear response function
E[Zt ]LR =
Zt
χ(t − s)f (s)ds.
(3.6)
0
The linear response function and the corresponding linear response (LR) of the conjugate
variable (3.6) can be easily calculated analytically if one of the noise-sources is switched
off, i.e. when either γ = 0 or σ = 0. In all other cases χ(t) has been obtained by averaging
over many simulations of the unperturbed SDE (cf. Eqs. (3.3) and (3.4)). Similarly, exact
responses according to Eq. (3.5) have been obtained by means of stochastic simulations
of the perturbed SDE.
3.1.3
Results
A comparison between exact (3.5) and linear (3.6) responses has been presented in
Figures 3 and 4 in the paper. It turns out
that predictions of the LR fit exact results for
4
γ=0
γ=0.1
γ=1
γ=10
weak perturbations only. This is related to
2
in turn causes Zt to solve a nonlinear SDE.
The exact result is recovered for γ = 0, i.e.
when Z(x) is linear. Since the linear response
Z(x)
the nonlinearity of Z(x) (cf. Fig. 3.1), which
0
−2
theory is expected to work only in the weak
perturbation limit, we conclude that it is indeed applicable in the investigated example
−4
−4
−2
0
x
2
4
and, supposedly, in other systems subject to Figure 3.1: The nonlinear transformation
defining the conjugate variable (3.3) for σ = 1
and different values of γ. Note that Z(x) is
surprising in view of the fact that the system non-monotonic for γ > 0.
weak α-stable noise. This observation is quite
is far from equilibrium, as can be seen directly
by checking that the detailed balance condition is broken (see Appendix C).
Although the model describes an overdamped particle in one dimension, it should be
possible to generalize it to higher dimensions. In this context, in the paper we demonstrate
basic features of multidimensional, spherically symmetric Lévy α-stable distributions and
discuss how the relaxation kinetics in 3-dimensional space is affected by α-stable noise.
34
3.2. PAPER II
3.2
3.2.1
Paper II
Motivation
In [Wan+02] authors have verified the fluctuation theorem (FT)2 with an optical tweezers
experiment, wherein a colloidal particle has been trapped in a harmonic potential induced
by laser light. Additionally, the minimum of the harmonic potential has been moved with a
constant velocity, effectively dragging the particle. Due to a heat bath (fluid) surrounding
the particle, dragging it requires work which is dissipated into the heat bath. Although
authors of [Wan+02] have claimed to measure the entropy production, they have in fact
measured total work done on the particle (hereafter denoted by Wt and called just work),
as explained in [VC03a]. The work related FT for a dragged Brownian particle takes the
form
ln
fWt (w)
= c(t)w,
fWt (−w)
(3.7)
where fWt is a PDF of work done on the particle up to time t and c(t) is some function
of time. Note that Eq. 3.7 is a generalized version of the FT and includes both transient
(TFT) and stationary state (SSFT) fluctuation theorems [VC03a].
In [Wan+05] authors have derived an exact form of the FT (including function c(t))
and have shown experimentally that it holds in a non-equilibrium steady state (NESS) for
all times. In our paper we follow their setup and focus on a one-dimensional stochastic
description of an overdamped particle dragged by a harmonic potential moving with a
constant velocity. The novelty of our model lies in an addition of a non-Gaussian (i.e.
Cauchy) noise, which can correspond either to a non-equilibrated environment (an internal
noise interpretation) or to an external random force. These two interpretations lead to
different work and heat definitions and thus to different work and heat fluctuations.
A similar model has already been studied before by Touchette and Cohen [TC07].
The main difference lies in the noise, which in the aforementioned article is taken to be
generated by a single α-stable process. Authors assume that this is an external random
force, arguing that, due to the infinite variance, Lévy noise cannot constitute an internal
noise, as it would violate the fluctuation-dissipation theorem. They do not, however,
include a separate Gaussian noise as a heat bath, which limits the thermodynamical
interpretation of the model. In contrast, we include both Gaussian and Lévy α-stable
2
Fluctuation theorems are also referred to as fluctuation relations (FR), which underlines that they
can be observed or hypothesized without being formally proven.
35
3.2. PAPER II
noises in our model. This corresponds to positive temperature of the heat bath in the
external random force interpretation. Additionally, a relative strength of the Cauchy noise
γ
σ
may be used to quantify the distance of the heat bath from equilibrium in the (more
controversial) internal Lévy noise interpretation. Note that the statistical temperature is
undefined in this case.
The statistics of heat fluctuations is generally more complicated to calculate than the
statistics of work fluctuations. Even in a simple case of the dragged Brownian particle
analytical treatment is hard. Some analytical results, including asymptotic forms of the
heat distribution, have been reported in [VC03b; VC04; Imp+07]. Here we only note that
the linear form of the logarithmic ratio as in Eq. (3.7) is recovered in the limit t → ∞
and for relatively small fluctuations.
3.2.2
Methodology
A stochastic process analyzed in the paper is defined by the following SDE:
dXt = −a(Xt − vt)dt +
√
2σdWt + γdCt ,
(3.8)
where (Wt )t≥0 and (Ct )t≥0 denote the Wiener process and the symmetric Cauchy process,
respectively. The overall process (Xt )t≥0 describes a one-dimensional dynamics of a particle subject to a quadratic potential, which is moved with a constant velocity v, and two
independent white noise sources. Due to the nonzero velocity of the dragged potential
the process does not possess a stationary PDF. It possesses, however, a NESS, i.e. a PDF
which moves steadily in time (with velocity v) but has a constant shape.
The aim of the paper is to calculate work and heat in the described system and to check
whether steady-state fluctuation relations can be extended to systems subject to nonGaussian noise. Note that there is an ongoing discussion on how to define work and heat at
the nanoscale [VR08; Sei12; Åbe13; Bra+15; GEW15]. In the paper, a stochastic calculus
of thermodynamic quantities as described by Sekimoto [Sek10] is followed. Definitions of
stochastic versions of work and heat in this framework stem from the total differential of
the energy of the particle
dH(Xt , λ(t)) =
∂H(Xt , λ(t))
∂H(Xt , λ(t))
dXt +
dλ(t),
∂Xt
∂λ(t)
(3.9)
where (Xt )t≥0 is the stochastic process, λ(t) is a parameter varied in time according
36
3.2. PAPER II
to some predefined (deterministic) protocol, and denotes a stochastic integral which
preserves standard rules of calculus3 . The first element of the sum is a differential of heat
transferred from a heat bath to the particle, whereas the second is a differential of work
performed on the particle. Since in our case the dynamics of the particle is analyzed in the
overdamped limit, no kinetic energy is involved, i.e. the energy is equal to the potential
energy U (Xt , λ(t)). Work can be thus written in the following form:
Wt =
Zt
0
∂U ∂λ(t0 ) 0
dt .
∂λ(t0 ) ∂t0
(3.10)
This is a ramification of the idea that work should represent an organized flow of energy.
Although the protocol is deterministic, work in this framework is a random variable.
Its expected value is equal to classical work, forasmuch as the thermodynamic limit in
this framework corresponds to the averaging over many independent realizations of the
process. From Eq. (3.8) the potential energy
U (Xt , λ(t)) =
1
(Xt − λ(t))2 ,
2
(3.11)
the protocol
λ(t) = vt,
(3.12)
and (mechanical) work
Wt = −av
Zt
(vt0 − Xt0 ) dt0
(3.13)
0
can be identified. Due to the presence of Cauchy noise, definition of heat becomes a
subtle issue. Since in Eq. (3.9) the standard chain rule has been assumed, so-called
Marcus integral should be used (for more information see Appendix B). Moreover, one
can either include or exclude the Cauchy term in a heat bath, leading to two different
definitions of heat. In the former case, the heat is given by the change of the internal
(potential) energy minus the work
= U (Xt , λ(t)) − U (X0 , λ(0)) − Wt ,
Qint
t
3
(3.14)
In [Sek10] Stratonovich integral is adopted. It preserves standard rules of calculus for white Gaussian
noise, but does not in the case of discontinuous noise-generating processes, cf. Appendix B.
37
3.2. PAPER II
whereas in the latter case, the energy flow due to the random Cauchy force is excluded
from the heat
where WtC = Wt + γ
Rt
0
= U (Xt , λ(t)) − U (X0 , λ(0)) − WtC ,
Qext
t
(3.15)
Ct0 dXt0 .
The work PDF has been calculated analytically with a method based on a characteristic functional of the noise [Các99; BC04] and a closed-form expression for a characteristic
function of work has been found. Many other distributions have been estimated numerically from stochastic simulations. In order to achieve stochastic integrals which preserve
the standard chain rule, a colored noise approximation has been applied (see Section
2.1.3).
Another method that could have been used is the Marcus canonical equation [CP14].
As explained in Section 2.1.3 and shown in examples in Appendix B, the Marcus equation
and the colored noise approximation lead to the same results in the limit of infinitesimal correlation time. The Marcus calculus, constructed independently in mathematical
[Mar78; Mar81] and engineering [DF93a; DF93b] literature, in the statistical physics community has been re-derived [KSH12] and finally noticed [LMW13] only quite recently.
A different approach to the problem of a standard chain rule preserving stochastic
integral has been proposed by Srokowski [Sro09; Sro10]. It is based on truncated Lévy
flights (TLFs) [MS94; Kop95; BP03], i.e. discrete-time stochastic processes constructed
by introducing some cut-off on the tails of α-stable increments. TLFs are similar to Lévy
flights for short times, but in a long run they converge to the Gaussian diffusion. Since
truncated Lévy flights have finite variance, the Stratonovich integral with respect to them
preserve standard rules of calculus. The idea is thus to approximate LFs with TLFs and
apply the Stratonovich integral.
3.2.3
Results
Due to the stability of Lévy α-stable distributions and linearity of Eq. (3.13), Wt can be
decomposed into a sum of two (Gaussian and Cauchy) independent random variables and
its characteristic function could be calculated analytically. Tails of the work distribution
are dominated by a power-law coming from the Cauchy component, thus the conventional
fluctuation relation (3.7) does not hold. Indeed, as already pointed out in [TC07], power-
38
3.2. PAPER II
law distributions lead to an anomalous fluctuation relation
lim ln
w→∞
fWt (w)
= 1,
fWt (−w)
(3.16)
meaning that large positive and negative fluctuations of the same magnitude attain the
same probability.
Due to the inability to perform analytical calculations, distributions of other quantities, i.e. W C , Qext , Qint , have been obtained by means of stochastic simulations. The
t
t
t
correctness of the integration algorithm could be partly checked by the comparison of
acquired results with analytical predictions for the distribution of Wt . Although they
have matched very well, the initial algorithm, based on the Stratonovich integral and a
simple Euler scheme, have not been converging for other quantities. A close inspection
has led to the conclusion that the multiplicative nature of Cauchy noise in definitions
of W C , Qext , Qint is not compatible with the implicit assumption of standard rules of
t
t
t
calculus. As mentioned before, in order to circumvent this issue, we have used a colored
noise approximation, in a similar manner to [KSH12]. As discussed in Appendix B, the
same solution could have been otherwise constructed by means of the Marcus integral.
Interestingly, neither standard FR (3.7) nor the FR of the form (3.16) is observed for
WtC . It seems that the positive tail is much stronger than the negative tail, yet their ratio
does not grow exponentially. Results regarding heat distributions are more problematic.
From [VC04] we know that in the pure Gaussian case (γ = 0) and in the limit of long times
the logarithm of the ratio of heat distribution tails is a linear function of time only for
small fluctuations, but then it saturates at some constant value. In our plots of long time
(t = 10) heat distributions, only the linear part of the fluctuation relation can be observed,
which suggests that the number of observations is insufficient to get a good estimate of the
(very rare) large positive heat events probabilities. The saturation, however, can be easily
seen for shorter times (t = 0.1). In conclusion, heavy-tailed noise severely affects fluxes
of work and heat in the system, leading to non-standard fluctuation relations. General
features of these anomalous fluctuation relations are still to be determined.
39
3.3. PAPER III
3.3
3.3.1
Paper III
Motivation
In a recent paper by Evans and Majumdar [EM11] a new model of random search has been
introduced. It combines the Brownian motion (Gaussian diffusion) with an instantaneous
return to the initial position (resetting mechanism), which is assumed to occur randomly
and homogeneously in time with a constant rate r. Resetting may be interpreted in
various ways, depending on the particular application of the model. One may for instance
think of a foraging bird regularly coming back to its nest or a person trying to locate a face
in a crowd with a natural tendency to return to the initial position after an unsuccessful
search.
In the model of free, unbounded diffusion, finding a single target may take a very long
time. Indeed, the mean first passage time (MFPT) is known to be infinite, although the
target will be reached with probability one4 . The efficiency of the search, as measured by
the MFPT5 , is so low due to the possibility of moving away from the target. Even though
the searcher will eventually come back, the average time needed for that to happen is infinite. The hope is that the resetting would serve as a mechanism for avoiding meandering
in the wrong direction, which should improve efficiency of the search. As a matter of fact,
such reasoning has already been intuitively recognized and applied by engineers, e.g. in
the graph mining algorithms[TFP06].
In their analysis Evans and Majumdar [EM11] have shown that resetting indeed makes
the MFPT finite. Of course, too large r may cause the searcher to get stuck close to the
initial position, hindering its efforts to reach the target. Since for both very small (r → 0)
and very large (r → ∞) resetting rates the MFPT approaches infinity, it is evident that
there exists some optimal r∗ for which the MFPT admits minimum. The optimal rate r∗
has been calculated in [EM11] as a function of a fixed position of the target.
In Paper III another variant of the random walk with stochastic resetting model is
proposed. The aforementioned concepts of Lévy flights and stochastic resetting are merged
and an optimization of the search strategy is performed. It has been expected that,
under some conditions, LFs would outperform the Gaussian random walk. Apart from
4
For proof, see e.g. [IM74; Gar+85; Red01].
We use the expected search time as a measure of the random search strategy efficiency. Some authors
advocate different efficiency measures, e.g. the expected inverse of the search time [PCM14] or the ratio
between number of visited target sites and the total distance traveled [Vis+99].
5
40
3.3. PAPER III
delineating which conditions promote LFs over the Gaussian random walk, the goal of the
project is to check if the resetting is still preferable when the searcher performs LFs.
3.3.2
Methodology
For the sake of simplicity, a discrete time version of a random walk with heavy-tailed
distributed jumps (Lévy flights) has been adopted. A continuous time model has also
been investigated—results are presented in Paper IV (see Section 3.4).
In the introduced model the searcher starts from an initial position x0 and moves at
discrete time steps, according to the following rule: with probability r it comes back to
the initial position, i.e. xn = x0 , and with probability 1 − r it randomly jumps to a new
position, i.e. xn = xn−1 + ηn , where jump lengths ηn are i.i.d. random variables, each
drawn from a given PDF f (η). The target is assumed to be immobile and its position
without loss of generality is fixed at the origin. The jump length distribution f (η) is taken
from a family of symmetric α-stable distributions, with the characteristic function given
α
by fˆ(k) = e−|k| . Note that in the paper the symbol µ is used in place of α. Here the
latter notation is applied, consistently with the rest of the main body of this thesis.
There are three parameters in the model: (x0 , r, α). Two of them—the resetting
probability r and the stability index α—describe the way the searcher move, hence each
choice of values (r, α) constitutes a search strategy. The initial position x0 , however,
is a feature of the situation which the searcher has been put in, therefore its value is
not included as a part of the search strategy. A first passage time is considered, which
implies that jumping over the target is interpreted as an event of actually finding it. As
the quality measure of the search strategy a mean first passage time (MFPT) has been
adopted. Analytical and numerical studies have been combined in order to optimize the
search strategy. The optimization has been performed for different values of x0 . Due
to the symmetry of the jump length distribution only positive x0 have been considered.
Values of the optimal parameters (r∗ , α∗ ) and the optimal MFPT have been plotted as a
function of x0 .
3.3.3
Results
With the help of the Pollaczek-Spitzer formula [Pol52; Spi56; Maj10] a general expression
for MFPT has been derived (cf. Eq. 6 and Eq. 7 in the paper). It involves, however,
rather complicated integrals which could not have been solved analytically. Thus, the
41
3.3. PAPER III
5.8
5.6
5.4
5.2
5
T
*
5.75
4.8
5.7
4.6
4.4
5.65
4.2
4
0.55
0
0.1
0.2
0.3
0.4
0.56
0.5
0.57
0.58
0.6
0.59
0.6
0.7
x0
Figure 3.2: The optimal MFPT as a function of a distance to the target. The horizontal line at
the top corresponds to the α → 0 limit. Colors indicate whether the MFPT is a global minimum
(green), a local minimum (blue) or neither of them (red). Data represented by points have been
obtained by means of numerical calculations: numerical stochastic simulations and numerical
optimization in the two-parameter space. The essential part around the transition is zoomed in
and shown in the inset plot.
optimization has been performed by means of numerical approximations, including averaging over many samples from stochastic simulations and numerical optimization, in
tandem with an analytical analysis in some special cases.
When the resetting is switched off (r = 0) the model simplifies to a standard discretetime random walk. In that case, if the jump length distribution is symmetric and continuous (i.e. its CDF is continuous), the survival probability is given by the Pollaczek-Spitzer
formula and the MFPT is infinite. Furthermore, surprising universality is observed if one
sets x0 at the origin6 : distribution of the first passage time does not depend on the jump
length distribution, as long as it is symmetric and continuous. This result is known as the
6
Note that this does not imply that the first passage time is equal to 0. The searcher has to actually
cross the target, i.e. move to the left from the origin.
42
3.3. PAPER III
0.25
1.2
0.245
1
0.24
r*
α*
0.8
0.6
0.235
0.4
0.23
0.2
0.225
0
0
0.2
0.4
0.22
0
0.6
0.2
x
0.4
0.6
x
0
0
(a) The optimal stability index
(b) The optimal resetting probability
Figure 3.3: Optimal parameters of the random search strategy as a function of a distance to the
target. The horizontal lines at the bottom correspond to the α → 0 limit. Colors have the same
meaning as in Fig. 3.2.
Sparre Andersen theorem7 , and can be easily derived from the Pollaczek-Spitzer formula
[Maj10].
As it turns out, similar universality is present in the model with resetting. In the
limit x0 → 0 the resulting distribution of the first passage time also does not depend on
the jump length distribution. In particular, the MFPT is given by an intriguingly simple
expression:
T (x0 = 0) = √
1
.
r−r
The optimization in that case is straightforward and leads to r∗ =
(3.17)
1
4
and T ∗ = 4 with
arbitrary α ∈ (0, 2]. One may be lured to try to derive these results in some simple fashion,
but it seems that all derivations end up being rather complicated, unless the PollaczekSpitzer formula is employed. Similarly, no simple proof of the Sparre Andersen theorem
exists (for some examples see [FF95] and references therein). As mentioned before, that
universal behavior of first passage times holds for continuous jump length distributions
only. Accordingly, it would be informative to check how the MFPT is affected if this
assumption is not fulfilled. In Appendix E a case of a lattice random walk (i.e. possible
jumps are ±1) is discussed in detail. Indeed the MFPT departs from Eq. (3.17), which
7
The Sparre Andersen theorem is, in fact, more general and is concerned with random walks with
symmetrically dependent (exchangeable) jumps of any distribution. The universal result for symmetric
and continuous distributions is its most widely recognizable corollary. The original proof [And54] is
extremely complicated, for a simpler derivation and a detailed discussion, see [Fel71].
43
3.3. PAPER III
demonstrates that the assumption of continuity is essential. Since analyzed generic lattice
random walk exhibits consistently higher values of MFPT than these of continuous jump
length distributions with the same r, a hypothesis is proposed that this is true for any
discontinuous jump length distribution.
A little bit more tricky limit α → 0 leads to another useful closed-form formula for the
MFPT (cf. Eq. 11 in the paper), which is a nontrivial function of r but independent of
x0 . Quite importantly, these two limits (x0 → 0 and α → 0) do not commute, hence the
formula is valid for x0 > 0. Roughly speaking, the independence follows from the fact that
the α-stable distribution in the limit α → 0 loses its length-scale. This limit is particularly
important in the context of the search strategy optimization. The corresponding MFPT
∗
. Nonetheless, the strategy
admits minimum at some nontrivial value of resetting rate r>
∗
, α = 0) is locally optimal only if x0 is larger than the critical value
(r>
xc0 = e−γE
(3.18)
where γE ≈ 0.5772 stands for Euler-Mascheroni constant. This bifurcation is essential for
the rich behavior of the model, which is described further below.
As already discussed, for x0 = 0 the MFPT is universal, i.e. independent of α. What
happens if the process starts not from the origin but from a close neighborhood of it?
In order to answer this question, the MFPT can be expanded in a Taylor series around
x0 . Even if only the linear term is taken into account, the condition for optimal α∗ is
nontrivial (see Eq. 10 in the paper), but can be easily solved numerically yielding the
value:
lim+ α∗ (x0 ) ≈ 1.289.
x0 →0
(3.19)
All results of the optimization have been presented in a concise form in Fig. 3.2 and
Fig. 3.3. The optimal strategy for a given x0 , corresponding to the global minimum of
the MFPT in (r, α) space, can be identified by looking at the green part of the plot. The
most intriguing part of the plots are linked with a coexistence of two minima. There
one can see two branches—one colored green (the global minimum) and the other blue (a
local minimum). The competition between these two branches leads to a discontinuity of
the α∗ (x0 ) and r∗ (x0 ) curves. This behavior bears a strong resemblance to a first-order
phase transition defined in the framework of statistical mechanics8 The position of the
8
This is not really a phase transition in the classical sense, since here we do not deal with a many-body
system and no order parameter can be defined.
44
3.4. PAPER IV
discontinuity (dashed black lines) has been obtained from stochastic simulations, whereas
the bifurcation point of α → 0 branch (red to blue transition) is given by Eq. (3.18).
One can easily see that in the whole range of possible values of x0 the optimal MFPT is
bounded from below (Tmin = 4) and from above:
Tmax = 2
e
+
e−1
r
e
e−1
≈ 5.68.
(3.20)
Analogously, the optimal parameters are bounded:
√ 2
e −e+1−e 1
r ∈
,
2
4
∗
3.4
3.4.1
(3.19)
and α∗ < lim+ α∗ (x0 ) ≈ 1.289.
x0 →0
(3.21)
Paper IV
Motivation
Similarly to Paper III this is a follow-up study expanding results from [EM11] to include
symmetric Lévy flights (LFs). In contrast to Paper III it deals with a continuous time
model. Moreover, the mean first arrival time (MFAT) is used instead of the mean first
passage time (MFPT). In the case of Lévy flights these notions are two different quantities,
as discussed in Appendix D. Depending on the phenomenon at hand one may be prompted
to use one or the other, though when modeling search processes the MFAT seems to
be more appropriate. If the MFAT is employed, leaping over a predefined target by a
discontinuous part of the trajectory does not count as an event of finding the target. In
particular, this assumption should fit well studies of search processes, in which motion
has much shorter time scales than observation, e.g. oculography [FW12; Ray97].
Another reason to study this model came from results described in Paper III. The
diagram of optimal parameters calculated there is surprisingly complex, including the
competition of two minima, and discontinuity of globally optimal parameters as functions
of a distance to the target. One may ask whether this discontinuity is an intrinsic property
of the search driven by Lévy flights, or perhaps shapes of the optimal parameters plots
depend strongly on a particular set-up of the model. In other words: how generic this
"first-order phase transition" is?
Sample paths of the Wiener process are continuous, thus its first arrival times and
first passage times are equal [Che+03b]. In contrast, sample paths of Lévy flights are
45
3.4. PAPER IV
discontinuous (see Theorem 2 in Section 2.2.2). Additionally, the target is defined to be
a point-like object, and in consequence it can be easily leaped over by the trajectory of
Lévy flights. Subsequently, natural questions arise: is it possible to hit a point in a finite
time? Can the MFAT be finite for discontinuous trajectories with a finite probability of
jumping over the target? If so, what conditions should be met in order for the MFAT to
be finite? These questions have been addressed in the paper.
In Paper III, as well as in [EM11], the position of the target is fixed. An optimal rate of
resetting r∗ is calculated assuming that the distance to the target is known, which seems
to be too simplified: if one knows the position of the target no search is needed at all.
Thus, another issue analyzed in the paper is an impact of a distribution of target positions
on the MFAT. This should not be confused with distribution of many targets: there is still
one target only, although for each sample path its position is chosen independently from
an even probability distribution function pXt (x) (Xt denotes the target position random
variable).
3.4.2
Methodology
A common way of dealing with the first passage or first arrival times problems in continuous time domain is to solve the corresponding Fokker-Planck equation with proper
boundary conditions. The target is modeled there by an absorbing boundary. The solution of that equation, f (x, t), gives the joint probability: density of being at point x and
that trajectory did not hit the target until time t. Hence, the solution is not normalized
in x, as would be for the free diffusion (i.e. without absorbing boundaries). By definition
R
the integral f (x, t)dx introduces the survival probability, i.e. the probability that the
first passage time is larger than t. The survival probability incorporates all information
about the distribution of first arrival times.
This approach, however, becomes cumbersome with Lévy flights and corresponding
fractional Fokker-Planck equation (FFPE). Due to the nonlocal nature of the fractional
derivative operator (which is, in fact, an integral operator) defining boundary conditions
is more subtle, e.g. in order to calculate the PDF of first passage times the whole halfline has to be set to zero. For the same reason the method of images cannot be applied
[Che+03b]. In the paper the FFPE with absorbing boundary has not been solved directly.
Instead, a trick Redner [Red01] has been employed. It turns out that—in order to calculate
the distribution of first arrival times to a point—one needs only the free propagator of
46
3.4. PAPER IV
the FFPE, i.e. the initial condition is a δ-function and no absorbing boundary has to
be imposed. Solving the FFPE for the free propagator is much simpler: it reduces to an
algebraic equation in the Fourier-Laplace space.
The propagator of a free symmetric Lévy flight process takes a very simple form in
the Fourier-Laplace space:
f (k, s) =
1
.
s + D|k|α
(3.22)
Nevertheless, in general f (x, t) can be expressed in terms of special functions only. Even
though inverting these transforms is not always possible one can learn quite a lot by
inspecting transforms itself. In the paper the MFAT is written in an integral form, which
comes from the inverse Fourier transform. This form allows for an analytical asymptotic
behavior analysis (with the Hardy-Littlewood tauberian theorem [Fel71]) as well as for a
fast numerical integration.
A large part of the paper is devoted to an optimization of the random search strategy.
The idea is quite natural: given a probability distribution over possible target positions,
the aim is to find the best strategy in the class of all possible strategies. The MFAT
has been chosen to serve as the objective function for the optimization. There are three
parameters describing Lévy flights with stochastic resetting: the stability index α, the
generalized diffusion constant D, and the resetting rate r. The unit of D depends on
α. Nevertheless, larger D always gives faster diffusion and shorter MFAT. The MFAT
has been optimized in two variables (r, α) with D = D(α) chosen properly to fix some
measure of average mobility.
3.4.3
Results
Quite surprisingly the MFAT turns out to be finite for α > 1 and r > 0. In contrast,
for α ≤ 1 the searcher will not reach the target at all9 . Results for α ≤ 1 have not been
discussed in the paper—for more details, including the proof of the above statement, see
Appendix F.
A few examples of symmetric target position distributions have been analyzed. In
each case optimal parameters (r∗ , α∗ ) and the corresponding optimal MFAT T ∗ have been
numerically calculated for different values of a mean distance to the target λt = h|Xt |i.
Accordingly, graphs of optimal parameters and the optimal MFAT in function of λt have
9
More precisely: with probability one the target will not be reached in a finite time.
47
3.4. PAPER IV
been plotted.
In the simplest case the target is set to be at a fixed position xt , which is equivalent to
the position distribution pXt (x) = 12 δ(x−xt )+ 21 δ(x+xt ). This is analogous to the set-up of
the model in Paper III and one may be tempted to assume that the results of optimization
in both cases should be qualitatively the same, but this is not the case. In Paper IV it
is shown that for small distances to the target Gaussian diffusion is optimal, while in the
discrete time model from Paper III the Gaussian diffusion is never optimal. Furthermore,
in contrast to the discontinuous transition found in Paper III, here the resulting plots of
optimal parameters in function of xt are continuous. Still, they are not smooth as there
exists bifurcation point x∗ , at which the Gaussian diffusion loses its optimality. Below x∗
the Gaussian diffusion (α = 2) is optimal, while above x∗ LFs (α < 2) are optimal.
The Laplace distribution with a fixed mean distance to the target λt has been studied.
The graphs of optimal parameters and the optimal MFAT as a function of λt are qualitatively the same as these for the fixed position. A noticeable difference between these two
cases lies in the behavior of the Gaussian diffusion as a function of r. In the case of the
fixed position the MFAT is finite for any r > 0, whereas the Laplace distribution is more
demanding: the MFAT is finite only in the range r ∈ (0, λD2 ). The Laplace distribution has
t
also been employed to exemplify the insufficiency of including only the expected distance
to the target in the strategy planning.
Heavy-tailed distributions have been additionally investigated. Power-law tails preclude the Gaussian diffusion from the set of admissible strategies, For α = 2 the MFAT
diverges if tails of the distribution decay slower than exponentially, thus power-law tails
preclude the Gaussian diffusion from the set of admissible strategies. LFs are in this case
superior to the Gaussian diffusion—there exist α’s for which the MFAT is finite if only
the variance of the distribution exists. Since the Gaussian diffusion is never optimal for
heavy-tailed Xt distributions the graphs of optimal parameters must be different from the
previous ones. As an example, Student’s t-distribution has been numerically studied for
two different values of ν (degrees of freedom) and, as expected, no bifurcation point is
present.
48
Chapter 4
Conclusion
Many extensions and modifications of the described models are possible. First, all of
the presented papers have been devoted to one-dimensional processes. A natural generalization of our approach would be to introduce more spatial dimensions. However,
components of α-stable random vectors invariant under rotations are not independent
(except for the Gaussian case), which complicates the analysis of processes based on them
[DS15]. Similarly, an underdamped scenario is more involved. The corresponding PDF
solves a fractional Klein-Kramers equation [KBD99; KRS08]. Interestingly, even in the
simplest case of a harmonic potential, the momentum and the position in the stationary
distribution are not independent [SED11].
Another promising modification would be to explore Lévy walks with resetting. Lévy
walks, in which a constant velocity of a walker is assumed, should in serve as more realistic
models of animal movement than LFs. Also, subdiffusion with stochastic resetting is worth
investigating, since subdiffusive processes has been suggested to properly describe the kinetics in overcrowded environments, e.g. the cytoplasm of living cells [Wei+04; Jeo+11].
An impact of bounded domains on the diffusion with resetting has been recently investigated numerically [CS15], but an analytical calculations are still to be performed. In
the context of the search strategy optimization, Evans et al. [EMM13] have compared the
(non-equilibrium) resetting strategy with an equilibrium strategy leading to the same stationary distribution. They have shown that better performance of the search is obtained
with the resetting strategy. Another question to be posed and analyzed is whether the
resetting strategy is more effective than any equilibrium dynamics.
With respect to Papers I and II, further analysis of FRs can be proposed. It may be
possible to generalize the Crooks and Jarzynski relations to systems amid white α-stable
noise. The detailed balance is broken in the stationary distributions of these systems (cf.
Appendix C). It should be thus possible to calculate the rates of entropy production and
probability currents, although the nonlocal character of the fractional operator begets
issues with their definitions. This will be the subject of our future studies.
49
Acknowledgments
I would like to express my gratitude to my supervisor, professor Ewa Gudowska-Nowak,
for her constant support and inexhaustible patience. Over the past few years she has
eagerly shared with me her knowledge, experience, and new ideas, without which this
thesis would have never came into existence and I would be a different person.
Special thanks go to my Mom, for pushing me to finish this thesis.
Finally, I am pleased to acknowledge financial support from National Science Center (Grant No. DEC-2014/13/B/ST2/02014).
50
Appendices
A
Papers
51
Vol. 44 (2013)
ACTA PHYSICA POLONICA B
No 5
ONSAGERS FLUCTUATION THEORY
AND NEW DEVELOPMENTS INCLUDING
NON-EQUILIBRIUM LÉVY FLUCTUATIONS∗
Ł. Kuśmierz
The M. Smoluchowski Institute of Physics, Jagiellonian University
Reymonta 4, 30-059 Kraków, Poland
W. Ebeling, I.M. Sokolov
Institute of Physics, Humboldt-University Berlin
Newtonstr. 15, 12489 Berlin, Germany
E. Gudowska-Nowak
The M. Smoluchowski Institute of Physics
and The M. Kac Complex Systems Research Center
Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland
(Received May 29, 2013)
The first part of the paper briefly reviews and explains basic ideas of the theory of Gaussian fluctuations and their relaxation developed in 1931 by Lars Onsager in the context of a general theory of irreversible processes. Motivated by Onsager’s approach, we extend the theory to fluctuations including Lévy processes.
We assume that deviations from Gaussian distributions, which are often observed
in non-equilibrium systems, may be described by convoluted Gauss–Lévy distributions and their relation to stationary states by generalized Smoluchowski equations. The central part of the distributions we study here is determined by the
Gaussian core with the wings decaying according to a power law characteristic for
a Lévy-type contribution to statistics. Furthermore, we develop a generalization
of Onsager’s theory of linear relaxation processes to those which include statistically independent Gaussian fluctuations and (non-equilibrium) Lévy noises. We
apply the generalized version of the fluctuation-dissipation theorem (FDT) to analyze regime of the linear response of the non-equilibrium system driven by Lévy
(Cauchy) white noise and subject to thermal (Gaussian) fluctuations. In the last
part, applications to non-Maxwellian velocity fluctuations and their relaxations
are investigated.
DOI:10.5506/APhysPolB.44.859
PACS numbers: 05.40.Fb, 02.50.–r, 87.16.Nn
∗
Presented at the XXV Marian Smoluchowski Symposium on Statistical Physics,
“Fluctuation Relations in Nonequilibrium Regime”, Kraków, Poland, September
10–13, 2012.
(859)
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Ł. Kuśmierz et al.
1. Introduction
On the basis of the early work by Boltzmann and Einstein, the modern theory of fluctuations was created in two fundamental works by Onsager [1, 2], which nearly 40 years later were pointed out when nominating
the Author to the Nobel Prize for “the discovery of the reciprocal relations
bearing his name, which are fundamental for the thermodynamics of irreversible processes”.
The basic ideas of Onsager’s fluctuation theory can be found in the second part of the eminent paper [2]. More than 20 years after its publication
in 1931, Onsager’s theory was extended and generalized in a fruitfull collaboration with Machlup [3].
In 2013 the 110th birthday anniversary of Lars Onsager takes place. On
that occasion, let us briefly recollect several biographical facts from Onsager’s works and life [4]. He was born in Oslo, November 27, 1903, to
parents Erling Onsager, the barrister of the Supreme Court of Norway and
Ingrid née Kirkeby. Young Lars Onsager received a quite liberal education:
His main interests were focused on classical literature, philosophy and music. In 1920, Onsager became matriculated at the Norges Tekniske Hoiskole,
Oslo in the field of chemical engineering. It is reported that beside the basic program of studies which he did not take too serious, his main interest
was to study a textbook on mathematical functions written by Whittaker
and Watson. In forthcoming years Onsager devoted himself to Debye’s theory of electrolytes and in 1925 he visited Debye in Zürich. The biography
sources say that he introduced himself to a prominent professor with the
words : “Professor Debye, your theory is incorrect”. Apparently, Debye indulged the impoliteness and offered him an assistenship. During the time
spent in Zürich, he worked on irreversible processes in electrolytes which,
at that time, were the best studied examples of linear irreversible processes.
Onsager succeeded to develop a new, more symmetric and in this way, also
more correct version of the Debye–Hückel theory, a work which found broad
recognition worlwide [5].
In 1928, Onsager was appointed an associate in chemistry at the Hopkins University, but he eventually failed and was fired. Over the period
1928–33 he worked on the theory of kinetic rates of chemical reactions employed by C.A. Kraus at the Department of Chemistry, Brown University.
In 1933 Onsager was appointed a postdoctoral Sterling and Gibbs Fellowship at Yale University. When the Chemistry Department found out that
he had no Ph.D., Onsager had to do something. An outline of his results on
the reciprocal relations submitted to the Trondheim University was rejected
for a doctorate. His colleagues at Yale suggested that for the thesis any
published work would do. However, Onsager felt that he should write some-
Onsagers Fluctuation Theory and New Developments Including . . .
861
thing new and submitted a thesis on the solutions of the Mathieu equations
which brought him the Ph.D. degree awarded in 1935. Already in 1934,
he was appointed an Assistent Professor in the Chemistry Department at
Yale, where he was to remain for the greater part of his life. Between 1936–
1939, Onsager published several works on dielectric properties (claimed to
be “unreadable”, according to Debye) and theory of turbulence. Over next
years, the subject of his scientific interest switched occasionally from theory of fluctuations to order–disorder transitions, theory of magnetization,
quantized vorticity and lattice Ising problem. In 1968 Onsager was awarded
the Nobel Prize in chemistry. A year later, he received the National Science
Medal and became a honorary member of The Bunsen Society for Physical
Chemistry. During spring 1970 he was Lorentz Professor in Leiden and in
1972–76, being already emeritus and living in Coral Gable/Miami he worked
actively on various biophysical problems [4].
Apart from an elegant solution to the lattice Ising problem, the major
Onsager’s achievement is the fluctuation theory based on the assumption
that the fluctuations follow linear laws and are described by Gaussian probability distribution functions. His theory predicted the symmetry relations
for the coefficients in the linear laws of relaxation to equilibrium and provided tools to study general irreversible processes. A possible generalization
of this point, specifically allowing for inclusion of non-Gaussian fluctuations
is a main objective of our studies.
In the recent experimental work, an accumulating evidence demonstrates
that in certain non-equilibrium systems, the distribution of fluctuating physical quantities possesses also, beside a Gaussian “core” part, a heavy-tailed
wing typical for of Lévy-type distributions [6, 7]. Here, we will present an
entirely phenomenological approach to this problem by including Lévy-type
terms into the phenomenological Onsager theory of linear relaxation processes. This way, we aim to describe some realistic situations where Lévy
flights are considered as external perturbations to weakly non-equilibrium
(i.e. influenced by Gaussian thermal fluctuations) thermodynamic states.
Our method is based on generalizations of the Smoluchowski–Fokker–Planck
equation (SFPE), which describes a normal diffusion under the influence
of an external force field, to situations modeling anomalous (super) diffusion [8–12]. The common SFPE can be then replaced by space-fractional
equation which governs evolution of the probability density p(x, t). This
type of equation can be derived either from the generalized Master Equation with long-range jump length statistics [11], or from a suitable Langevin
equation with additive, white Lévy noises [9].
There exist several theoretical approaches which connect Lévy-type distributions with Langevin– and Fokker–Planck equations [8, 9, 13–23]. We
follow here a different way of reasoning. First, we start with fairly gen-
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Ł. Kuśmierz et al.
eral Onsager’s theory of linear relaxation and fluctuation processes [24, 25],
and include two statistically independent white-noise sources. Apart from
discussing relaxation properties of the system in terms of Gaussian thermal fluctuations [26], we analyze its response to additional external white
Cauchy noise [24, 27].
Notably, the problem of various noise sources in a classical Langevin
equation is well investigated, both for Gaussian, as well as for more general Lévy-type random forces [8, 24, 27]. Usually, non-Gaussian Lévy noise
sources are characterized by a stability parameter 0 < α < 2 which determines their asymptotic properties, i.e. power law decay of the probability
distribution function (PDF), Lα,β (y) ∝ |y|−(α+1) . Self-affine Lévy PDF are
found ubiquitous in nature: Examples of super-diffusion include motion of
fluid particles in fully developed turbulence, ion transport in tokamak plasmas [13], tracer particles in vortex arrays in a rotating flow [28], layered
velocity fields [29], and Richardson diffusion [30]. Lévy superdiffusion and
the so-called truncated Lévy flights, in which arbitrarily large steps are eliminated by a cutoff [31, 32], have been also used extensively to model stock
markets. Accordingly, they have finite variance and are more suitable to address diffusive transport in physical systems, in which an unavoidable cutoff
seems to be always present, due to e.g. finite size of the system. Moreover,
in molecular spectroscopy and atmospheric radiative transfer, the combined
effects of Doppler and pressure broadening lead to the so-called Voigt profile function which is the convolution of Gaussian (representing the Doppler
broadening) and the Lorentzian (responsible for pressure broadening) distributions [33, 34].
Since the Gaussian and Maxwell distributions are the key distributions
in equilibrium statistical mechanics, clearly use of more general, Léve-type
PDFs requires extension to non-equilibrium situations [8, 9, 13, 16, 18–20].
In particular, the problem of velocity and energy distribution for exploding
Coulomb clusters by using combination of Gaussian and Lévy-type stochastic forces has been addressed [17].
The goal of this work was stimulated by several observations that in
many non-equilibrium fluids and plasmas, in particular in turbulent systems
and in cell kinetics, non-Gaussian distributions and anomalous diffusion are
observed [6, 7, 13, 15, 35–39]. One of possible causes of this could be,
from our point of view, noise sources with a Gaussian and a non-Gaussian
component. We will consider here systems with two noise sources with
Gaussian and Lévy-type probability distribution functions. As a specific
example, we will study Cauchy distributions, which are analytically most
simple Lévy PDF. We consider the convolution of two distributions and the
solution of Langevin and Smoluchowski equations with two noise sources.
Onsagers Fluctuation Theory and New Developments Including . . .
863
2. Onsager’s theory of fluctuations
and linear relaxation processes in a nutshell
2.1. One relaxation variable
According to Einstein’s postulate, any macroscopic quantity x may be
considered as a fluctuating variable, which is determined by a certain probability distribution function w(x). We consider a Gaussian distribution
r
Λ
Λ(x − x0 )2
p(x) =
exp −
.
(1)
2π
2
The mean value
R is given by the first moment of the probability distribution
x0 = hxi = x · w(x)dx. In a stationary state, without loss of generality,
one may shift the origin and assume x0 = 0. The dispersion is then given
by hx2 i = Λ1 . In view of the fact that at equilibrium state (x = 0) entropy
assumes a maximum, the following relations hold
S(x = 0) = max ;
2 ∂ S
∂S
= 0;
≤ 0.
∂x x=0
∂x2 x=0
(2)
(3)
According to Onsagers’s view, the relaxation dynamics of the variable x is
determined by the first derivative of the entropy, which is different from zero
outside equilibrium. Starting from a deviation from the equilibrium (with
entropy value below its maximum), the spontaneous irreversible processes
should drive the system towards increasing entropy, so that
d
∂S dx
S(x) =
·
≥ 0.
dt
∂x dt
(4)
In this expression two factors appear which were interpreted by Onsager in
a quite ingenious way. First, the derivative
X=−
∂S
∂x
(5)
is considered — in the spirit of the Second Law — as the driving force of
the relaxation to equilibrium. In irreversible thermodynamics this term is
named in analogy to mechanics the thermodynamic force conjugated to a
dynamic variable x. This analogy suggests that the (negative) entropy takes
over the role of a potential. The second term
J =−
dx
dt
(6)
864
Ł. Kuśmierz et al.
is considered as the thermodynamic flux or thermodynamic flow. Onsager
(1931) postulated a linear relation
J = LX = −ẋ .
(7)
The idea behind is that the thermodynamics force is the cause of the thermodynamic flow and both should disappear at the same time. The coefficient L
is called Onsager’s phenomenological coefficient, or Onsager’s kinetic coefficient. From the Second Law it follows that the Onsager-coefficients are
strictly positive
P ≡
∂S
d
S(x) = ẋ
= J · X = L · X2 ≥ 0 .
dt
∂x
(8)
Onsager’s postulate about a linear connection between thermodynamic forces
and fluxes has been the origin of the development of the thermodynamics
of linear dissipative system, termed also linear irreversible thermodynamics.
A remarkable property of the theory is the bilinearity of the entropy production P = J · X. Moreover, since p(x)dx is proportional to the number of
accessible microscopic states of the system, by use of the Boltzmann identity
p(x)dx = const × eS(x)/kB , we find for the neighborhood of the equilibrium
state the relation
∂S
X=−
= kB Λx .
(9)
∂x
Using Eq. (7), we get finally the following linear relaxation dynamics
ẋ = −LX = −kB LΛx = −λx ,
(10)
where λ = LkB Λ stands for the so-called relaxation coefficient of the quantity x. Accordingly, the entropy function can be expressed as a Taylor series
around the equilbrium value x0 = 0
S(x) = S(x0 ) − 21 kB Λ(x − x0 )2 + . . .
(11)
The linear kinetic equation (10) describes the relaxation of a thermodynamic
system brought initially out of equilibrium. Starting with the initial state
x(0), dynamics of the variable x follows the trajectory
x(t) = x(0) exp[−λt] .
(12)
We see that t0 = λ−1 plays the role of the decay time of the linear deviations from the equilibrium. This way, we arrived for the first time at the
so-called fluctuation-dissipation relation. In his approach to relaxation dynamics, Onsager assumed that deviations of macroscopic observables from
their equilibrium values and spontaneous fluctuations around the equilibrium
state follow the same kinetics.
Onsagers Fluctuation Theory and New Developments Including . . .
865
The Onsager theory for one fluctuating variable may be formulated in a
compact form by use of the Smoluchowski equation which we devise in two
steps. First, we assume a continuity equation for the probability density
−
∂
p(x, t) = ∇x j(x, t) .
∂t
(13)
Here, p(x, t) can be interpreted as a concentration of particles at a given
position x and time t. In the next step, we assume
j = −λxp(x, t) − D∇x p(x, t) ,
(14)
where D stands for the diffusion coefficient. The meaning of this relation is
that there is at first a deterministic flow ẋp = −λxp into the direction of the
equilibrium state and an opposite compensating “diffusional flow” following
the gradient of the concentration (alternatively, the gradient of the probability density). With these assumptions, we get the standard Smoluchowski
equation which describes the relaxation of the probability density function
to the stationary distribution
∂
p(x, t) = ∇x (λxp(x, t) + D∇x p(x, t))
∂t
(15)
with ∇x = ∂/∂x. The stationary solution (t → ∞) to the Smoluchowski
equation reads
r
λ
λx2
pss (x, t) =
exp −
,
(16)
2πD
2D
and the corresponding stationary dispersion (we have assumed x0 = 0) is
hx2 i = D/λ. In order to be compatible with the Onsager approach, we
identify
λ
Λ=
.
(17)
D
Formula (17) expresses a fluctuation-dissipation relation between the dispersion of fluctuations around equilibrium and the linear transport coefficient of
the relaxation problem. The relaxation of the distribution to the stationary
one is described by the Smoluchowski diffusion equation. At the same time,
by use of Eq. (16), oneR canRdetermine the time derivative of the correlation
function hx(t)x(0)i = dx dx0 xx0 p(x, t|x0 , 0)pss (x0 )
∂
hx(t)x(0)i = −λ hx(t)x(0)i .
(18)
∂t
R
With the (equilibrium) initial condition hx(0)2 i = dxx2 pss (x), the solution
to the above equation yields hx(t)x(0)i = D
λ exp(−λt).
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Ł. Kuśmierz et al.
2.2. Many coupled relaxation variables
The formalism of the preceding section can be easily extended to systems
described by n forces Xi and conjugated displacements xi which, in most
general case, are cross-coupled and have to vanish at the state of equilibrium.
Retaining terms up to the second order only, we get for the entropy
X
S(x1 , . . . , xn ) = Smax − 12 kB
Λij xi xj .
(19)
i,j
Following the Onsager approach, we get the forces and flows
X
Xi = kB
Λij xj ,
Ji = −ẋi .
(20)
The generalized linear Onsager-ansatz reads
X
X
Ji =
Lij Xj ,
Lij Xi Xj ≥ 0 .
(21)
j
j
The positive definiteness of the matrix Lij follows from the positivity of
the entropy production which is now a bilinear expression in the fluxes and
thermodynamic forces appearing in phenomenological equations for which
the Onsager relations hold
X
P =
Ji Xi ≥ 0 .
(22)
i
By using Eqs. (20), (21), we get
ẋi = −
X
Lij Xj ,
(23)
j
and after introducing the matrix of relaxation coefficients, we end up with
X
ẋi = −
λij xj ,
(24)
j
λij = kB
X
k
Lik · Λkj .
(25)
Since the matrix Λij determines the dispersion of the stationary fluctuations,
we have found again a close relation between fluctuations and dissipation,
i.e. we have got a fluctuation-dissipation relation for a set of fluctuating and
relaxing variables.
Onsagers Fluctuation Theory and New Developments Including . . .
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The Smoluchowski equation assumes now the form
X
∂
p(x1 , ..., xn , t) =
∇i (λij xj p(x1 , ...xn , t) + Dij ∇j p(x1 , ...xn , t)) (26)
∂t
ij
with the stationary solution

ps (x, t) = C exp − 21
X
ij

Λij xi xj  .
(27)
The fluctuation-dissipation relations between the matrices is given by
Eq. (25). The linear kinetic equation (24) describes the relaxation processes
which are characterized typically by an exponential decay in time. Further,
the Smoluchowski equation (26) describes the relaxation of the probability
distributions to the stationary solution. We underline that the information about the relaxation process contained in the Smoluchowski equation
is more extended than the information contained in the relaxation dynamic
(24), since the Smoluchowski equation describes the mean as well as the
fluctuations. We will show that this description is the appropriate form to
be generalized to fluctuations of a Lévy type.
3. Including Lévy flights into the relaxation-fluctuation theory
3.1. Linear response theorem
Let us first briefly review the linear relaxation theory in the context
of fluctuation-dissipation theorem (FDT), as discussed for Markovian processes [24, 25, 40]. The theorem applies to any Markov process x(t) whose
dynamics depends on a set of parameters and for which a well-defined (nonequilibrium) stationary state exists.
We would like to study the linear response of the system to (weak) perturbations f (t) = f0 Θ(−t) switched on at some distant past time and switched
off at time t0 . Evolution of the dynamic variable x(t) is governed by the
propagator p(x0 , t|x, 0)
Z
Z
0
hx(t)i =
dx
dx x0 p(x0 , t|x, 0)p̃(x, 0) ,
p̃(x, 0) = R
e−β[H0 (x)+xf0 ]
e−βH(x)
R
=
dx0 e−βH(x0 )
dx0 e−βH(x0 )
(28)
with p̃(x, 0) given by a canonical form, in which perturbation forces coupled
to dynamic variable x(t) contribute to the energy of the system and β stands
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Ł. Kuśmierz et al.
for the reciprocal temperature β = 1/kB T . By performing approximation
e−βxf0 ≈ 1 − βxf0 ,
p̃(x, 0) ≈ p0 (x)(1 − βf0 (x − hxi)) = p0 (x)(1 − βf0 x) ,
the average value of x(t) is given by
Z
Z
0
hx(t)i =
dx
dx x0 p(x0 , t|x, 0)p0 (x)(1 − βf0 x)
= hxi0 − βf0 hx(t)x(0)i0 .
(29)
(30)
Here, subscript 0 denotes average taken with unperturbed form of the distri−βH0 (x)
bution p(x) = R e 0 −βH
0 . We expect that response to (weak) perturbation
0 (x )
dx e
can be expressed via linear relation
hx(t)i = hxi0 +
Zt
−∞
(31)
f (τ )χ(t − τ )dτ ,
in which RHS integrated over time and compared with Eq. (30) leads to the
identity
f0
Z∞
0
dτ Θ(τ − t)χ(τ ) = βf0 hx(t)x(0)i0 ,
−χ(t) = β
d
hx(t)x(0)i0 .
dt
(32)
Hence the fluctuation-dissipation theorem (FDT) relates susceptibility χ(t)
to correlations measured in the reference unperturbed state. In general, the
variable conjugate to perturbations fγ can be defined as
∂ ln pss x; f~ ∂φ
Xγ (x) = −
=
.
(33)
∂fγ
∂fγ
f~=f~0
Note that in the above definition φ ≡ − ln pss stands for a non-equilibrium
potential [24, 25]. If the reference state is the Gibbs equilibrium state corresponding to a temperature kB T = β −1 and a Hamiltonian H(x; f~ ), the stationary PDF pss (x; f~ ) assumes the form pss (x; f~ ) = exp[−βH(x; f~ )]/Z(β, f~ )
and the conjugate variable reads
h i ~
~
1 ∂ H x; f − F β, f Xγ (x) =
,
(34)
kB T
∂fγ
~ ~
f = f0
Onsagers Fluctuation Theory and New Developments Including . . .
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where F = −kB T ln Z stands for the free energy. Accordingly, X
γ can be
~
~
f )
f0 )
≡ ∂H(x;
interpreted as the fluctuation of the quantity ∂H(x;
∂fγ
∂fγ ~ ~
f = f0
Xγ (x) =

1 
kB T
∂H x; f~0
∂fγ
−
*
+ 
∂H x; f~0
.
∂fγ
(35)
0
We see that if the control parameter responsible for deviations from stationary state appears in the Hamiltonian as −fγ xγ , then the conjugate variable
Xγ = −(xγ − hxγ i)/(kB T ) represents fluctuations of xγ . Following the Onsager theory, the above relation holds true generally for any pair of conjugate
thermodynamic variables (Xγ , fγ ) provided one can assume that the perturbation of the equilibrium state is linear.
In general, if the reference state pss (x; f~0 ) is not an equilibrium state,
the conjugate variables defined by Eq. (33) do not have any straightforward
physical interpretation [24, 25, 41]. They do not have also any particular
signature with respect to the time reversal [40]. In practice, it is therefore
by no means a trivial problem to identify correct choice of generalized forces
and “displacements” [25, 42].
We proceed to discuss response of the linear system (over-damped Lévy–
Brownian particle) modeled by the corresponding Langevin equation
ẋ(t) = µ0 − λx + f (t) + ξC (t) + ξG (t) ,
(36)
where ξC (t) and ξG (t) stand for symmetric Lévy noises with stability indices
αC = 1 (for Cauchy) and αG = 2 (for Gaussian case). Here, f (t) represents
additional deterministic, time-dependent force field. The noises are defined
as the time derivatives of stationary Lévy processes, both described by means
of characteristic functions
2
2
ϕG (k, t) = e−σ0 |k| t ,
(37)
ϕC (k, t) = e−γ0 |k|t .
(38)
Since the Langevin Eq. (36) is linear, its solution depends linearly on two
independent stable processes. Accordingly, PDF of x(t) attains the form
of the convolution of two Lévy PDFs. The corresponding characteristic
function is then a product of two characteristic functions
p̂(k, t) = eikµ(t)−σ
2 (t)|k|2 −γ(t)|k|
.
(39)
In the above equation µ(t) stands for the time-dependent location parameter,
whereas σ 2 (t), γ(t) represent scale parameters (intensities) of the Gaussian
and Lévy–Cauchy noises, accordingly.
870
Ł. Kuśmierz et al.
Note, that Eq. (36) can be associated with fractional Fokker–Planck
equation [14, 17–23] in which Gaussian and Cauchy terms appear as independent contributions
∂p(x, t)
∂
= −
[µ0 − λx + f (t)] p(x, t)
∂t
∂x
∂
∂2
p(x, t) .
+σ02 2 p(x, t) + γ0
∂x
∂|x|
(40)
Here, the
h fractional
i (Riesz–Weyl) derivative is defined by its Fourier transα
∂
α
form F ∂|x|
α f (x) = −|k| F [f (x)]. Accordingly, Eq. (40) has the following
Fourier representation
∂ p̂(k, t)
∂
= −λk p̂(k, t) + ik [µ0 + f (t)] p̂(k, t)
∂t
∂k
2
−σ0 |k|2 p̂(k, t) − γ0 |k|p̂(k, t) ,
(41)
where p̂(k, t) = F [p(x, t)]. For µ0 = γ0 = f (t) = 0 and σ 2 ≡ D, Eq. (40)
attains a form of a standard Smoluchowski equation Eq. (15) discussed in
Section 2.
We note that now the relation between the fluctuations and the transport
coefficient is more complicated and cannot be expressed by a simple relation
as Eq. (17). The Lévy jump length statistics leads to an inherently nonequilibrium situation: the stationary probability distribution of the process
is of Lévy type with the inverse power-law character in the tails. Accordingly, the stationary PDF deviates in such cases from a common exponential
Gibbs–Boltzmann form.
In particular, we have to take into account that the dispersion hx2 i does
not exist anymore. The distribution of fluctuations is characterized by a
bulk Gaussian body and a heavy tail. Asuming an initial condition f (k, 0) =
δ(k − k0 ), i.e. a wave-like perturbation, and µ0 = 0, we show in Fig. 1 how
the Fourier modes of the distribution decay in time.
By using the ansatz Eq. (39) and FFPE (41), we can easily obtain evolution equations for parameters
µ̇(t) = −λµ(t) + f (t) ,
γ̇(t) = γ0 − λγ(t) ,
dσ(t)2
= σ02 − 2λσ(t)2 .
dt
(42)
(43)
(44)
Onsagers Fluctuation Theory and New Developments Including . . .
871
Fig. 1. Convoluted Cauchy–Gauss probability distribution functions compared
at different times. Lower panel represents the logarithm of the non-equilibrium
pseudo-potential φ(x, t) (cf. Eq. (52)).
In order to analyze the generalized susceptibility, it is sufficient to derive a
stationary solution to FFPE for a constant force f
f
,
t→∞
λ
γ0
:= lim γ(t) =
,
t→∞
λ
σ2
:= lim σ(t) = 0 .
t→∞
2λ
µ∞ := lim µ(t) =
(45)
γ∞
(46)
2
σ∞
(47)
These results are the same as for the case with only one noise source (stable
process) [24].
872
Ł. Kuśmierz et al.
3.2. Stationary PDF
The characteristic function for a force-free case (hence, µ∞ = 0) reads
2
p̂ss (k) = e−σ∞ |k|
2 −γ |k|
∞
(48)
.
Although the corresponding PDF
1
pss (x) =
2π
Z∞
dk p̂s (k)e−ikx
(49)
−∞
cannot be expressed in terms of elementary functions, it can be nevertheless
rewritten using the Faddeeva function (also known as the complex error
function)
2
w(x) := e−x erfc(−ix) ,
(50)
where erfc(x) is the complementary error function. Accordingly,
−x + iγ∞
1
pss (x) = √
<w
.
2σ∞
2 πσ∞
1
(51)
σ=1, γ=0
σ=0.8, γ=0.5
σ=0, γ=1.8
0.9
0.8
p(x)/p(0)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
7
x
Fig. 2. Comparison of a Gauss distribution (fast decay of extreme cases) with a
Cauchy distribution (slow decay) and a convoluted Gauss–Cauchy distribution.
Graphs represent the ratio of p(x) with respect to the maximal value at the origin.
The case of the convoluted Gauss–Cauchy distribution describes a typical Voigt
profile of a spectral line, see the main text [33, 34].
Onsagers Fluctuation Theory and New Developments Including . . .
873
In Fig. 2, we show a comparison of (time-independent) Cauchy, Gauss
and mixed Cauchy–Gauss distributions. We see, that in a convolution of
Gauss and Lévy distributions the tail is determined by the Lévy part and the
body by the Gauss part. Such distributions are widely used in spectroscopy
for the description of line profiles [43]. If ions are at a non-zero temperature,
their profiles are widened in accordance with the Doppler effect. The Doppler
effect is associated with temperature; particles moving in random directions
due to thermal motion away and towards the observing optics will produce
a Gaussian profile. In addition to temperature broadening, the Stark effect
also causes lines to broaden. The Stark effect results from the electric field
imposed on the radiating particle by the charged particles surrounding it and
causes spectral lines to have a Lorentzian (Cauchy) profile. If the line profile
results from the convolution of two (statistically independent) broadening
mechanisms, one observes typically Voigt profiles.
3.3. Conjugate variable
Non-equlibrium pseudo-potential of the system reads
(52)
φ := − ln pss (x) ,
where pss (x) is the stationary PDF (the stationary solution to the corresponding Langevin equation for a constant force f ). Accordingly, the conjugate variable takes the form
∂φ ∂ ln pss (x)
X :=
=X=
.
(53)
∂f f =0
∂x
Using formula (51), it is easy to obtain the conjugate variable for the system
subject to Cauchy and Gauss noises simultaneously
XGC = −
−x+iγ∞
2σ∞
γ∞ = w
x
.
−
2 λ
2 λ
−x+iγ∞
2σ∞
2σ∞
<w
2σ∞
(54)
The limit of pure Gauss (Cauchy) driving can be obtained by taking the
limit
x
x
=− 2,
2
2λσ∞
σ0
2λx
.
= − 2
γ0 + λ2 x2
lim XGC = −
(55)
lim XGC
(56)
γ0 →0
σ0 →0
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Ł. Kuśmierz et al.
In order to consider the effect of a weak Cauchy (Gauss) noise, one can
expand (54) in series of σ10 ( γ10 )
XGC
q
γ0 π2 x γ0 2 (−2 + π)x
x
−
= − 2+ √
σ0
λπσ0 4
λσ0 3
6
γ0 x −3γ0 2 (−4 + π) + λ2 πx2
1
√
+
+O
.
σ0
3 2λ3/2 π 3/2 σ0 5
By expanding (54) in series of
XGC
1
γ0 ,
(57)
one gets
6
1
2(λx) 2λ2 x 5σ0 2 + λx2
+
+
O
.
=−
γ0 2
γ0 4
γ0
(58)
3.4. Susceptibility and response
We are now at position to compare the response of the system to external
perturbation as calculated directly from the definition
hX(t)i =
Z∞
X(x)p(x, t)dx ,
(59)
−∞
or, otherwise determined by the generalized susceptibility
χ(t) =
d
hX(t)X(0)i0
dt
within linear response theory
hX(t)iLR =
Zt
0
χ(t − s)f (s)ds .
(60)
The time-dependent average (59) can be calculated exactly with the
probability density
p(x, t) =
Z∞
p(x, t|x0 , 0)p(x0 )dx0 ,
(61)
−∞
where p(x0 ) ≡ ps (x = x0 ) is an initial stationary distribution at zero force
(f (t) = 0) and p(x, t|x0 , 0) is given by the inverse Fourier transform of the
solution to the FFPE Eq. (41).
Onsagers Fluctuation Theory and New Developments Including . . .
875
On the other hand, the FDT relates the susceptibility with the autocorrelation of the conjugate variables in the reference state, i.e., for f = 0. The
autocorrelation is defined as
hX(t)GC X(0)GC i0
(62)
with X(t)GC given by Eq. (54). From the above, the generalized susceptibility can be derived by differentiation with respect to time
χ(t) =
d
hX(t)X(0)i0 .
dt
(63)
Note that, unlike Eq. (32) which describes relaxation of the system after
a stepwise external field has been switched off, in the above formula, the
response is related to the changes of quantity X(t) when an external field is
applied. The causality criterion for both cases is reflected in different sign
of the derivative with respect to time, cf. Eqs. (32), (63).
Fig. 3. Comparison of response to external driving f (t) = sin(t)/10 + t/100 evaluated by Monte Carlo solution to Langevin equation and by use of the linear
response theory. Deterministic relaxation time of the linear system has been set
up to 1 (λ = 1).
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Ł. Kuśmierz et al.
For the mixed Gauss–Cauchy case, the integrals Eq. (60) have been evaluated numerically and the results are presented in Figs. 3, 4. Direct inspection
of the above figures indicates that the range of the linear response is finite
and “exact” (Monte Carlo) and LRT-approximated curves of response diverge
for times t > 50, when the external perturbation f (t) = sin(t)/10 + t/100
becomes stronger (all units of time in our model have been set up to 1).
The results resemble closely the ones analyzed in [24], however, inclusion of
convoluted noises results in “tuning” of the response caused by the action
of relatively strong Cauchy noise (cf. upper and lower panels in Fig. 4).
Fig. 4. Response of the system to external driving f (t) = sin(t)/10 + t/100 evaluated by Monte Carlo solution to Langevin equation and by use of the linear
response theory. Graphs represent results obtained for a constant scale parameter
of Gaussian noise σ 2 and various intensities of the scale parameter γ characterizing
Lévy–Cauchy noise. Deterministic relaxation time λ−1 has been set up to 1.
Onsagers Fluctuation Theory and New Developments Including . . .
877
Closer examination of a relative error in approximated and exact response
exhibits damping of those differences for increasing γ (results not shown).
This observation can be traced back to a nonlinear character of the conjugate variable X(t)GC coupled to the driving force f (t) and will be a subject
of a seperate analysis.
4. Applications to fluctuations of the velocity, its modulus
and energy distributions of Lévy-type
4.1. Distributions of velocity and kinetic energy
One of the first distributions studied in physics was the Maxwell distribution of velocities of molecules of an ideal gas. In units in which the mass of
a molecule is set m = 1, each component of the velocity vector vi , i = 1, 2, 3
is distributed according to the PDF
p
pM (v) = β/2π exp −βvi2 /2 ,
(64)
where β = 1/kB T . The corresponding distribution of the modulus of the
velocity v = |v| reads
pM (|v|) = const|v|2 exp −β|v|2 /2 ,
(65)
and the kinetic energy = mv 2 /2 is, accordingly, distributed as
√
pM () = const exp(−β) .
(66)
However, deviations from Maxwell distributions are quite ubiquitously
observed in nature: Inelastic collisions between particles in granular matter
create corrrelations which are responsible for velocity distributions departing
from the standard Maxwell PDF and, in consequence, overpopulation of
high energy tails [44]. Temperature fluctuations of the cosmic microwave
background radiation [45] have been reported to follow PDF with algebraic
tails and experimental investigation of the edge turbulence in the fusion
devices showed that plasma is characterized with non-Gaussian statistics and
non-Maxwellian velocity distribution [13, 17, 39]. The Gaussian distribution,
which we find most often under equilibrium conditions, is a special case of
a more general class of Lévy-type distributions. Hereafter we assume that
in appropriate units the components of the velocities may be distributed
according to a Lévy alpha-stable, symmetric distribution in velocity space
characterized by and index α and defined by the form
1
pL (v) =
2π
Z∞
−∞
1
exp [−itv − |t| ] dt =
π
α
Z∞
0
cos(vt) exp [−tα ] dt .
(67)
878
Ł. Kuśmierz et al.
The asymptotics of this distribution for large v and α < 2 reads
pL (v) ∼
α sin(πα/2)Γ (α)
.
π|v|α+1
(68)
For α = 1, the above formula represents the Cauchy–Lorentz distribution
pC (x) =
1 1
,
π v2 + 1
(69)
whereas for α = 2, it reduces to a Maxwellian
1
pM (v) =
π
Z∞
0
1
cos(vt) exp −t2 dt = √ exp −v 2 /4 .
2 π
(70)
We consider now a more complicated case. As discussed in preceding sections, the sum of two statistically independent random variables ξ and η is
distributed according to a convolution. In other words, the sum ζ = ξ + η
has the PDF
+∞
Z
pζ (y) =
pξ (y − x)pη (x)dη .
(71)
−∞
Since convolution corresponds to a multiplication of the characteristic functions in Fourier domain, the convoluted Gauss–Lévy distribution of a velocity component attains the form [15, 16]
const
pGC (v) =
π
+∞
Z
cos(vt) exp −γtα − σ 2 t2 dt .
(72)
−∞
Here, the coefficients γ, σ 2 give the strength of each Lévy-type component.
This example was analyzed in the context of a distribution in plasmas
[15, 16]. Since the integrals are — in the general case — quite complicated, we consider in the following the simplest case of a Gauss–Cauchy
distribution α = 1. Then the integral may be expressed by error functions
(cf. Eqs. (50), (51))
1
−v + iγ
pGC (v) = √ < w
.
(73)
2σ
2 πσ
From the asymptotic representations of the error function, we get for v → ∞
the asymptotic wing of the distribution in the form of a Cauchy distribution
γ
pC (v) '
.
(74)
2
π (γ + v 2 )
Onsagers Fluctuation Theory and New Developments Including . . .
879
So far, we have considered only a one-component problem by looking just
at one component of the fluctuating quantity, say e.g. a component of the
velocity. However, in many applications we need to know besides the distribution of the components, also the distribution of the modulus and the
distribution of the energy (m = 1, n = 2, 3)
q
|v| = v12 + . . . + vn2 .
(75)
Finding an appropriate PDF is then not as trivial as for the Gaussian case. In
calculating the distribution of the absolute value of the field, its components
cannot be considered as independent [26]. We proceed here as follows: The
distribution discussed above gives us a projection of the velocity field v onto
an arbitrary direction defined by its unit vector e, i.e. p(y) with y = v cos θ
where θ = arccos (v · e/v). We are interested, however, in the distribution
of the absolute value x = v ≥ 0.
If the probability density p(x) of x is known, it is not complicated to
calculate the PDF of y assuming the angles between v and e are taken at
random, i.e. e is homogeneously distributed on a unit sphere. In this case,
1
py (y) =
4π
Z2π
0
1
=
2
1
=
2
dφ
Zπ/2
0
−π/2
Z1
d cos θ
Z1
dξ
p
ξ
−1
−1
dθ sin θ
Z∞
Z∞
dxp(x)
0
dx p(x)δ(y − x cos θ)
y
1
δ
−x
cos θ cos θ
y
.
ξ
Note that since p(x) = 0 for x < 0, the integrand vanishes on the left halfaxis for y > 0 and on the right half-axis for y < 0. Since the distribution
py (y) is symmetric, it is enough to consider y > 0, i.e. to write
1
py (y) =
2
Z1
0
dξ
p
ξ
y
ξ
for y > 0. Now we can make the change of variables in our integral, z = y/ξ,
to get
Z∞
1
dz
py (y) =
p (z) .
2
z
y
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Ł. Kuśmierz et al.
The rest is simple: Differentiating both parts of the equation in y, we get
d
1
py (y) = − p (y)
dy
2y
and, therefore,
p(x) = −2x
d
d
p (x) = −2x pL (x, α) ,
dx
dx
(76)
where the corresponding Lévy distribution is considered only on the positive half-axis. At this stage, we can also check the normalization of the
distribution
Z∞
0
p(x)dx = −2
=
Z∞
x
d
pL (x, α)
dx
0
−2xpL (x, α)|10
+2
Z∞
pL (x, α) dx = 1 ,
0
where we performed the integration by parts and used the fact of the symmetry of the Lévy distribution and its integrability.
Using Eq. (76) and the asymptotic form of the Lévy distribution Eq. (68),
we get the following asymptotic decay form for the far tail of the distribution
of the absolute value of the velocity
p(x) '
2 sin(πα/2)Γ (α + 2)
π|v|(α+1)
with the same power-law asymptotics as the distribution of a single component.
Using these results, we get for the distribution of the modulus of the
velocity
p(|v|) = const
Z∞
0
dk(k|v|) sin(k|v|) exp −γk α − σ 2 k 2 .
(77)
The asymptotics is for large v and α < 2 given by the Lévy part of the
distribution
p(|v|; µ) ∼
const
.
α|v|α+1
(78)
Onsagers Fluctuation Theory and New Developments Including . . .
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Figure 5 displays the stationary distributions of the modulus of the velocity
for the case of pure Cauchy noise, pure Gaussian noise, and mixed, Cauchy–
Gaussian convoluted noise. The corresponding distribution of kinetic energy
= mv 2 /2 is given by
p(; α) = const
Z∞
0
√
√
dk k 2m sin k 2m exp −γk α − σ 2 k 2 .
(79)
The tail of this distribution of kinetic energy is given by
p(; α) ∼
const
.
αα+1/2
(80)
The corresponding cumulative distribution for finding energies higher than
e0 is
p0 ( > 0 ; µ) ∼
const
.
α(α + 1)α−1/2
(81)
For many physical processes, in particular for rate processes, the tail of the
energy distribution and the cumulative distribution may be quite important [17].
0.45
σ=1, γ=0
σ=0.8, γ=0.5
σ=0, γ=1.5
0.4
0.35
p(|v|)
0.3
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
6
7
8
|v|
Fig. 5. Probability distribution of the modulus of the velocity ps (|v|) for the Cauchy,
Gaussian and mixed Cauchy–Gaussian noises.
882
Ł. Kuśmierz et al.
4.2. General relaxation kinetics
Lévy-type processes are most conveniently described in the Fourier-space.
We introduce the 3d-Fourier transform by
Z
1
dk exp[ik · v]p̂(k, t) ,
(82)
p(v, t) =
(2π)3
where k denotes the Fourier vector. Then, in the case with only one relaxation time (τ = λ−1 ) in conformity to the results of Section 3.1, we get
the phenomenological Smoluchowski-type equation for the evolution of the
probability density function in the k-space
∂
∂
p̂(k, t) + λk ·
p̂(k, t) = −D2 k2 p̂(k, t) − |k|α Dα p̂(k, t) .
∂t
∂k
(83)
The solution is
p̂(k, t) = exp −d2 (t)D2 k2 − dα (t)Dα |k|α .
(84)
By generalizing the many-component relaxation equation Eq. (26) at this
level of description, we get for the Fourier transform of the distribution
function p(v1 , . . . , vn , t)
n
X
∂
∂
p(k1 , . . . , kn , t) +
ki λij
p(k1 , . . . , kn , t) = R(k1 , . . . , kn , t) . (85)
∂t
∂kj
i,j=1
For the generalized r.h.s. of the stochastic equation including Levy terms,
exist several possibilities, the easiest one is
R(k1 , . . . , kn , t) = −D2 k 2 p(k1 , . . . , kn , t) − |k|α Dα p(k1 , . . . , kn , t) .
(86)
A more complex “ansatz” with a tensorial diffusion reads
R(k1 , . . . , kn , t) = −
n
X
i,j=1
ki kj Dij p(k1 , . . . , kn , t) − |k|α Dα p(k1 , . . . , kn , t) .
(87)
4.3. Relaxation of 3-dimensional velocities including external forces
We study now the velocity relaxation of a particle in a fluid starting from
a representation of the Langevin equation in velocity Fourier space [8, 9, 13,
16, 17, 22]. For the 3d-Fourier transform of a process including a constant
Onsagers Fluctuation Theory and New Developments Including . . .
883
external force F 0 and a friction constant λ (k denotes the Fourier vector in
the velocity space), we find [16, 17]
∂
1
∂
p̂(k, t) = −i k · F 0 p̂(k, t) − λk ·
p̂(k, t) − Dα |k|α p̂(k, t) − D2 k 2 p̂(k, t) .
∂t
m
∂k
(88)
In the simplest case when F0 = 0 and D2 = 0 the explicit solution to the
above equation reads
p̂(k, t) = exp [−dα (t)Dα |k|α ] ,
1
dα (t) =
(1 − exp(−αλt)) .
αλ
(89)
(90)
There are two easily obtained limits: the short time distribution for t 1/λ, where the term proportional to λ may be neglected and the stationary
solution for t → ∞
Dα |k|α
α
p̂(k, t) = exp [−Dα |k| t] ;
p̂ss (k) = exp −
.
(91)
αλ
In the general case, the time-dependent solution assumes the form
1
α
2
p̂(k, t) = exp −i k · F 0 d1 (t) − Dα |k| dα (t) − D2 k d2 (t) .
m
For short times, this solution becomes
1
α
2
p̂(k, t) = exp −Dα |k| t − D2 k t − i k · F 0 t .
m
(92)
(93)
We see that the tails have the longest relaxation times. The stationary
distribution is the limit of long times and we find by back transformation
again the stationary distribution in the velocity space
Z
1
1
Dα |k|α D2 k2
pss (v) =
dk exp ik · v −
K 0 exp −
−
.
(2π)3
mγ
αγ
2γ
(94)
This distribution conforms with the results of Section 3, where we have
analyzed a propagator of the generalized Ornstein–Uhlenbeck process driven
by independent Gauss and Cauchy white noises. As we discussed already
above, in such case, the velocity distribution has a diverging mean square,
and correspondingly is characterized by a long, slowly decaying tail. Figure 5
shows the stationary distributions of the modulus of the velocity for the case
of pure Cauchy noise, pure Gaussian noise, and mixed noise.
884
Ł. Kuśmierz et al.
At high velocities and correspondingly high energies which are large in
comparison to the mean value, the slowly decaying tails determine the behavior. The long tails decay with the slowest relaxation time. This might be
of interest, in particular, for rate processes. We must underline that, strictly
speaking, all the distributions given above correspond to non-equilibrium.
Since proper thermodynamic equilibrium is characterized by Gaussian
Maxwell distributions, the PDFs obtained here may correspond only to stationary stationary stages of non-equilibrium processes. In some cases, the
distribution may be considered as quasi-stationary, where the constants in
the distributions are slowly changing time-dependent quantities. In particular, the friction λ may be a slowly changing phenomenological quantity.
The most remarkable result are the long velocity tails for α < 2. As demonstrated in [15, 16], long velocity tails may be responsible for a strong increase
of reaction rates and in particular this refers to special fusion processes.
5. Discussion
After repeating briefly the main topics of Onsagers theory of Gaussian
fluctuations and relaxation processes near to equilibrium including Smoluchowski equations, we have discussed extensions to Lévy-type distributions
and the relaxation to stationary states. For clarity, let us summarize the
main points of our strategy in extending the Onsager theory:
(i) The Gaussian equilibrium distributions in Onsagers theory are replaced by stationary distributions which are convolutions of Gauss and
Lévy distributions. The Gaussian fluctuations determine the body and
the Lévy contributions determine the tail of the distribution, i.e. the
large fluctuations.
(ii) The role of entropy is taken over by a non-equilibrium potential, also
called stochastic potential, which is (up to a constant) the log of the
stationary non-equilibrium distribution.
(iii) The Onsager forces are replaced by derivatives of the stochastic potential.
(iv) Onsagers linear laws for the relaxation of fluctuations are replaced
by Smoluchowski-type equations for the relaxation of the distribution
function to the stationary distribution.
(v) The relaxation times are not fixed as in Onsagers theory but depend on
the amplitude of the fluctuations. The Lévy fluctuations corresponding to the tail decay are much slower than the Gaussian fluctuation
corresponding to the body of the distribution.
Onsagers Fluctuation Theory and New Developments Including . . .
885
We have discussed several special cases and, in particular, mixed Gauss–
Cauchy distributions. The body of the distribution is determined by the
Gaussian component and the tails by Cauchy contributions to the fluctuations. The relaxation to stationary states is studied by solving generalized
Smoluchowski equations in the Fourier space. We have shown that, in comparison to the Gaussian contributions around the center body of the PDF,
the fluctuations in the tails have longer relaxation times. This is an essential difference to the Onsager theory, where all fluctuations have the same
relaxation time (at least in the one-component case).
In the last part, we have studied distributions of the components of velocity vectors, the velocity modulus and the kinetic energy. We have investigated the relaxation of these fluctuations by solving Smoluchowski equations
in the 3d-velocity Fourier space including in a systematic way Lévy-type fluctuations. We have shown that under certain conditions, in reality, long tails
in the velocity distribution might exist. The appearance of high-energetic
tails in the velocity and energy distribution, which we observe here as well
may contribute to the understanding of several high rate processes which
are determined by the tails in the velocity and the energy distribution. We
underline again that strictly speaking, all the distributions given above are
valid only for stationary non-equilibrium systems since fluctuations around
the proper thermodynamic equilibrium are always characterized by Gaussian
distributions of the fluctuating quantities. For that reason, the PDFs obtained here clearly correspond non-equilibrium processes. Our distributions
and relaxation processes are related to fluctuations around certain stationary
states. Looking at the time dependence of the PDFs, we see the relaxation
of the non-equilibrium Lévy fluctuations characterizing the far tails of the
fluctuations decay in a longer time (λα)−1 , in comparison to the decay time
(λ)−1 for Gaussian fluctuations.
Altogether, we conclude that Lévy statistics and, in particular, Lévy
flights are suitable models of random phenomena where rare and large (extreme) events are important. Such statistics are e.g. relevant for description
of turbulence and radio-wave scattering in the interstellar plasma. Admixture of Lévy component in white Gaussian noises driving dynamic systems at
hand, results in scaling of temporal broadening of spectral profile in which
large seldom fluctuations are observed to decay slower in time than their
Gaussian analogues.
The Authors acknowledge support by the European Science Foundation
via the research networking programme EPSD (Exploring Physics of Small
Devices). Special thanks are directed to M.Yu. Romanovsky and J.M. Rubi
for many fruitful discussions.
886
Ł. Kuśmierz et al.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
L. Onsager, Phys. Rev. 37, 405 (1931).
L. Onsager, Phys. Rev. 38, 2265 (1931).
L. Onsager, S. Machlup, Phys. Rev. 91, 772 (1953).
P.C. Hemmer, H. Holden, S. Kjelstrup Ratkje, Collected Works Lars Onsager
(With commentaries), World Scientific, 1996.
W. Ebeling, J.-C. Justice, Lars Onsager’s Contribution to Electrolyte
Theory, in: P.C. Hemmer et al., Collected Works Lars Onsager (With
commentaries), World Scientific, Singapore 1996.
K. Tamura, Y. Hidaka, Y. Yusuf, S. Kai, Physica A 306, 157 (2002).
Y. Hidaka et al., Physica D 239, 735 (2009).
H.C. Fogedby, Phys. Rev. Lett. 73, 2517 (1994).
S. Jespersen, R. Metzler, H.C. Fogedby, Phys. Rev. E59, 2736 (1999).
R. Metzler, E. Barkai, J. Klafter, Europhys. Lett. 46, 431 (1999).
R. Metzler, T.F. Nonnenmacher, Chem. Phys. 284, 67 (2002).
M. Magdziarz, A. Weron, Phys. Rev. E75, 056702 (2007).
A.V. Chechkin, V.Yu. Gonchar, M. Szydlowski, Phys. Plasmas 9, 78 (2002).
A.V. Chechkin et al., Phys. Rev. E66, 046129 (2002); Phys. Rev. E78,
021111 (2008).
M. Romanovsky, W. Ebeling, Contr. Plasma Phys. 46, 195 (2006).
W. Ebeling, M. Romanovsky, Contr. Plasma Phys. 49, 477 (2009).
W. Ebeling, M. Romanovsky, I. Sokolov, I.A. Valuev, Eur. Phys. J. Special
Topics 187, 157 (2010).
B. Dybiec, E. Gudowska-Nowak, I.M. Sokolov, Phys. Rev. E78, 011117
(2008).
B. Dybiec, E. Gudowska-Nowak, Phys. Rev. E80, 061122 (2009).
B. Dybiec, E. Gudowska-Nowak, J. Stat. Mech. 05, P05004 (2009).
S.A. Trigger, Phys. Lett. A372, 1288 (2008).
S.A. Trigger, Phys. Lett. A374, 134 (2009).
S.A. Trigger, G.J.F. van Heijst, P.P.J.M. Schram, Physica A 347, 77 (2005).
B. Dybiec, J.M.R. Parrondo, E. Gudowska-Nowak, Europhys. Lett. 98,
50006 (2012).
J. Prost, J.-F. Joanny, J.M.R. Parrondo, Phys. Rev. Lett. 103, 090601
(2009).
S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943).
P. Garbaczewski, R. Olkiewicz, J. Math. Phys. 40, 1057 (1999).
E.R. Weeks, H.L. Swinney, Phys. Rev. E57, 4915 (1998).
G. Zumofen, J. Klafter, A. Blumen, J. Stat. Phys. 65, 991 (1991).
Onsagers Fluctuation Theory and New Developments Including . . .
887
[30] L.F. Richardson, Proc. Roy. Soc. 110, 709 (1926).
[31] J.-P. Bouchaud, M. Potters, Theory of Financial Risk and Derivative
Pricing, Cambridge University Press, Cambridge, UK, 2003.
[32] M.Y. Romanovsky, Mod. Prob. Stat. Phys. 6, 65 (2007).
[33] J. He, Q. Zhang, J. Opt. A9, 565 (2007).
[34] E.P. Petrov, J. Quant. Spectrosc. Radiat. Transfer 103, 272 (2007).
[35] V.V. Begun, M. Gazdzicki, M.I. Gorenstein, Phys. Rev. C78, 024904 (2008).
[36] B. Dybiec, E. Gudowska-Nowak, Fluct. Noise Lett. 4, L273 (2004).
[37] A. Scher, E.W. Montroll, Phys. Rev. B12, 2455 (1975).
[38] I.M. Sokolov, Phys. Rev. E63, 011104 (2000).
[39] H.U. Bödeker, C. Beta, T.D. Frank, E. Bodenschatz, Europhys. Lett. 90,
28005 (2010).
[40] U.M.B. Marconi, A. Puglisi, L. Rondoni, A. Vulpiani, Phys. Rep. 461, 111
(2008).
[41] P. Hänggi, H. Thomas, Phys. Rep. 88, 207 (1982).
[42] U.M.B. Marconi, A. Puglisi, L. Rondoni, A. Vulpiani, Phys. Rep. 461, 111
(2008).
[43] J.J. Olivero, R.L. Longbothum, J. Quant. Spectr. Rad. Transfer 17, 233
(1977).
[44] T. Pöschel, N.V. Brilliantov, Phys. Rev. E61, 2809 (2000).
[45] A. Bernui, C. Tsallis, T. Villela, Europhys. Lett. 78, 19001 (2007).
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J
ournal of Statistical Mechanics: Theory and Experiment
Heat and work distributions for mixed
Gauss–Cauchy process
1
Marian Smoluchowski Institute of Physics and Mark Kac Center for
­Complex Systems Research, Jagellonian University, ul. Reymonta 4, 30-059
Kraków, Poland
2
Departament de Física Fonamental, Facultat de Física, Diagonal 647, 08028
Barcelona, Spain
E-mail: [email protected], [email protected] and
[email protected]@ub.edu
Received 20 May 2014
Accepted for publication 30 June 2014
Published 8 September 2014
Online at stacks.iop.org/JSTAT/2014/P09002
doi:10.1088/1742-5468/2014/09/P09002
Abstract. We analyze energetics of a non-Gaussian process described by a
stochastic differential equation of the Langevin type. The process represents a
paradigmatic model of a nonequilibrium system subject to thermal fluctuations
and additional external noise, with both sources of perturbations considered
as additive and statistically independent forcings. We define thermodynamic
quantities for trajectories of the process and analyze contributions to mechanical
work and heat. As a working example we consider a particle subjected to a
drag force and two statistically independent Lévy white noises with stability
indices α = 2 and α = 1. The fluctuations of dissipated energy (heat) and
distribution of work performed by the force acting on the system are addressed
by examining contributions of Cauchy fluctuations (α = 1) to either bath or
external force acting on the system.
Keywords: transport processes/heat transfer (theory), fluctuations (theory),
stochastic processes
© 2014 IOP Publishing Ltd and SISSA Medialab srl
1742-5468/14/P09002+24$33.00
J. Stat. Mech. (2014) P09002
Łukasz Kuśmierz1, J Miguel Rubi2 and
Ewa Gudowska-Nowak1
Heat and work distributions for mixed Gauss–Cauchy process
Contents
1. Introduction2
2. Thermodynamic description: the entropy production
in a system subjected to external driving forces4
3. Irreversible thermodynamics for continuous time
stochastic Markov process7
5. Analytical solutions for γ ≠ 011
6. Numerical schemes used for generating distributions
of heat and work13
7. Fluctuation relations16
8. Summary19
Acknowledgments20
Appendix20
References
22
1. Introduction
Superlinear and sublinear diffusional transport and nonexponential relaxation kinetics
are ubiquitous in nature and have been observed and analyzed in a number of systems
ranging from hydrodynamic flows and plasma physics to transport in a crowded environment inside cells and price variations in the economy [1]. Usually, a starting point
of description of systems exhibiting anomalous transport properties involves use of continuous time random walk (CTRW) and fractional differential diffusion equations of
the Fokker–Planck type (FFPE). Although both concepts turned out to be extremely
powerful in applications, there is still lack of full understanding of their links with first
principles of mechanics and thermodynamics [2–4].
The celebrated Smoluchowski–Fokker–Planck equation [5] provides a tool to quantify
propagation of randomness in stochastic nonlinear dynamical systems and in a standard
derived from the stochastic differential equations driven by Gaussian noises. However, by
virtue of the generalized central limit theorem put forward by Paul Lévy [6], the Gaussian
distribution is a special case of more fundamental ‘stable statistics’ which form the class
of distributions possesing selfscaling properties under convolution. In particular, summing
independent random variables having this distribution results in a random variable with
a similar distribution to the original ones. This attraction property of stable distributions
doi:10.1088/1742-5468/2014/09/P09002
2
J. Stat. Mech. (2014) P09002
4. Heat and work distribution in the presence of Gaussian
noise (γ = 0)9
Heat and work distributions for mixed Gauss–Cauchy process
x(1)
˙(t ) = − a(x (t ) − vt ) + 2 σξG(t ) + γξC (t ).
In the above equation x represents a position at time t, a is a parameter related to
the strength of the harmonic potential and friction coefficient, and ξG(t), ξC(t) stand
for independent white Gaussian (α = 2) and Cauchy (α = 1) stable noises. Since
the dynamics is analyzed in the overdamped limit, the particle has fully equilibrated
(Maxwellian distributed) velocities and its internal energy is solely given by the potential energy which includes a comoving frame, i.e. U(x, t) = (x(t) − vt)2/2. Models of that
type with only Gaussian noise term entering equation (1) have been previously successfully applied to describe motion of colloidal particles in optical tweezers and unfolding
of biopolymers [15, 24–26]. There have also been test arrangements used in asserting
the character of fluctuation theorems for measurable quantities and their symmetry.
For thermostated dynamical systems of that type, interacting with thermal reservoirs,
generalization of the fundamental second law of thermodynamics has led to derivation
of so-called fluctuation theorems (fluctuation relations) [27]. The latter express largedeviation symmetry properties of the probability density functions for physical observables considered in nonequilibrium situations. They have been thoroughly analyzed for
doi:10.1088/1742-5468/2014/09/P09002
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J. Stat. Mech. (2014) P09002
is characterized by the stability index α, (0 < α  2) which describes the exponent of
the power-law of probability distribution tails: p(ξ)dξ ≈ |ξ|−(1 + α)dξ. In analogy to the
Wiener process representing realization of diffusive Brownian motion, the Lévy process
{X(t), t  0} can be then described as a continuous, homogeneous stochastic process with
independent increments sampled from the stable Lévy distribution and converging to the
common Gaussian case for α = 2. Accordingly, formal time derivatives of Lévy processes
can be understood as examples of white (Markov) noises with (in general) non-Gaussian
(α ≠ 2) distributions of jumps. In consequence, similarly to the correspondence between
Smoluchowski–Fokker–Planck equation and stochastic differential equations driven by
Gaussian noises, derivation of fractional Fokker–Planck equation [1, 7] establishes the link
between the Langevin-type equation with non-Gaussian stable white noise and temporal
evolution of probability distribution of the ensemble particles whose motion it describes.
Among various problems addressed in the field of Lévy-noise driven dynamics is
the origin of superdiffusive transport in momentum space investigated in a number of
studies [8–11]. In particular, based on the Langevin model with linear friction proportional to velocity and non-Gaussian noise, superdiffusive transport in velocity space
for a magnetized plasma has been analyzed [8]. The resulting FFPE has been further
shown to describe the evolution of the velocity distribution function of strongly nonequilibrium and hot plasmas obtained in tokamaks [10].
On the other hand, anomalous transport is also relevant for analysis of signal transmission and detection in biological systems [12, 13]. In this field Brownian-ratchet
models provide mechanisms by which operation of biomolecular motors is investigated
[14–16]. In recent analysis it has been shown that the minimal setups of simple ratcheting potentials and additive white symmetric Lévy (non-Gaussian) noise are sufficient
to produce directional transport [17–20] and inversion of currents can be obtained by
considering a time periodic modulation of the chirality of the Lévy noise [21–23].
Here we continue this line of research and analyze thermodynamic energetics of a
linear system subject to Lévy fluctuations. We assume that the motion of a test particle
confined by a harmonic potential is described by a Langevin equation
Heat and work distributions for mixed Gauss–Cauchy process
2. Thermodynamic description: the entropy production in a system subjected to
external driving forces
Following derivation of de Groot and Mazur [31–33], we first note that the external field
F acting on the system increases its internal energy U by an amount dU = F · dx, where
x denotes a state variable coupled to the action of the F force. The differential of entropy
can be expanded in variables x and U (both assumed to be independent variables) yielding
 ∂S 


 dU +  ∂S  dx =  F − g · xdx
d
S = 
(2)
 T

 ∂U x
 ∂x U
In the linear regime of nonequilibrium thermodynamics we identify stationary states
of the system with thermal equilibrium for which probability p(x)dx that a system
is in a state for which x takes on values between x and x + dx can be expressed
as p(x )dx = C exp [S (x )/ k B] = C exp [− 12 k B−1∑gij x j x i ]. Accordingly, g stands for the
∂S
matrix whose elements are gij = (∂2S/∂xi∂xj)U and we introduce Xi = ∂x = − g · x, an
i
intensive thermodynamic state variable (generalized thermodynamic force) coupled
to an extensive variable x. For driving forces which are constant in time, the linear
response [11] allows us to establish a relation for the mean value x(t ) = χ^(0)F which
is identified with the most probable value of x in the stationary ensemble [31]. Based
on equation (2), the entropy production in the system is described by
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J. Stat. Mech. (2014) P09002
systems operating close to equilibrium, when fluctuations of thermodynamic variables
follow the Gaussian law of statistics. Extensions to non-Gaussian white noise and correlated Gaussian fluctuations have been also recently addressed [27–30]. In particular,
for linear systems perturbed by Lévy white noise, it was found that the ratio of the
probabilities of positive and negative work fluctuations of equal magnitude behaves in
an anomalous nonlinear way, developing a convergence on the one for large fluctuations
[27, 29, 30]. Hence, negative fluctuations of the work performed on the particle are just
as likely to happen as large positive work fluctuations of equal magnitude. This unusual
property strongly departs from the Gaussian case, where the validity of a standard FR
implies that positive fluctuations are exponentially more probable than negative ones.
Our analysis further generalizes those findings and refers to a situation when the source
of fluctuations is modeled by two independent white noises: Cauchy noise of intensity γ
and Gaussian noise of intensity σ, both entering the system as random additive forces.
The paper is organized as follows: section 2 provides a brief recapitulation of the
thermodynamic interpretation of a standard Langevin equation, in which the source of
random forces is represented by a generalized derivative of a Gaussian Wiener process
(white noise). In section 3 we devise analysis of thermodynamic potentials based on the
concept of the ensemble average of the system trajectories, i.e. utilizing the probability density function of a corresponding Fokker–Planck–Smoluchowski equation. The
dynamic behavior of the nonequilibrium system is further investigated in sections 4 and
5 in terms of fluctuating thermodynamic components of energy such as heat, mechanical and dissipated work and internal energy. Crucial for this analysis is the statistical
balance of total energy (section 6) and fluctuation relations for thermodynamic quantities, discussed in section 7. Section 8 brings about a summary of the results.
Heat and work distributions for mixed Gauss–Cauchy process
F
 dx
dS
=  + X .
(3)
 dt
 T
dt
By use of the the first and second laws of thermodynamics expressed for quasi-stationary processes, the entropy production under constant temperature can be rewritten as
dS
1 d
1 d
=
{dU − DW + p dV − µdN } =
{dU − DW }
,
(4)
dt
T dt
T dt
V ,N
x(5)
¨ = −Γx˙ − kx + F + ξ˜G(t ).
Here Γ stands for the friction constant related to the correlation function of Gaussian
white noise
ξ˜G(t )ξ˜G(t ′) = 2Γk BTδ(t − t ′).
(6)
In accordance with equation (3), the thermodynamic force coupled to the coordinate x
is of the Hooke’s law type X = −kx/T. Examination of time variations of mechanical
energy of the system yields:
d 1 2
x2 
dx
 x˙ + k  = −Γx˙ + F + ξ˜G(t )
(7)


dt  2
2 
dt
By combining equations (7) and (5), we identify heat transferred from the system to
the heat bath as
 dx

−
DQ ≡ Γ
− ξ˜G(t )dx ,
(8)

 dt
and work performed on the system DW ≡ Fdx. In the overdamped limit, the e­ quation of
motion for variable x takes on the form
k
F
1
x(9)
+ ξ˜G(t ),
˙=− x+
Γ
Γ
Γ
so that 1/Γ = τ represents the time scale of relaxation of velocities to the stationary
state. Accordingly, the energy conservation law can be expressed by multiplying the
forces in equation (9) by an incremental change of the state dx and identifying heat
discarded by the system into the bath DQ (or heat transferred from the bath into the
system, [25, 34]), internal energy dU and work performed by the external force dW:
((10)
−Γx˙ + ξ˜G(t ) − kx + F )dx = DQ − dU + dW = 0
doi:10.1088/1742-5468/2014/09/P09002
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J. Stat. Mech. (2014) P09002
where the symbol DW (and in the forthcoming paragraphs, DQ) has been used to
denote the general Pfaffian differential forms (to be distinguished from the exact differential, which they can be turned into by finding an appropriate integrating factor).
As an exemplary case, we consider further time variations of the variable x described
by the Langevin equation for the one-dimensional damped harmonic oscillator of a
unit mass:
Heat and work distributions for mixed Gauss–Cauchy process
In the absence of additional Cauchy random force, equation (9) transforms (γ = 0) into
(1) by identifying a = kΓ−1, ξ˜(t )Γ −1 = 2 σξ(t ) and FΓ−1 = avt = kΓ−1vt. Now, the
Hooke’s force stems from the potential pulled with a constant velocity v. In a comoving
frame y = x − vt, the model system equation (1) can be further reduced to
y(11)
˙(t ) = −ay − v + 2 σξG(t ) + γξC (t )
˙ = − 1 [ay(y˙ + v )],
S(12)
T
with the heat transferred to the system identified as
∫
Q
≡ T dt S˙(y˙, y )
(13)
where T stands for the ambient temperature. When averaged over an ensemble of
trajectories (i.e. with respect to the PDF p(x, t) = 〈x(t) − x〉ξG(t)), equations (12) and
1
(13) reflect an increase of the entropy in the medium S˙m = − T ay(y˙ + v ) p(x , t ), in
line with the definition of stochastic energetics [25, 34]. It has to be stressed that the
problem of studying the action of additive Gaussian white noise on a Brownian particle, as described by a standard Langevin equation, has a well documented universal
statistical physics approach which, by use of the fluctuation-dissipation theorem,
relates thermal noise intensity to the friction coefficient of temperature of the system
[25, 31, 34]. This is however, not the case of white noise with a heavy tailed distribution of impulse intensities which are qualitatively very different from standard noise
characterized by probability distribution densities with finite cumulants (moments)
[27–30]. In contrast to regular, Gaussian-like noise, the variance of more general Lévy
noise is infinite and its power spectral density does not exist. Clearly, this feature
violates the classical fluctuation-dissipation theorem and obscures thermodynamic
interpretation of the stochastic Langevin equation by posing dubiety on the definition
of the temperature of the heat bath and its relation to fluctuations imparted to the
system [11, 25, 31, 35]. Such censure of Lévy fluctuations is clearly legitimate if the
noise is treated as internal (arising from the environment directly surrounding the
test particle). If, on the other hand, the Lévy noise is considered to be external, the
time dependent driving force, the friction and external noise are decoupled and the
fluctuation–dissipation relation equation (6) has to be fulfilled solely by the thermal
(internal) part of the noise.
In what follows we will reconsider contributions to entropy production in a linear
system driven by independent Gauss and Cauchy noises by analyzing, instead of direct
use of the relation equation (13), the balance of energy with fluctuating work and heat
being exchanged in the system.
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with the coefficient a established by the relation a = k/Γ. Consequently, for negligible Cauchy noise contribution (γ = 0), in line with the definition equation (3) the
instanteneous entropy production can be defined at the trajectory level and written
in the form of
Heat and work distributions for mixed Gauss–Cauchy process
3. Irreversible thermodynamics for continuous time stochastic Markov process
Time evolution of the probability density function (PDF) of the system described by
equation (11) is associated with the fractional FFPE:
∂p(y, t )
∂
∂2
∂
p(y, t ),
= − [−ay − v ]p(y, t ) + σ 2 2 p(y, t ) + γ
(14)
∂t
∂y
∂y
∂y
∂p^(k , t )
∂
= −ak p^(k , t ) + ikvp^(k , t ) − σ 2|k|2 p^(k , t ) − γ|k|p^(k , t ),
(15)
∂k
∂t
where p^(k , t ) = F [p(y, t )]. Note, that in equation (14) the strength of the white
Gaussian noise σ2 is related to the temperature of the thermal bath via the usual
relation 〈ξG(t)ξG(t′)〉 = 2kBTΓδ(t − t′) = 2σ2δ(t − t′) with the viscous drag coefficient
Γ ≡ 1 referring to the time units t = 1/Γ. By contrast, the intensity γ of the external
fluctuating force ξC is an arbitrary parameter of the model.
Due to linearity of the equation (11), the PDF of the process attains the form of the
convolution of Lévy distributions (with unknown, so far, time-dependent parameters).
The corresponding characteristic function is then:
^(k , t ) = eikµ(t ) − σ (t ) | k | −γ(t ) | k|.
p(16)
2
2
Using this ansatz and FFPE (equation (14)) we can easily obtain evolution equations for parameters:
µ
˙(t ) = −aµ(t ) + v
(17)
γ
˙(t ) = γ − aγ (t )
(18)
dσ 2(t )
= σ 2 − 2aσ 2(t ).
(19)
dt
Accordingly, the stationary solution to the FFPE subject to conditions
µ˙(t ) = σ˙ 2(t ) = γ˙(t ) = 0 for a constant v attains the form [35]:
v
µ
(20)
∞: = lim µ(t ) =
t →∞
a
γ
γ
(21)
∞: = lim γ(t ) =
t →∞
a
σ2
2
σ
:
=
lim
σ
(
t
)
=
(22)
∞
t →∞
2a
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which can be derived by use of the cumulant generating function of the motion defined
by the Langevin equation [7, 8]. Here, the fractional derivative is understood in the

 ∂α
Riesz-Weyl sense [7] and defined by its Fourier transform F  ∂ | x|α f ( y )  = −| k |α F [ f ( y ) ].


Consequently, equation (14) has the following Fourier representation:
Heat and work distributions for mixed Gauss–Cauchy process
Stationary PDF can be expressed by a reverse Fourier transform of the characteristic
function:
v
p^s(k ) = e−σ∞| k | −γ∞| k |+ik a
2
2
∞
(23)
1
ps(y ) =
dk p^s(k )e−iky .
2π −∞
∫
 − y + iγ∞ 
1

p(24)
Re w
s (y ) =
 2σ∞ 
2 π σ∞
and assumes a Gibbsian, equilibrium profile with well defined first and second moments
only in the absence of the Cauchy-noise contribution, i.e. for γ = 0. Despite that observation, for the Markovian system described by equation (11), a general version of the
H-theorem has been shown to hold [36], after identifying the H-functional with the relative entropy. Considering a correspondence with stochastic energetics under Gaussian
noise [34, 37–39], one is prompted to define a fluctuating entropy function following the
Shannon–Gibbs prescription S[p(y, t)] = − ∫ p(y, t) log p(y, t)dy. By an analogy to the
Gibssian equilibrium state the functional Φ(y) ≡− T log ps(y) can be then interpreted
as a nonequilibrium pseudo-potential [40, 41] with equivalents of internal and free energies of the system given by
∫
U
= Φ(y ) p(y, t ) = − T p(y , t ) log ps(y )dy
(25)
and
F
= U − TS = Φ(y ) p(y, t ) −T log p(y , t ) p(y, t ),
(26)
respectively. It follows then that all transient solutions to equation (14) tend towards
the stationary nonequilibrium state described by ps(x). It should be stressed however,
that such thermodynamic analogy lacks full integrity with statistical interpretation of
thermodynamic quantities and their relation to variations of entropy and precise meaning of the stationary solution.
The interplay of Gaussian and Lévy noises in the dynamics of the linear system
equation (11) results in scaling properties of the PDF p(y, t ) = 1/ C (t )p˜(y /C (t )), cf [11].
Accordingly, the dynamic entropy
∫
S(27)
[p(y , t )] = − p˜(z ) log p˜(z )dz + log C (t ) z = y / C (t )
increases in time and its production attains the form
C˙ (t )
dS
=
.
(28)
dt
C (t )
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Although it cannot be expressed in terms of elementary functions, for v = 0 it can be
2
rewritten [11] in a closed form by use of the Faddeeva function w(x ): = e−x erfc( − ix )
with erfc(x) standing for the complementary error function. Accordingly, the stationary
PDF solution takes on the form
Heat and work distributions for mixed Gauss–Cauchy process
In particular, for systems driven by Lévy white noise described by the stability
index α, the variation of entropy is governed by a scaling function C(t) ∝ t1/α, i.e.
S[p(y, t )] = 1/ α log t + const and accordingly,
dS
= (αt )−1 ,
(29)
dt
4. Heat and work distribution in the presence of Gaussian noise (γ = 0)
Let us first review the heat and work distribution function for a generic Brownian
particle system in a confining harmonic potential and in contact with a heat reservoir.
Such a scenario is used as a typical model of manipulated nano-systems (e.g. motion
of colloidal particles in opticcal tweezers) and biomolecules (unfolding of biopolymers)
[15, 24, 25]. Analytical results can be easily obtained in a steady state regime when
1
the harmonic potential is dragged for a long time t 0 >> a before any observables of
(
)
interest have been measured. In such a case the position of the particle is a Gaussian
random variable with corresponding mean and variance given by
v
⟨(30)
y0⟩ = −
a
3
Lévy white fluctuations have been represented in this case by a compound Poisson process ξ ( t ) = ∑ N ηm ( t ) with
m
the number of summands N given by a Poisson distribution and selfsimilar statistics of impulse intensities f(η)
dη = const × |η|−(1 + α)dη, 2 > α > 0.
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so that for increasing α the entropy rate decreases [42]. This observation has been
further used as a concept in diffusion entropy analysis, DEA applied among others,
to characterize the statistical properties of the natural time series (as e.g. the dynamics of electroencephalogram recordings [43]) and maximizing information exchange in
complex networks [44]. For Lévy jump diffusion3 in a force field, Vlad et al [36] performed relaxation analysis of a Lyapunov function—equivalent to the entropy in equation (27)—and discusssed existence and stability of the nonequilibrium steady state. In
a different context, a general discussion on ‘measures of irreversibility’ and time evolution of relative Shannon and Tsallis entropies constructed for PDFs governed by spacefractional differential operators has been initiated by Prehl et al [42] who investigated
the ‘entropy production paradox’ (i.e. aforementioned decrease of the entropy rate
with increasing scaling parameter α) for anomalous superdiffusive processes. All those
studies point to the significant differences of statistical properties of systems driven
by Brownian and Lévy noises while stressing uncertainty of rigorous thermodynamic
interpretation of entropy.
Accordingly, in the forthcoming sections we do not adhere to a specific formulation
of nonequilbrium entropy. As an alternative, we choose direct statistical analysis of
fluctuations in work performed on the system and heat transferred to the environment,
thus providing the insight into the analogue of entropy production in systems operating
close to equilibrium.
Heat and work distributions for mixed Gauss–Cauchy process
σ2
⟨(31)
y02⟩ − ⟨y0⟩2 =
.
2a
In the absence of dragging (v = 0) no mechanical work is done on the system. Therefore,
dissipated heat is equal to the internal energy change of the particle:
a 2
(yt − y02).
Q
t = Ut − U0 =
(32)
2
In order to derive the formula for heat PDF we start from the characteristic function:
∞
∫
∫
a
−∞
2
2
−∞
where p(yt, y0) = p(yt|y0)p(y0) is a joint PDF expressing probability of finding the
particle at a position y0 at time t = 0 and finding it at yt at time t. The propagator is
given by
(yt − y0e−at )2
− 2
1
2 σa (1 − e−2at ) .
p(34)
(yt|y0) =
e
2
σ
2π a (1 − e−2at )
After integration the closed-form formula for the heat characteristic function reads
1
G
Qt(k ) =
(35)
4
1 + σ (1 − e−2at )k 2
and its inverse Fourier transform yields a final result


q
1

P
K0
(36)
Qt(q ) =

−2at
2
−
2
2
at
 σ 1 − e

πσ 1 − e
where K0(x) stands for the zeroth order modified Bessel function of the second kind.
This result is complementary to former derivations: in [45] the Fourier transform
of the distribution of heat fluctuations is obtained for a constant velocity v by
observing that Wt is Gaussian distributed, while Qt = Wt − U(x, t), through definition of U(x, t) is quadratic in trajectory x. In turn, works [46, 47] have presented
long-time heat PDF under assumption of equlibrium (Boltzmann) distribution of
initial positions and v = 0. Here the time-dependent PDF for the heat equation (36)
has been derived by assuming stationary Gaussian form of p(x0) with parameters
­equations (30) and (31).
In the case when v ≠ 0 mechanical work performed on the system is given by the
formula:
t
t
∫
∫
0
0
W
ds(xs − vs ) = −av dsys.
(37)
t = −av
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∞
G
〉=
dyt
dy0eik 2 (yt − y0 )p(yt , y0)
Qt(k ) = 〈e
(33)
iQtk
Heat and work distributions for mixed Gauss–Cauchy process
t=0.1
t=0.2
t=1
1
t
log P(Q )
0
−1
−2
−3
−2
−1
0
Q
1
2
t
Figure 1. Comparison of the heat PDFs obtained with analytical formula,
equation (36) (lines) and numerical estimation (points) for t = {0.1, 0.2, 1}. Other
parameters: v = 0, a = 1, σ = 1, γ = 0. The integration step has been set
up dt = 0.1. Results have been derived for the ensemble of N = 106 sampling
trajectories.
Again, due to linearity of the Langevin equation governing evolution of y(t) and
the above definition, at any instant of time Wt is a Gaussian random variable with
parameters:
〈(38)
Wt 〉 = v 2t
2v 2σ 2
⟨(39)
Wt2⟩ − ⟨Wt ⟩2 =
(at + e−at − 1).
a
Figures 1 and 2 display congruence between analytical and numerical results, thus validating the numerical algorithm used in our evaluation of heat and work distributions
under action of combined Gauss and Cauchy noises.
5. Analytical solutions for γ ≠ 0
Mechanical work done by the potential dragging force is still given by equation (37).
By using the method of characteristic functional [29, 30, 48–50] of the process xt,
we derive (for details, see appendix) the characteristic function of work distribution
GWt(w ) ≡ exp (iwWt ) :


1
ln GWt(k ) = iv 2kt − σ 2v 2k 2t + (e−at − 1) − γv|k|t .
(40)


a
We can also obtain a similar result for a non-steady state, e.g. for x0 not fulfilling
the requirements equations (30) and (31). For initial condition sampled from the delta
distribution p(x, t = 0) = δ(x − x0) one obtains
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Heat and work distributions for mixed Gauss–Cauchy process
7
Analytical formula
Numerical estimator
6
VarWt
5
4
3
2
0
0
0.5
1
1.5
2
2.5
t
3
3.5
4
Figure 2. Comparison of work variance as a function of time obtained from
analytical formula, equation (39) (line) and by numerical estimation (points).
Parameters: v = 1, a = 1, σ = 1, γ = 0, dt = 0.1, number of trajectories: N = 105.
t=0.1
t=0.5
t=2
t=2, Gaussian noise
0
−1
t
log P(W )
−2
−3
−4
−5
−6
−7
−20
−15
−10
−5
0
5
10
15
20
W
t
Figure 3. Comparison of work PDFs obtained with analytical formula, equation (40)
(lines) and numerical estimation (points) at different times t = {0.1, 0.5, 2}
represented by red, green and blue symbols, respectively. Initial conditions have
been sampled from the stationary distribution (in a comoving frame). Separated
dashed line is an analytical result for the Gaussian case (γ = 0 and t = 2). Other
parameters: v = 1, a = 1, σ = 1, γ = 1, dt = 0.1, N = 106.



v
ln GWt(k ) = ik v 2t − v x 0 + (1 − e−at )




a
(41)




2
1
1
−γv|k|t − (1 − e−at )−σ 2v 2k 2 t +
(1 − e−2at )− (1 − e−at ).




a
2a
a
As shown in figures 3 and 4, the asymptotic form of work PDF P(Wt) is governed
by tails of the Cauchy distribution behaving like |Wt|−2. Note that, as explained in the
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1
Heat and work distributions for mixed Gauss–Cauchy process
t=0.1
t=0.5
t=2
t=2, Gaussian noise
1
0
t
log P(W )
−1
−2
−3
−4
−6
−8
−6
−4
−2
0
Wt
2
4
6
Figure 4. Comparison of work PDFs obtained with analytical formula, equation (42)
(lines) and numerical estimation (points) for t = {0.2, 0.5, 1.5} under non-steady
state conditions (see the main text). Separated dashed line displays an analytical
result for Gaussian case (γ = 0 and t = 1.5). Other parameters: v = 1, a = 1,
π
σ = 1, γ = 1, x 0 = , dt = 0.01, N = 105.
2
following section, our calculations refer to the situation when additive Gaussian and
Cauchy noises acting on the system are assumed here to be generated by the surroundings and do not contribute to the definition of mechanical work [11, 22, 23].
6. Numerical schemes used for generating distributions of heat and work
Unlike Gaussian noise, which is pertinent to situations close to equilibrium, where
fluctuations of the components of the Gibbs entropy are governed by Gaussian law,
more general infinitely divisible probability laws are known to account accurately for
long-range interactions and anomalous fluctuations of physical observables as encountered in strictly non-equilibrium complex systems [3, 51, 52]. Typical examples are
stochastic fields acting on charged particles in plasma and random gravitational forces
in clustered systems [3, 9, 10], both described by Holtsmark distributions of energy
and velocities exhibiting heavy long tails. In a recent work [11], we have shown that
mixed (convoluted) Gauss–Lévy distributions are candidates for describing observed
deviations from Maxwell distributions in plasmas and other systems [9, 10]. They are
also proper distributions describing effects of energy transfer in radiation resulting in so
called Voigt profile (convolution of Doppler and Lorenz line profiles) of spectral lines.
As an addition to those works, in the forthcoming paragraphs we present evaluation
of work and heat distributions constrained by the assumption that (1) Cauchy noise is
an external driving random force acting on the system, otherwise subject to Gaussian
(equilibrated bath) fluctuations and (2) Cauchy noise is a component of a nonequilibrium bath. In both cases our major concern is analysis of the balance of energy and
energy exchanged between the system and its reservoir.
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Heat and work distributions for mixed Gauss–Cauchy process
Following definitions of section 2 expressing stochastic energetics [25] at the level
of trajectories described in terms of a Langevin equation, we further investigate thermodynamic quantities like work and dissipated heat by exploring their PDFs. That
concept can be studied by formulating discretized versions of corresponding stochastic
differential equations which we interpret according to Stratonovich. In particular, for a
system subject to internal Cauchy noise, discretized formulas for heat and work read:
• Internal Cauchy noise
where wtG, t + h and wtC, t + h stand for (Gaussian- or Cauchy-distributed) increments
of the (stationary) Wiener process. Alternatively, heat exchange can be obtained
by direct use of the energy balance formula ΔQ = ΔU − ΔW with the change in
internal energy due to the mechanical work given by
∆
Wt = − kv(x t − vt )h
(43)
Both methods of evaluation have been tested for the problem at hand and produced identical results. In turn, for a corresponding situation with external
Cauchy–Lévy noise driving we refer to a discretization scheme:
• External Cauchy noise
1
∆
Qtext =  2 σwtG, t + h − (x t + h − x t )(x t + h − x t )
(44)

h
x
− xt
∆
Wtc = ∆Wt + γwtC, t + h t + h
(45)
h
Note, that in both cases the Cauchy random force does not contribute directly to evaluation of the mechanical work Wt. When considering ξc(t) as an explicit random force
acting on the system, we assess the ‘total work’ PDF by analyzing random quantity
Wtc = Wt + Wtr , i.e. we add extra (random) energy Wtr from the Cauchy noise to the
overall definition of work performed on the system.
In order to achieve convergence for estimated PDF for heat, a correct integration
method requires that:
dU (x t*)
0(46)
=−
h − (x t + h − x t ) + w t , t + h
dx
x +x
where x t* = t 2 t + h , i.e. the Stratonovich rule of integration has to be applied [25].
Moreover, we assume that a proper non-trivial spatio-temporal structure of environmental fluctuations can be preferentially modeled by a smooth colored noise process
which achieves the limit of a somewhat ill-defined idealization, namely the white noise,
as the correlation time of the approximation tends to zero. Indeed, according to the
Wong–Zakai theorem [53–57] in this limit the smoothed stochastic integral converges
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1
∆
Qtint =  2 σwtG, t + h + γwtC, t + h − (x t + h − x t ) (x t + h − x t )
(42)

h
Heat and work distributions for mixed Gauss–Cauchy process
to the Stratonovich stochastic integral. Naively, one could think that e.g. in order to
achieve a white noise limit of the Ornstein–Uhlenbeck process, one should scale the
corresponding SDE
ξ
σ
d
ξ(t ) = − 2 dt + dW (t ) ≡ −Γξ dt = σ^ dW (t )
(47)
ε
ε
with the parameter ε → 0. Such scaling yields formula for the correlation function of
the process
with the characteristic correlation time of the noise given by Γ−1. The spectral density
of the rescaled process is then
1
σ^ 2
σ 2ε2
S(49)
(ν ) ≡
e−iντ C (τ )dτ =
=
2π
2π(ν 2 + Γ 2)
2π(ν 2ε4 + 1)
∫
and tends to 0 for ε → 0 which indicates that we end up with the ‘noiseless’ limit. In
contrast, the proper white noise limit can be achieved when at the same time σ → ∞
and Γ → ∞ in such a way that σ2/2Γ → const. Under these circumstances the power
density S(ν) tends to a constant which is the signature of white noise. In other words,
by taking e.g. the rescaled version of the Gaussian colored noise ξG(t/ε2) in the Langevin
equation with multiplicative noise term
f (x )
ξG(t / ε2),
ε
(50)
 t−s 
1

ξG(t )ξG(s ) = exp −

ε2 
2
x˙ = a(x ) +
in the limit of ε → 0 the result of integration converges weakly to x^(t ) which satisfies SDE
d
x^ = a(x^)dt + f (x^) dw(t )
(51)
where w(t) stands for a standard 1-dim Wiener process. Since our analysis of (stochastic) work and heat functionals relies on estimation of multiplicative white noise sources
entering equations (42)–(45), we adjust to the aforementioned scheme, in which the
white noise limit solutions are achieved as limits of integrals performed with respect to
colored noise sources.
Figures 6 and 5 summarize our findings. First, by using the aforementioned definitions of stochastic heat and work (equations (44)–(46), the overall balance of energy
at the level of a single realization of the stochastic process is acomplished. Second,
distribution of energy at the ensemble level changes profoundly, depending on whether
the random Cauchy noise is treated as external forcing contributing to the evaluation
of the total work performed on the system, or included in the definition of heat. The
mechanical part of the energy can be evaluated analytically and numerical results corroborate correctly with formulae derived in the appendix (cf upper panel of figure 5).
On the other hand, estimation of heat or dissipated work can be achieved either by
evaluation of stochastic integrals or by using the energy balance formula of the first
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 t−s 
σ^ 2
σ 2ε2

C
( t − s ) ≡ ⟨ξ(t )ξ(s )⟩ =
exp (−Γ t − s )=
exp −
(48)

2Γ
2
ε2 
Heat and work distributions for mixed Gauss–Cauchy process
0
t
log p(Qint)
−5
−5
−10
−10
−15
−20
−10
Qint
0
10
γ=0
γ=0.1
γ=0.5
0.1
0
−200 −150 −100 −50
int
Q10
50
0
50
−5
t
log p(Qext)
−5
−10
−10
−15
−15
−20
−10
ext
Q0.1
0
10
−200 −150 −100 −50
Qext
10
Figure 5. Short time (t = 0.1, left panel) and long time (t = 10, right panel)
heat PDFs for the system subject to mixed Gaussian–Cauchy fluctuations. Plots
represent PDFs for different intensities γ of the Cauchy additive noise. Parameters
of the model: v = 1, a = 1, t = 0.1, σ = 1, correlation time of the noise ε = 0.01.
Histograms have been collected on N = 106 trajectories, time step of simulations
Δt = 10−3. Upper panels refer to Cauchy noise included in the bath, equation (42)
whereas lower panels depict heat PDFs for external Cauchy noise, equation (44)
treated as additional random force acting on the system.
law. In both cases, for finite time integration, we obtain almost indistinguishable histograms of derived PDFs.
7. Fluctuation relations
Fluctuations around out-of-equilibrium steady states are frequently described in terms
of the probability ratio of a time-integrated observable (like entropy production, heat
absorbed by driven Brownian particles or the current of the zero-range processes) evaluated for the forward and reverse changes a system undergoes in course of its evolution.
The ratio has been named fluctuation relation (FR) and by definition serves as a measure
of the symmetry in the distribution of fluctuations [45, 58–60]. In particular, a typical
arrangement of a Brownian particle dragged by a spring through a thermal environment
modelled by Gaussian white noises has been exploited both experimentally and theoretically to deduce symmetry of FR. It has been shown [15, 45, 50] that the work performed
on the particle satisfies a standard or conventional FR P(Wτ = w)/P(Wτ = − w) ≈ eρ(w)τ
with ρ(w) being a linear function of w. By contrast, the heat fluctuations follow and
extended the fluctuation relation P(Qτ = q)/P(Qτ = − q) ≈ eρ(q)τ where the exponential
doi:10.1088/1742-5468/2014/09/P09002
16
J. Stat. Mech. (2014) P09002
0
Heat and work distributions for mixed Gauss–Cauchy process
γ=0
γ=0.1
γ=0.5
t
log p(W )
0
−5
−5
−10
−10
−15
−2
0
2
4
−100
−50
0
0.1
100
150
50
100
150
10
t
−5
−5
−10
−10
−15
−2
0
Wc
2
0.1
4
−100
−50
0
c
W10
Figure 6. Short time (t = 0.1, left panel) and long time (t = 10, right panel)
work PDFs for the system driven by a symmetric internal Gaussian (σ = 1) noise
and an additional Cauchy noise of various intensities γ. Parameters of the model:
v = 1, a = 1, t = 0.1 correlation time of the noise ε = 0.01. Histograms have been
collected on N = 106 trajectories, time step of simulations Δt = 10−3. Plots in the
upper row refer to estimation of the mechanical work equation (43) and lower row
presents PDFs of the total work under action of steady dragging and additional
Cauchy random force, equation (45). Solid lines indicate analytical results, see
equation (40).
weight that relates the probability of positive and negative fluctuations does not scale linearly with the heat variable, i.e. ρ(q) is a non-linear function of q. As discussed elsewhere
[27, 30, 45, 59], conventional or extended FRs are valid when the underlying probability
distributions obey (for long times τ → ∞) a large deviation principle P(Wτ = w) ≈ e−τk(w)
with some rate function k(w). On the other hand, the large deviation principle is violated
in self-similar systems, where power law distributions of fluctuations govern the statistics
[27, 30]. In such systems the ratio of probabilities of positive and negative work fluctuations of equal magnitude shows anomalous behavior (P(W)/P(−W) ≈ 1) exhibiting the
same probability of occurrence for large positive and large negative fluctuations [27].
Figures 7 and 8 display work and heat fluctuations asymmetry for short and long
time limit evaluated for internal Gaussian noise, representing fluctuations of the heat
bath and an external Cauchy noise standing for additional random forcing. In the analysis, the aforementioned Cauchy noise is assumed as either an additional noise in the
surroundings or a random force contributing to the definition of work ∆Wtc performed
on the system, see equation (45).
For a linear system driven solely by Gaussian fluctuations represented by an
additive white noise, conventional fluctuation relation is detected, in line with the
large deviation probability [29, 45, 50]. By contrast, in the case when the system
doi:10.1088/1742-5468/2014/09/P09002
17
J. Stat. Mech. (2014) P09002
0
log p(Wc)
50
W
W
Heat and work distributions for mixed Gauss–Cauchy process
log(p(Wt)/p(−Wt))
6
10
γ=0
γ=0.1
γ=0.5
4
8
6
4
2
2
0
0
0.5
1
0
1.5
0
20
40
60
W10
80
0
20
40
Wc
80
W0.1
10
c
8
4
6
4
2
2
0
0
0.5
Wc0.1
1
0
1.5
60
10
Figure 7. Work fluctuation (a)symmetry for short time (t = 0.1, left panel) and
long time (t = 10, right panel). Increments of the mechanical work have been
evaluated according to equations (43) and (45). Histograms collected on N = 106
trajectories, a = 1, v = 1, time step of simulations Δt = 10−2. Solid lines represent
analytical results obtained by use of equation (40).
t
10
1.5
t
log(p(−Qint)/p(Qint))
2
1
γ=0
γ=0.1
γ=0.5
0.5
log(p(−Qext
)/p(Qext
))
t
t
0
0
2
int
Q0.1
4
6
5
0
0
20
Qint
40
60
40
60
10
6
10
4
5
2
0
0
2
Qext
0.1
4
6
0
0
20
Qext
10
Figure 8. Heat fluctuation (a)symmetry for short time (t = 0.1, left panel) and
long time (t = 10, right panel). Histograms collected on N = 106 trajectories,
a = 1, v = 1, time step of simulations Δt = 10−2.
doi:10.1088/1742-5468/2014/09/P09002
18
J. Stat. Mech. (2014) P09002
log(p(Wct)/p(−Wt ))
6
Heat and work distributions for mixed Gauss–Cauchy process
8. Summary
We have examined stochastic linear systems subject to action of two independent
stable noises, one of which exhibits a power law decay of large deviations. In a comoving frame our system represents a generalization of the Ornstein–Uhlenbeck process
with convoluted Gaussian-Cauchy fluctuations. Following standard interpretations of
the thermodynamics at the level of the Langevin equation, we have derived formulae for heat and work distribution in such system for various interpretations of the
Cauchy noise which enters dynamics either as a contribution to (nonequilibrated)
thermal bath, or as an additional external random driving. In order to conform
with Stratonovich rules of integration, for external Cauchy forces we have adopted
white-noise limit equations (51) and (51) of a colored noise. For Gaussian additive
noises, a standard fluctuation relation implies that positive fluctuations are exponentially more probable than negative ones [2, 29]. This scenario becomes severely
affected by the presence of white but otherwise non-Gaussian noise. The interplay
of the power-law distributed Lévy fluctuations causes a strong assymetry of the heat
PDF and, in line with previous studies [2, 27] leads to an anomalous structure of the
P(Wt)/P(−Wt) ratio.
The infinite power of the Cauchy noise spectrum results in a breakdown of the standard fluctuation-dissipation relationship (FDR) between friction force and correlation
function of fluctuations. Nevertheless, when an appropriate conjugate (dynamic) variable is chosen [35] in the form of a derivative of the stochastic entropy in the stationary
doi:10.1088/1742-5468/2014/09/P09002
19
J. Stat. Mech. (2014) P09002
becomes perturbed by additional Cauchy noise, FR deviates from the symmetric form
P(Wτ)/P(−Wτ) ≈ eAτw and the assymetry increases with the strength of the Cauchy
component in external forcing. For stronger Lévy noise, corresponding PDFs do not
decay exponentially (reflecting loss of the large deviation principle) and the work
fluctuation relation assumes the anomalous limit P(Wt)/P(−Wt) ≈ 1. The location
of the maximum of the P(W)/P(−W) ratio depends on the intensity of the Cauchy
noise and time of observation: the assymetry is linear in W, it deviates from linearity
for increasing values of W and then saturates to a constant value. For longer times
(t = 10), the turnover of the P(Wt)/P(−Wt) ratio is observed at higher values of W
and the slope of the linear constituent in the ratio is bigger than for short times.
Altogether, these observations stay in compliance with the character of the PDFs of
increments ΔW where domination of very large fluctuations leads to the aforementioned asymmetry.
Inclusion of the Cauchy white noise into definition of work performed on the system equation (45) results in heavy tailed, asymmetric PDF of work (see figure 5)
resembling the mirror-image of heat PDF for system subject to mixed GaussianCauchy fluctuations in the bath (see figure 6). Also in this case, the fluctuation ratio
P(W)/P(−W) reflects the generic asymmetry of work PDF with violation of the
exponential large deviation form and recovery of the transient fluctuation relation
P(Wτ)/P(−Wτ) ≈ eAτw for γ = 0.
Heat and work distributions for mixed Gauss–Cauchy process
state Xc = − T
∂Φ(x )
∂f
f =0
with respect to external driving f(t), the proper form of FDR
can be recast for sufficiently weak operating noises. This issue, analyzed for the linear
system perturbed by Gaussian and Cauchy noises will be the object of our forthcoming
studies.
Acknowledgments
Appendix
Let us recall (see equation (11)) that in the model w ( t ) =
a standard Brownian motion (Wiener process) and wα ( t ) =
t
∫0 dt ′ ξG ( t ′ ) stands
for
t
∫0 dt ′ ξC ( t ′ ) is
a Lévy
α-stable process with the stability index α = 1. We assume that increments of both
processes are statistically independent. Our aim is to calculate the characteristic function GWt(k ) = 〈eikWt 〉 of the process:
t
t
∫
∫
0
0
W
(A.1)
t = −av ds(Xs − vs ) = −av dsys.
We start with the observation that the characteristic functional of the total noise
[36, 48, 49] entering equation (11) is given by the relation:
∞
∞
∫ dt | k(t ) |2 −γ ∫ dt | k(t )|
(A.2)
Gξ[k ] ≡ G
2 σξG + γξC [k ]
=G
−σ 2
2 σξG[k ]GγξC [k ]=e
0
0
.
.
Next, we substitute ξ = yt + ayt + v in the definition of the characteristic function and
.
integrate the expression with yt by parts:
∞
i
∫
.
dtk(t )(yt + ayt + v )
∞
i(k ∞y∞− k 0y0) + i
∫ dt[vk(t ) + (ak(t ) − k˙(t ))yt ]
0
G
e 0
=
e
.
(A.3)
ξ [k ] =
Assuming that y0 ≡ y0 is a number (i.e. py0(y ) = δ(y − y0)) and that the function k(t)
decays sufficiently fast (securing k(∞) = 0), we arrive at:
∞
∫ dtk(t )
(A.4)
˙
−ik 0y0 + iv
Gξ[k ] = e
0
Gy | y0[−k + ak ].
doi:10.1088/1742-5468/2014/09/P09002
20
J. Stat. Mech. (2014) P09002
The authors acknowledge the support by the European Science Foundation (EFS)
through Exploring Physics of Small Devices (EPSD) program.
Heat and work distributions for mixed Gauss–Cauchy process
In the next step we introduce a new function m(t ) ≡ − k˙(t ) + ak(t ). It is easy to
reverse this relation yielding:
∞
t
∫
at
−as
∫
k(A.5)
(t ) = k(0)e − e
ds e m(s ) = e
ds e−as m(s )
at
at
0
t
∞
with k (0) =
∫ ds e−asm ( s ). This identyfication together with equation (A.4) allows us
0
∞
i
∫
∞
ds e−as m(s )z 0 − iv
∫
∞
dt eat
∞
∫ ds e−asm(s)
∫
(A.6)
0
t
G
× Gξ[eat ds e−as m(s )].
y | y0[m ] = e 0
t
Accordingly,
∞
ln Gy | y0 [ m ] = i
∞
∞
∫ ds e−asm ( s ) y0 − iv ∫ dt eat ∫ ds e−asm ( s )
0
0
t
∞
∞
∞
∞
(A.7)
− σ2
∫ dt | eat ∫ ds e−asm ( s ) |2 − γ ∫ dt | eat ∫ ds e−asm ( s ) |
0
t
0
t
Work characteristic functional GWt | y0(k ) can be now obtained by choosing a proper test
function m(t′):
m
(t ′) = − avqΘ(t − t ′)
(A.8)
where Θ(·) denotes the Heaviside step function. Indeed, by plugging it into formula
Gy [ k ] ≡ ⟨ exp ( i
∞
∫ dsk ( s ) ys ) ⟩ one can easily verify that this choice leads to GW (k ).
t
0
The very last step is to calculate integrals in equation (A.8) with m (·) given by
equation (A.8) which leads to the work characteristic function for the dynamics equation (11) with an initial condition y0 = x0:



v
ln GWt | x 0(k ) = ik v 2t − vx 0 + (1 − e−at )




a
(A.9)




1
1
2
(1 − e−2at )− (1 − e−at )
−γv|k|t − (1 − e−at )−σ 2v 2k 2 t +
a
2a
a




In order to get the GWt(k ) in a steady state regime we have to average GWt x 0(k ) over
steady state initial conditions at time t = 0. It is worthy noticing that GWt x 0(k ) can be
rewritten as
−ix 0b
G
(A.10)
Wt x 0(k ) = g(q )e
doi:10.1088/1742-5468/2014/09/P09002
21
J. Stat. Mech. (2014) P09002
to express the Gy | y0[m ] in terms of the known noise characteristic functional:
Heat and work distributions for mixed Gauss–Cauchy process
−at
where b = v q (1 − e ) and g(q) does not depend on x0. Thanks to that feature the
final formula contains just the Fourier transform of the initial (steady state) PDF:
∫
∫
G
dxGWt x (k )ps(x ) = g(q ) dx e−ixb ps(x ) = g(q )p˜s( − b),
(A.11)
Wt (k ) =
and
v
γ
σ2
p(A.12)
˜s(s ) = e−i a s − a | s |− 2a s .
2
With the help of equations (A.10)–(A.12) we arrive at the final result, equation (40),
References
[1] Klafter J, Metzler R and Lim S C 2011 Fractional Dynamics: Recent Advances (Singapore: World
Scientific)
[2] Klages R, Radons G and Sokolov I M 2008 Anomalous Transport: Foundations and Applications
(Weinheim: Wiley)
[3] West B J, Geneston E L and Grigolini P 2008 V-Langevin equations, continuous time random walks and
fractional diffusion Phys. Rep. 468 1–99
[4] Marconi U M B, Puglisi A, Rondoni L and Vulpiani A 2008 Fluctuation–dissipation: response theory in
statistical physics Phys. Rep. 461 111–95
[5] Risken H 1989 The Fokker–Planck Equation: Methods of Solution and Applications (Berlin: Springer)
[6]Lévy P 1937 Theorie de l'addition des Variables Aléatoires (Paris: Gauthier-Villars)
[7] Schertzer D, Larchev^eque M, Duan J, Yanovsky V V and Lovejoy S 2001 Fractional Fokker–Planck equation for nonlinear stochastic differential equations driven by non-Gaussian Lévy stable noises J. Math.
Phys. 42 200–12
[8] Chechkin A V, Gonchar V Y and Szydlowski M 2002 Fractional kinetics for relaxation and superdiffusion in
a magnetic field Phys. Plasmas 9 78–88
[9] Trigger S A 2009 Anomalous diffusion in velocity space Phys. Lett. A 374 134–8
[10] Ebeling W, Romanovsky M Y, Sokolov I M and Valuev I A 2010 Convoluted Gauss-Lévy distributions and
exploding Coulomb clusters Eur. Phys. J. Spec. Top. 187 157–70
[11] Kuśmierz L, Ebeling W, Sokolov I M and Gudowska-Nowak E 2013 Onsagers fluctuation theory and new
developments including non-equlibrium lévy fluctuations Acta Phys. Pol. B 44 859–80
[12] Stauffer D, Schulze C and Heermann D W 2007 Superdiffusion in a model for diffusion in a molecularly
crowded environment J. Biol. Phys. 33 305–12
[13] Mattháus F, Jagodič M and Dobnikar J 2009 E-coli superdiffusion and chemotaxissearch strategy, precision,
and motility Biophys. J. 97 946–57
[14] Bier M 2008 The energetics, chemistry, and mechanics of a processive motor protein BioSystems 93 23–8
[15] Ciliberto S, Joubaud S and Petrosyan A 2010 Fluctuations in out-of-equilibrium systems: from theory to
experiment J. Stat. Mech. 2010 P12003
[16] Fiasconaro A, Ebeling W and Gudowska-Nowak E 2008 Active Brownian motion models and application to
ratchets Eur. Phys. J. B 65 403–14
[17] Dybiec B and Gudowska-Nowak E 2004 Resonant activation driven by strongly non-Gaussian noises Fluctuat. Noise Lett. 4 L273–85
[18] Dybiec B, Gudowska-Nowak E and Sokolov I M 2007 Stationary states in Langevin dynamics under asymmetric Lévy noises Phys. Rev. E 76 041122
[19] del Castillo-Negrete D, Gonchar V Y and Chechkin A V 2008 Fluctuation-driven directed transport in the
presence of Lévy flights Physica A 387 6693–704
[20] Pavlyukevich I, Dybiec B, Chechkin A V and Sokolov I M 2010 Lévy ratchet in a weak noise limit: theory
and simulation Eur. Phys. J. Spec. Top. 191 223–37
doi:10.1088/1742-5468/2014/09/P09002
22
J. Stat. Mech. (2014) P09002
σ 2v 2 2
ln GWt(k ) = iv 2kt − γv|k|t −
k (at + e−at − 1).
(A.13)
a
Heat and work distributions for mixed Gauss–Cauchy process
doi:10.1088/1742-5468/2014/09/P09002
23
J. Stat. Mech. (2014) P09002
[21] Ebeling W, Gudowska-Nowak E and Fiasconaro A 2008 Statistical distributions for Hamiltonian systems
coupled to energy reservoirs and applications to molecular energy conversion Acta Phys. Pol. B 39
1251–72
[22] Dybiec B and Gudowska-Nowak E 2009 Lévy stable noise-induced transitions: stochastic resonance, resonant
activation and dynamic hysteresis J. Stat. Mech. 2009 P05004
[23] Dybiec B 2008 Current inversion in the Lévy ratchet Phys. Rev. E 78 061120
[24] Bustamante C, Liphardt J and Ritort F 2005 The nonequilibrium thermodynamics of small systems Phys.
Today 58 43–8
[25] Sekimoto K 2010 Stochastic Energetics (Berlin: Springer)
[26] Reimann P, Van den Broeck C, Linke H, Hánggi P, Rubi J M and Pérez-Madrid A 2001 Giant acceleration
of free diffusion by use of tilted periodic potentials Phys. Rev. Lett. 87 010602
[27] Chechkin A V and Klages R 2009 Fluctuation relations for anomalous dynamics J. Stat. Mech. 2009 L03002
[28] Chechkin A V, Lenz F and Klages R 2012 Normal and anomalous fluctuation relations for Gaussian
­stochastic dynamics J. Stat. Mech. 2012 L11001
[29] Touchette H and Cohen E G D 2009 Anomalous fluctuation properties Phys. Rev. E 80 011114
[30] Touchette H and Cohen E G D 2007 Fluctuation relation for a Lévy particle Phys. Rev. E 76 020101
[31] De Groot S R and Mazur P 1969 Nonequilibrium Thermodynamics (Amsterdam: North Holland)
[32] Vainstein M H and Rubi J M 2007 Gaussian noise and time-reversal symmetry in nonequilibrium Langevin
models Phys. Rev. E 75 031106
[33] Rubi J M 2012 Mesoscopic thermodynamics Phys. Scr. 2012 014027
[34] Seifert U 2012 Stochastic thermodynamics, fluctuation theorems and molecular machines Rep. Prog. Phys.
75 126001
[35] Dybiec B, Parrondo J M R and Gudowska-Nowak E 2012 Fluctuation-dissipation relations under Lévy noises
Europhys. Lett. 98 50006
[36] Vlad M O, Ross J and Schneider F W 2000 Lévy diffusion in a force field, Huber relaxation kinetics, and
nonequilibrium thermodynamics: H theorem for enhanced diffusion with Lévy white noise Phys. Rev. E
62 1743
[37] Hatano T and Sasa S 2001 Steady-state thermodynamics of Langevin systems Phys. Rev. Lett. 86 3463
[38] Campisi M and Hánggi P 2011 Fluctuation, dissipation and the arrow of time Entropy 13 2024–35
[39] Maes C 2012 Nonequilibrium entropies Phys. Scr. 86 058509
[40] Kubo R, Matsuo K and Kitahara K 1973 Fluctuation and relaxation of macrovariables J. Stat. Phys.
9 51–96
[41] Graham R, Roekaerts D and Tél T 1985 Integrability of Hamiltonians associated with fokker-planck equations Phys. Rev. A 31 3364
[42] Prehl J, Boldt F, Essex C and Hoffmann K H 2013 Time evolution of relative entropies for anomalous
­diffusion Entropy 15 2989–3006
[43] Ignaccolo M, Latka M, Jernajczyk W, Grigolini P and West B J 2010 Dynamics of electroencephalogram
entropy and pitfalls of scaling detection Phys. Rev. E 81 031909
[44] Balescu R 2007 Maximizing information exchange between complex networks Chaos Solitons Fractals
34 62–80
[45] Van Zon R and Cohen E G D 2003 Stationary and transient work-fluctuation theorems for a dragged
­Brownian particle Phys. Rev. E 67 046102
[46] Fogedby H C and Imparato A 2009 Heat distribution function for motion in a general potential at low
temperature J. Phys. A: Math. Theor. 42 475004
[47] Imparato A, Peliti L, Pesce G, Rusciano G and Sasso A 2007 Work and heat probability distribution of an
optically driven Brownian particle: theory and experiments Phys. Rev. E 76 050101
[48] Budini A and Cáceres M O 2004 Functional characterization of generalized Langevin equations J. Phys. A:
Math. Gen. 37 5959
[49] Cáceres M O 1999 Lévy noise, Lévy flights, Lévy fluctuations J. Phys. A: Math. Gen. 32 6009
[50] Sabhapandit S 2011 Work fluctuations for a harmonic oscillator driven by an external random force Europhys. Lett. 96 20005
[51] Garbaczewski P and Olkiewicz R 1999 Cauchy noise and affiliated stochastic processes J. Math. Phys.
40 1057–73
[52] Garbaczewski P 2005 Differential entropy and time Entropy 7 253–99
[53] Wong E and Zakai M 1965 On the convergence of ordinary integrals to stochastic integrals Ann. Math. Stat.
36 1560–4
[54] Kupferman R, Pavliotis G A and Stuart A M 2004 Itô versus Stratonovich white-noise limits for systems
with inertia and colored multiplicative noise Phys. Rev. E 70 036120
Heat and work distributions for mixed Gauss–Cauchy process
[55] Sussmann H J et al 1978 On the gap between deterministic and stochastic ordinary differential equations
Ann. Probab. 6 19–41
[56] Mannella R and McClintock P V E 2011 Comment on ‘influence of noise on force measurements’ Phys. Rev.
Lett. 107 078901
[57] Kanazawa K, Sagawa T and Hayakawa H 2012 Stochastic energetics for non-Gaussian processes Phys. Rev.
Lett. 108 210601
[58] Gallavotti G and Cohen E G D 1995 Dynamical ensembles in stationary states J. Stat. Phys. 80 931–70
[59] Kurchan J 1998 Fluctuation theorem for stochastic dynamics J. Phys. A: Math. Gen. 31 3719
[60] Jarzynski C and Mazonka O 1999 Feynman's ratchet and pawl: an exactly solvable model Phys. Rev. E
59 6448
J. Stat. Mech. (2014) P09002
doi:10.1088/1742-5468/2014/09/P09002
24
PHYSICAL REVIEW LETTERS
PRL 113, 220602 (2014)
week ending
28 NOVEMBER 2014
First Order Transition for the Optimal Search Time of Lévy Flights with Resetting
Lukasz Kusmierz,1,2 Satya N. Majumdar,3 Sanjib Sabhapandit,4 and Grégory Schehr3
1
2
Institute of Physics, UJ, Reymonta 4, 30-059 Krakow, Poland
Department of Automatics and Biomedical Engineering, AGH, Aleja Mickiewicza 30, 30-059 Krakow, Poland
3
Université Paris-Sud, CNRS, LPTMS, UMR 8626, Orsay F-91405, France
4
Raman Research Institute, Bangalore 560080, India
(Received 18 July 2014; published 26 November 2014)
We study analytically an intermittent search process in one dimension. There is an immobile target at the
origin and a searcher undergoes a discrete time jump process starting at x0 ≥ 0, where successive jumps are
drawn independently from an arbitrary jump distribution fðηÞ. In addition, with a probability 0 ≤ r < 1, the
position of the searcher is reset to its initial position x0 . The efficiency of the search strategy is characterized
by the mean time to find the target, i.e., the mean first passage time (MFPT) to the origin. For arbitrary jump
distribution fðηÞ, initial position x0 and resetting probability r, we compute analytically the MFPT. For the
heavy-tailed Lévy stable jump distribution characterized by the Lévy index 0 < μ < 2, we show that, for
any given x0 , the MFPT has a global minimum in the ðμ; rÞ plane at (μ ðx0 Þ; r ðx0 Þ). We find a remarkable
first-order phase transition as x0 crosses a critical value x0 at which the optimal parameters change
discontinuously. Our analytical results are in good agreement with numerical simulations.
DOI: 10.1103/PhysRevLett.113.220602
PACS numbers: 05.40.-a, 02.50.-r
The study of search strategies has generated tremendous
interest in the last few years, as they have found a wide
variety of applications in various areas of science. For
instance, they play an important role in diffusion-controlled
reactions [1]—with implications in the context of genomic
transcription in cells [2]—or in computer science [3], like
in the quest of solution of hard optimization problem.
More recently, search processes have been intensively
studied in behavioral ecology [4]. In that context, searching
for a target is a crucial task for living beings to obtain food
or find a shelter [4]. In this case, the survival of a species is
conditioned, to a large extent, to the optimization of the
search time. Hence, the characterization of the efficiency of
search algorithms has generated a huge interest during the
last few years, both experimentally [4–6] and theoretically
[7–12].
When studying animal movements during their search or
foraging period, it has proven to be useful to model their
outwardly unpredictable dynamics by random walks (RWs)
[1,7–12]. The increasing number of experimental data for
various animals [4–6], have stimulated the study of several
search strategies based on RWs. In particular, multiple
scales RWs, where phases of local diffusion alternate with
long range nonlocal moves, have been put forward as a
viable and efficient search strategy. For instance, these
nonlocal moves can be modeled by Lévy flights [7,8], or by
the so called “intermittent” RWs [9].
Recently, an intermittent strategy, where a locally diffusive searcher is reset randomly with a constant rate to its
initial position, has been introduced and demonstrated to be
rather efficient in searching a fixed target located at the
origin in all dimensions [11–16]. In particular, it was shown
0031-9007=14=113(22)=220602(5)
that the mean capture time of the target, a natural measure
of the efficiency of the search process, is finite and becomes
minimal for an optimal choice of the resetting rate. Apart
from the issue of search, this resetting move also drives the
system to a nonequilibrium stationary state which has been
characterized fully both for a single Brownian motion
[11,12,15] and spatially extended systems including fluctuating interfaces [17] or reaction-diffusion systems [18] (in
the latter case with a different resetting procedure). In the
last years, stochastic processes with random restarts have
FIG. 1 (color online). Illustration of the search strategy which
combines long jumps (Lévy flights) and random resettings, with
probability r, at the initial position x0 . Here, the search time,
i.e., the first passage time in 0 where the target is located is
T x0 ðμ; rÞ ¼ 9 while there have been two resettings, at step 4 and
7. The integers n and m with n ¼ 6 and m ¼ 2 here illustrate the
notations in the renewal equation in Eq. (4).
220602-1
© 2014 American Physical Society
PRL 113, 220602 (2014)
also been used in computer science as a useful strategy to
optimize search algorithms in hard combinatorial problems
as well as in simulated annealing [3,19].
In all these situations discussed above, the local exploration process is typically diffusive. However animal
movements on a local scale are not always diffusive
[7,8] and the jump distribution between two successive
positions may itself have heavy tails, such as in Lévy
flights. It is then natural to ask, for such jump processes
with heavy tails, whether resetting to the initial position
also makes the search of a target more efficient. In this
Letter, we introduce a simple model that combines jump
processes with heavy tails and random resetting to the
initial position. Indeed, we demonstrate that resetting is an
efficient search strategy even when the local moves are not
Brownian, but are instead heavy tailed. In particular, our
analytical results demonstrate that this model has a rather
rich behavior even in the simple one-dimensional setting,
where it exhibits a rather surprising first-order phase
transition. Even though our results concern the onedimensional case, they are of interest given that 1D
searching is highly relevant to biological applications, in
particular to the process of finding location of specific
DNA sequences by proteins [1,20].
For simplicity, we define the model in one dimension.
Higher-dimensional generalizations of the model are
straightforward. In our model, the searcher moves in
discrete time on a line, starting from the initial position
x0 ≥ 0. The target is located at the origin. At time step n,
the current location xn of the searcher is updated via the
following rules (see Fig. 1):
xn ¼
week ending
28 NOVEMBER 2014
PHYSICAL REVIEW LETTERS
x0
with probability r;
xn−1 þ ηn
with probability 1 − r;
hT x0 ðμ; rÞi, which depends on x0 , μ, and r. In this
Letter, we obtain an exact expression for hT x0 ðμ; rÞi given
in Eq. (7). For a fixed x0 , we then optimize hT x0 ðμ; rÞi with
respect to the two parameters μ and r and find the optimal
parameters μ ðx0 Þ and r ðx0 Þ as a function of x0 . Naively,
one might have expected that the optimal parameters are
μ ¼ r ¼ 0, independently of x0 . Instead we find, quite
remarkably, that these optimal values μ ðx0 Þ and r ðx0 Þ
exhibit a rather rich and surprising behavior, as functions
of x0 . We show indeed that there exists a critical value
x0 ≃ 0.58 (its value determined numerically) such that the
optimal strategy depends crucially on whether x0 > x0 or
x0 < x0 . When x0 > x0 , the optimal parameters are independent of x0 , and are given by
μ ðx0 > x0 Þ ¼ 0;
where
pffiffiffiffiffiffiffiffiffiffiffi e − 1 pffiffiffi pffiffiffiffiffiffiffiffiffiffiffi
e − e − 1 ¼ 0.22145…:
¼
2
ð2aÞ
ð2bÞ
In Eq. (2a), μ ¼ 0 actually means the limit μ → 0.
On the other hand, for x0 < x0, the optimal values μ ðx0 Þ
and r ðx0 Þ depend continuously on x0 , both of them being
monotonically decreasing functions of x0 . In particular, in
the limit where x0 → 0þ , we find
ð1Þ
where 0 < r < 1 denotes the probability of a resetting
event and the jump lengths ηn ’s are independent and
identically distributed (i.i.d.) random variables each drawn
from a probability density function (PDF) fðηÞ with a
heavy tail fðηÞ ∼ jηj−1−μ for large jηj, with a Lévy index
0 < μ < 2. Here, we consider the class of Lévy stable
processes for which the Fourier transform of the jump
R þ∞ ikη
μ
distribution is given by f̂ðkÞ ¼ −∞
e fðηÞdη ¼ e−jakj ,
where a sets the scale of the jumps (we set a ¼ 1 in
the following). The heavy tail is reflected in the small
k behavior of f̂ðkÞ ∼ 1 − jkjμ þ as k → 0. The case
μ ¼ 2 corresponds to ordinary random walks, while
μ < 2 describes Lévy flights where the jumps are typically
very large [21].
In the following, we consider the case of “myopic
search” where the search ends when the walker crosses
the origin (the location of the immobile target) for the first
time, (see Fig. 1). The efficiency of the search process is
conveniently characterized by the average search time
r>
r ðx0 > x0 Þ ¼ r> ;
r ðx0 → 0þ Þ ¼ r0 ¼ 1=4;
ð3aÞ
μ ðx0 → 0þ Þ ¼ μ0 ¼ 1.2893…;
ð3bÞ
where μ0 is the solution of a transcendental equation given
in Eq. (10). Moreover, we find that the optimal parameters
μ ðx0 Þ and r ðx0 Þ exhibit a discontinuity as x0 crosses the
value x0 (see Fig. 3). This behavior is typically a characteristic of a first order transition at x0 .
In order to compute the mean search time, or the mean
first passage time (MFPT), hT x0 ðμ; rÞi, to the origin (x ¼ 0),
we introduce the cumulative distribution function (CDF)
Qx0 ðr; nÞ ¼ Proba:½T x0 ðμ; rÞ > n. The CDF Qx0 ðr; nÞ is
thus the survival probability, i.e., probability that the walker
starting from x0 does not cross the origin up to step n in
presence of resetting. Obviously, one has hT x0 ðμ; rÞi ¼
P
n≥0 Qx0 ðr; nÞ. To compute Qx0 ðr; nÞ, we write a recursion
relation for this quantity, by using the fact that the resetting
dynamics in Eq. (1) is Markovian. At a fixed time step n, we
denote by m the number of steps elapsed since the last
resetting (see Fig. 1). The probability of a reset at step n − m
followed by no resetting during m steps is rð1 − rÞm . Using
this fact, we get (see [17] for the derivation of a similar
equation in a continuous time setting)
220602-2
Qx0 ðr; nÞ ¼
n−1
X
rð1 − rÞm Qx0 ðr; n − m − 1ÞQx0 ð0; mÞ
m¼0
þ ð1 − rÞn Qx0 ð0; nÞ;
ð4Þ
where Qx0 ð0; nÞ is the survival probability in the absence
of resetting (i.e., r ¼ 0). The first term on the right hand side
of Eq. (4) accounts for the event where the last resetting
before step n takes place at step n − m (see Fig. 1) with
0 ≤ m ≤ n − 1. The evolution from step n − m to step n
occurs without resetting and the survival probability during
this period is Qx0 ð0; mÞ, while Qx0 ðr; n − m − 1Þ accounts
for the survival probability from step 1 to step n − m − 1 in
presence of resetting. The last term in Eq. (4) corresponds to
the case where there is no resetting event at all up to step n,
which occurs with probability ð1 − rÞn.
To solve Eq. (4), we introduce its generating function
P
~
Qx0 ðr; zÞ ¼ n≥0 Qx0 ðr; nÞzn . Multiplying both sides of
Eq. (4) by zn and summing over n, we arrive at the result
~ x ðr; zÞ ¼
Q
0
~ x (0; ð1 − rÞz)
Q
0
:
~ x (0; ð1 − rÞz)
1 − rzQ
ð5Þ
0
This formula [Eq. (5)] relates the survival probability in
presence of resetting (r ≥ 0) to the one without resetting
(r ¼ 0). A relation, similar in spirit to Eq. (5), was derived
in the context of intermittent search in a confined system
[22], though the actual dynamics there is quite different
from the present model. Fortunately, for any continuous
and symmetric jump distribution fðηÞ, the Laplace trans~ x ð0; zÞ with respect to x0 (the case of no
form (LT) of Q
0
resetting), can be explicitly computed using the so-called
Pollaczek-Spitzer formula [23–25]:
Z
0
∞
1
~ x ð0; zÞe−λx0 dx0 ¼ pffiffiffiffiffiffiffiffiffiffi
φðz; λÞ;
Q
0
λ 1−z
Z
λ ∞ dk
φðz; λÞ ¼ exp −
ln (1 − zf̂ðkÞ) :
π 0 λ2 þ k2
ð6aÞ
ð6bÞ
Hence, Eq. (5) together with Eq. (6) allow us to compute
the CDF of the search time T x0 ðμ; rÞ. Note that Eqs. (5)
and (6) are actually valid for arbitrary jump distributions
fðηÞ, including, in particular, the Lévy case in which we are
interested.
A useful characteristic of the full PDF of T x0 ðμ; rÞ is
its first moment, on which we now focus. Noting the
~ x ðr; 1Þ, one obtains from
simple identity hT x0 ðμ; rÞi ¼ Q
0
Eq. (5)
~ x ðr; 1Þ ¼
hT x0 ðμ; rÞi ¼ Q
0
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28 NOVEMBER 2014
PHYSICAL REVIEW LETTERS
PRL 113, 220602 (2014)
~ x ð0; 1 − rÞ
Q
0
;
~ x ð0; 1 − rÞ
1 − rQ
0
ð7Þ
~ x ð0; 1 − rÞ can, in principle, be computed from
where Q
0
Eq. (6). The first observation is that when x0 ¼ 0, the MFPT
is totally independent of the jump distribution fðηÞ. Indeed,
~ x ¼0 ð0; zÞ is given by the Sparre Andersen
in this limit, Q
0
pffiffiffiffiffiffiffiffiffiffi
~ x ¼0 ð0; zÞ ¼ 1= 1 − z.
theorem [21,26], which states that Q
0
Therefore, for x0 ¼ 0, one obtains a universal result
1
;
hT x0 ¼0 ðμ; rÞi ¼ pffiffiffi
r−r
ð8Þ
which is independent of μ and has a minimum at
r ðx0 ¼ 0Þ ¼ 1=4, where the minimal MFPT is
T ðx0 ¼ 0Þ ¼ 4. The question is: what happens
when x0 > 0?
To get some insights for x0 > 0, we first perform a small
x0 expansion of hT x0 ðμ; rÞi. This requires the large λ
~ x ðr; zÞ in Eq. (6). For the case
expansion of the LT of Q
0
μ
of purely stable jumps, i.e., f̂ðkÞ ¼ e−jkj , this yields to
lowest nontrivial order (see Supplemental Material [27]):
1
x
1
pffiffiffi − pffiffiffi 0 pffiffiffi 2
hT x0 ðμ; rÞi ¼ pffiffiffi
rð1 − rÞ
rð1 − rÞ π
Z ∞
μ
×
ln ½1 − ð1 − rÞe−k dk þ Oðx20 Þ:
0
ð9Þ
We can now look for the optimal parameters r ðx0 Þ and
μ ðx0 Þ that minimize hT x0 ðμ; rÞi in Eq. (9), for a fixed
(small) x0 . To the lowest order, we find r ðx0 Þ ¼
1=4 þ Oðx0 Þ while limx0 →0þ μ ðx0 Þ ¼ μ0 , where μ0 is the
unique solution, on the interval (0,2), of the equation:
Z
0
∞
kμ0 ln k
dk ¼ 0:
expðkμ0 Þ − 3=4
ð10Þ
Solving Eq. (10) via MATHEMATICA yields μ0 ¼ 1.2893…,
as announced in Eq. (3). From Eq. (9), one finds that the
optimal MFPT is given by T ðx0 Þ ¼ hT x0 ðμ ; r Þi ¼
4 þ Oðx0 Þ. This perturbative calculation for small x0
demonstrates the nontrivial fact that, for small x0 , there
exists a nontrivial optimal set of parameters (r ðx0 Þ;
μ ðx0 Þ) given in Eq. (3). This leading order perturbation
theory can, in principle, be extended to higher orders in x0 .
To proceed beyond the perturbative calculation presented
above, we perform numerical simulations of the resetting
dynamics in Eq. (1). For a given x0 , we compute numerically hT x0 ðμ; rÞi by sampling 107 to 9 × 107 (depending on
x0 ) independent realizations of the resetting dynamics
[Eq. (1)], for different values of the parameters r and μ.
In Fig. 2, we show hT x0 ðμ; rÞi as a function of μ and r for
three different values of x0 . As shown in Fig. 2(a), for
x0 < x0 ≈ 0.58, hT x0 ðμ; rÞi exhibits a global minimum at a
nontrivial value of μ ðx0 Þ and r ðx0 Þ which are both
decreasing functions of x0 [see Figs. 3(a) and 3(b),
respectively]. In the limit x0 → 0, these curves converge
220602-3
T
T
5.70
5.76
5.74
5.68
5.66
5.64
5.62
0.20
T
5.72
5.70
1.2
0.24
0.56
r
0.22
0.8
b x0
0.6
0.58
6.00
5.95
5.90
5.85
5.80
5.75
0.22
r
0.5
0.24
0.3
0.26
1.3
1.1
0.9
0.7
0.20
0.5
0.24
0.26
0.28
1.3
1.1
0.9
0.7
5.68
0.18
0.20
1.0
0.22
a x0
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28 NOVEMBER 2014
PHYSICAL REVIEW LETTERS
PRL 113, 220602 (2014)
0.1
c x0
0.28
0.65
r
0.3
0.26
0.28
0.1
FIG. 2 (color online). 2D plots of the average search time hT x0 ðμ; rÞi, computed using numerical simulations, in the ðμ; rÞ plane for
different values of the initial position x0 : (a) x0 ¼ 0.56 < x0 , (b) x0 ¼ x0 ≃ 0.58, and (c) x0 ¼ 0.65 > x0 .
to our exact results in Eqs. (3). In contrast, for x0 > x0,
our simulations show [see Fig. 2(c)] that the minimum
of hT x0 ðμ; rÞi is instead reached at μ ðx0 > x0 Þ ¼ 0.
Figure 2(b) shows the critical case x0 ¼ x0 . Quite remarkably, the values of the optimal parameters μ ðx0 Þ and
r ðx0 Þ exhibit a sharp discontinuity as x0 crosses the
critical value x0 ≈ 0.58, as shown in Fig. 3.
The case x0 > x0 .—Our numerical simulations clearly
indicate that, for x0 > x0, μ ðx0 > x0 Þ ¼ 0 but r ðx0 > x0 Þ
is a nontrivial constant independent of x0 . We can actually
compute this constant analytically (see Supplemental
Material [27]). Using the hint from the simulations that
μ ¼ 0, we analyze Eq. (6) in the limit μ → 0. In this limit,
we find [27]
1
limhT x0 ðμ; rÞi ¼ pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
:
r 1 − ð1 − rÞ=e − r
ð11Þ
μ→0
As a function of r, hT x0 ðμ → 0; rÞi in Eq. (11) has a unique
minimum at the optimal value r> given in Eq. (2b).
Substituting r ¼ r> in Eq. (11) gives the optimal value
of the MFPT
pffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffi
2 eð e þ e − 1Þ
T ðx0 > x0 Þ ¼
¼ 5.6794…: ð12Þ
e−1
In Fig. 4, we show a plot of hT x0 ðμ; rÞi for x0 ¼ 1 > x0,
computed numerically, as a function of r and for different
small values of μ. This plot confirms that, as μ → 0,
1.4
0.255
1.2
0.250
1.0
0.245
0.8
0.240
0.6
0.235
0.4
0.230
0.2
0.0
0.0
a
0.225
0.2
0.4
x0
0.6
0.8
0.220
0.0
b
0.2
0.4
0.6
0.8
x0
FIG. 3 (color online). Plot of the optimal parameters μ ðx0 Þ and
r ðx0 Þ, obtained from numerical simulations, as a function of x0 .
Both of them exhibit a clear discontinuity for x0 ¼ x0 ≃ 0.58,
reminiscent of a first order phase transition.
the numerical data do converge to our exact results in
Eqs. (11) and (12).
Interestingly, our numerics also reveal the existence of a
second special value xc0 ≈ 0.56 < x0 , suggesting the following scenario as x0 is increased from 0 to ∞. As x0
increases, starting from 0, hT x0 ðμ; rÞi admits a single global
ð1Þ
minimum at Xmin ¼ ðμ ðx0 Þ > 0; r ðx0 ÞÞ [see Fig. 2(a)]
until x0 reaches the value x0 ¼ xc0 . At this point, a second
ð2Þ
local minimum appears at Xmin ¼ ðμ ¼ 0; r> Þ. The value of
ð2Þ
hT x0 ðμ; rÞi at this local minimum at Xmin however remains
ð1Þ
higher than the one at Xmin until x0 > x0 . Therefore, in this
range, when xc0 < x0 < x0 , there are two competing local
ð1Þ
ð2Þ
minima with Xmin being the global minimum, and Xmin
being a metastable minimum [see Fig. 2(b)]. When x0
ð2Þ
increases beyond x0 , then Xmin becomes the global minimum [see Fig. 2(c)]. This is then a typical scenario for a
first order phase transition, as clearly illustrated in Fig. 3.
This second value xc0 can actually be estimated analytically
ð2Þ
by studying the stability of the local minimum Xmin starting
from large x0 where it is also a global minimum. We compute
the sign of the derivative of ∂hT x0 ðμ; rÞi=∂μ evaluated at
μ ¼ 0 and r ¼ r> given in Eq. (2b). A straightforward
computation, using Eqs. (6) and (7) shows that
∂hT x0 ðμ; rÞi sgn
¼ sgn½lnðx0 Þ þ γ E ;
∂μ
r¼r> ;μ¼0
ð13Þ
where γ E ¼ 0.57721… is the Euler constant. The slope does
change sign from positive to negative as x0 crosses from
above the value xc0 ¼ e−γE ¼ 0.56146… < x0 ≈ 0.58. Our
numerical estimate of xc0 is fully in agreement with the exact
value xc0 ¼ e−γE .
To summarize, for a searcher undergoing stable Lévy
jump processes with resetting in one dimension, we showed
that the MFPT to a fixed target at the origin has a rich phase
diagram as a function of the Lévy index μ, the resetting
probability r, and the starting position x0 . In particular, the
optimal parameters (μ ðx0 Þ; r ðx0 Þ) that minimize the
MFPT exhibit a surprising first-order phase transition at a
critical value x0 . Our study leads to several open questions.
For example, how generic is this first-order phase transition?
220602-4
PHYSICAL REVIEW LETTERS
PRL 113, 220602 (2014)
Analytical prediction for μ → 0
Simulation results, μ=0.03
Simulation results, μ=0.06
Simulation results, μ=0.09
6.4
6.3
6.1
0
〈 Tx (μ,r) 〉
6.2
6
5.9
5.8
5.7
0.1
0.15
0.2
0.25
0.3
0.35
0.4
r
FIG. 4 (color online). hT x0 ðμ; rÞi vs r—comparison between
numerical results for small μ and analytical prediction for μ → 0.
Does it depend only on the tail or on other details of the
jump distribution? Also, does this transition exist in higher
dimensions and in presence of multiple searchers? These
questions remain outstanding for future studies.
SM and GS are supported by the ANR Grant No. 2011BS04-013-01 WALKMAT. SM, SS, and GS are partially
supported by the Indo-French Center for the Promotion of
Advanced Research under Project No. 4604-3. GS also
acknowledges support from Labex PALM (Project
RANDMAT). SM, SS, and GS thank the Galileo Galilei
Institute for Theoretical Physics, Florence, Italy for the
hospitality and the INFN for partial support during the
completion of this work.
[1] O. Bénichou, C. Loverdo, M. Moreau, and R. Voituriez,
Rev. Mod. Phys. 83, 81 (2011).
[2] O. G. Berg, R. B. Winter, and P. H. Von Hippel, Biochemistry 20, 6929 (1981).
[3] L. Lovasz, Random Walks on Graphs: A Survey in Combinatronics (Bolyai Society for Mathematical Studies,
Budapest, 1996), Vol. 2, p. 1.
[4] J. W. Bell, Searching Behaviour, the Behavioural Ecology
of Finding Resources, Animal Behaviour Series (Chapman
and Hall, London, 1991).
[5] W. J. O’Brien, H. I. Browman, and B. I. Evans, Am. Sci. 78,
152 (1990).
week ending
28 NOVEMBER 2014
[6] N. E. Humphries, H. Weimerskirch, N. Queiroz, E. J.
Southall, and D. W. Sims, Proc. Natl. Acad. Sci. U.S.A.
109, 7169 (2012).
[7] G. Viswanathan, V. Afanasyev, S. V. Buldyrev, E. J.
Murphy, P. A. Prince, and H. E. Stanley, Nature (London)
381, 413 (1996).
[8] G. Viswanathan, S. V. Buldyrev, S. Havlin, M. Da Luz,
E. Raposo, and H. E. Stanley, Nature (London) 401, 911
(1999).
[9] O. Bénichou, M. Coppey, M. Moreau, P.-H. Suet, and
R. Voituriez, Phys. Rev. Lett. 94, 198101 (2005).
[10] G. M. Viswanathan, M. G. E. da Luz, E. P. Raposo, and
H. E. Stanley, The Physics of Foraging (Cambridge,
Cambridge, 2011).
[11] M. R. Evans and S. N. Majumdar, Phys. Rev. Lett. 106,
160601 (2011).
[12] M. R. Evans and S. N. Majumdar, J. Phys. A 44, 435001
(2011).
[13] M. R. Evans, S. N. Majumdar, and K. Mallick, J. Phys. A
46, 185001 (2013).
[14] J. Whitehouse, M. R. Evans, and S. N. Majumdar, Phys.
Rev. E 87, 022118 (2013).
[15] M. R. Evans and S. N. Majumdar, J. Phys. A 47, 285001
(2014).
[16] D. Boyer and C. Solis-Salas, Phys. Rev. Lett. 112, 240601
(2014).
[17] S. Gupta, S. N. Majumdar, and G. Schehr, Phys. Rev. Lett.
112, 220601 (2014).
[18] X. Durang, M. Henkel, and H. Park, J. Phys. A 47, 045002
(2014).
[19] A. Montanari and R. Zecchina, Phys. Rev. Lett. 88, 178701
(2002).
[20] J. Gorman and E. C. Greene, Nat. Struct. Mol. Biol. 15, 768
(2008).
[21] W. Feller, An Introduction to Probability Theory and its
Applications (Wiley, New York, 1968).
[22] M. Coppey, O. Bénichou, R. Voituriez, and M. Moreau,
Biophys. J. 87, 1640 (2004).
[23] F. Pollaczeck, C.R. Hebd. Seances Acad. Sci., 234, 2334
(1952); J. Appl. Probab. 12, 390 (1975).
[24] F. Spitzer, Trans. Am. Math. Soc. 82, 323 (1956).
[25] For a review, see S. N. Majumdar, Physica (Amsterdam)
A389, 4299 (2010).
[26] E. Sparre Andersen, Math. Scand. 1, 263 (1953); Math.
Scand. 2, 195 (1954).
[27] See Supplemental Material at http://link.aps.org/
supplemental/10.1103/PhysRevLett.113.220602 for more
details.
220602-5
First order transition for the optimal search time of Lévy flights with resetting:
supplementary material
Lukasz Kusmierz,1, 2 Satya N. Majumdar,3 Sanjib Sabhapandit,4 and Grégory Schehr3
1
2
Institute of Physics, UJ, Reymonta 4, 30059 Krakow, Poland
Institute of Automatics, AGH, Al. Mickiewicza 30, 30-059, Krakow, Poland
3
Univ. Paris-Sud, CNRS, LPTMS, UMR 8626, Orsay F-01405, France
4
Raman Research Institute, Bangalore 560080, India
We give the technical details of the calculations yielding some of the results presented in the
manuscript of the Letter.
2
SMALL x0 EXPANSION OF hTx0 (µ, r)i
We compute the mean first passage time (MFPT) in the limit where the initial position x0 is small. The starting
point of the analysis is the following exact result, which is Eq. (7) of the Letter:
Q̃x0 (0, 1 − r)
,
1 − rQ̃x0 (0, 1 − r)
hTx0 (µ, r)i = Q̃x0 (r, 1) =
(1)
where Q̃x0 (0, 1 − r) is given by the Pollaczeck-Spitzer formula [1–3], given in Eqs. (6a) and (6b) of the Letter:
Z
∞
1
Q̃x0 (0, z)e−λx0 dx0 = √
ϕ(z, λ) ,
λ
1
−z
0
Z
λ ∞ dk
ˆ(k) .
ϕ(z, λ) = exp −
ln
1
−
z
f
π 0 λ2 + k 2
(2a)
(2b)
First
we notice that in the limit x0 → 0, Q̃x0 =0 (0, z) is given by the Sparre Andersen formula [4], Q̃x0 =0 (0, z) =
√
1/ 1 − z, which gives immediately, using (1), the result given in Eq. (8) of the Letter:
hTx0 =0 (µ, r)i = √
1
.
r−r
(3)
The small x0 expansion of the MFPT requires the small x0 expansion of Q̃x0 (0, z) which we assume to be regular of
the form
Q̃x0 (0, z) = √
1
+ x0 F1 (z) + O(x20 ) ,
1−z
(4)
and we want to determine the function F1 (z). This can be done by studying the Pollaczeck-Spitzer formula (2) in the
large λ limit. Exanding the left hand side of Eq. (2) for λ → ∞, using the expansion (4), we obtain:
Z
∞
Q̃x0 (0, z)e−λx0 dx0 =
0
1
1
1
√
+ 2 F1 (z) + O(λ−3 ) ,
λ 1−z
λ
(5)
which can be obtained by performing the change of variable y = λx0 in the above integral (5). On the other hand,
the large λ expansion of the right hand side of the Pollaczeck-Spitzer formula (2) reads
Z
Z ∞
1
1
1
1
λ ∞ dk
1
ˆ
√
ln 1 − z f (k) = √
exp −
− 2 √
ln (1 − z fˆ(k))dk + O(λ−3 ) . (6)
π 0 λ2 + k 2
λ 1−z
λ π 1−z 0
λ 1−z
Identifying the terms of order λ−2 in Eqs (5) and (6) yields the following expression for F1 (z):
1
F1 (z) = − √
π 1−z
Z
∞
0
ln (1 − z fˆ(k))dk .
(7)
Finally, using the relation (1) between hTx0 (µ, r)i and Qx0 (µ, r) together with the small x0 expansion of Qx0 (µ, r) in
(4) we obtain:
hTx0 (µ, r)i = √
x0
1
√ F1 (1 − r) + O(x20 ) .
+
r − r (1 − r)2
(8)
Finally, using the explicit expression of F1 (z) in (7), we obtain:
hTx0 (µ, r)i = √
1
x0
√
− √
r − r π r(1 − r)2
Z
0
∞
ln (1 − (1 − r)fˆ(k))dk + O(x20 ) ,
µ
which, for the special case of stable jumps, fˆ(k) = e−|k| yields the equation (9) in the Letter.
(9)
3
ASYMPTOTIC ANALYSIS OF THE MFPT IN THE LIMIT µ → 0
When x0 > x∗0 , our numerical simulations indicate that the optimal value of Lévy index is µ∗ → 0. Hence to obtain
µ
r (x0 > x∗0 ), we analyze Eq. (2) in the limit µ → 0. In this limit, one can show that fˆ(k) = e−|k| is almost flat
in the k space, with fˆ(k) ≈ e−1 , valid for e−1/µ |k| e1/µ . Substituting fˆ(k) ≈ e−1 in Eq. (2), we find that
Q̃x0 (r = 0, z) takes the simple expression:
∗
lim Q̃x0 (r = 0, z) = √
µ→0
1
p
,
1 − z 1 − z/e
(10)
which is independent of x0 . From Eq. (10) together with Eq. (1) one obtains
1
lim hTx0 (µ, r)i = √ p
,
r 1 − (1 − r)/e − r
µ→0
which is Eq. (11) given in the Letter.
[1]
[2]
[3]
[4]
F. Pollaczeck, C. R. Acad. Sci. Paris, 234, 2334 (1952); J. Appl. Probab. 12(2), 390 (1975).
F. Spitzer, Trans. Am. Math. Soc. 82, 323 (1956).
For a review see S. N. Majumdar, Physica A 389, 4299 (2010).
E. Sparre Andersen, Math. Scand. 1, 263 (1953); Math. Scand. 2, 195 (1954).
(11)
Optimal first arrival times in Lévy random walk with resetting
Lukasz Kuśmierz
Marian Smoluchowski Institute of Physics, Jagiellonian University,
ul. Lojasiewicza 11, 30-348 Kraków, Poland and
AGH, Department of Automatics and Biomedical Engineering, Al. Mickiewicza 30, 30-059, Kraków, Poland
arXiv:1508.03184v1 [cond-mat.stat-mech] 13 Aug 2015
Ewa Gudowska-Nowak
Marian Smoluchowski Institute of Physics, Jagiellonian University,
ul. Lojasiewicza 11, 30-348 Kraków, Poland and
Mark Kac Complex Systems Research Center, Jagiellonian University, Kraków, Poland
We consider diffusive motion of a particle performing a random walk with Lévy distributed jump
lengths and subject to resetting mechanism bringing the walker to an initial position at uniformly
distributed times. In the limit of infinite number of steps and for long times, the process converges
to a super-diffusive motion with replenishment. We derive formula for a mean first arrival time
(MFAT) to a predefined target position reached by a meandering particle and analyze efficiency of
the proposed searching strategy by investigating criteria for an optimal (a shortest possible) MFAT.
PACS numbers: 05.40.Fb, 05.10.Gg, 02.50.-r, 02.50.Ey,
I.
INTRODUCTION
Limited random walks, constrained to additional conditions, like e.g. sudden termination of a trajectory
by absorbing targets in porous media, biological tissues,
composite materials or dynamic networks are frequently
analyzed restrictions in description of extreme, catastrophic events like gambler’s ruin, chemical reactions and
species extinction [1–5]. Quite often however, the absorption events and disappearance of trajectories are followed
up by resets or restart activities of the system, reminiscent of e.g. relocation of searching paths in animals’ foraging, seeking for target location by repair proteins or
returning to initial position after unsuccessful search of
the address by an individual lost in a vast city [6, 7]. A
random walk with restart is also known as a graph mining
technique widely used in machine learning community for
page-ranking or web search models and cryptology [8–11].
In this approach the frequency of visits paid to a given
node can be analyzed as a random walk on a graph. It is
described as an ordered sequence of visits to vertices with
a source (initial) vertex probability p~0 . For Markov chain
models of transitions between subsequent locations on a
graph described by a matrix Π, reset events inject additional randomness to the walk p~i+1 = (1 − c)Π~
pi + c~si
with c being probability of resetting per a step and ~si
representing an arbitrary probability vector added at resetting. In a somewhat different context, similar random walks with stochastic resets have been analyzed by
Durang et al. [12] who posed the problem of interacting particles subject to a stochastic return to its initial
configuration in the coagulation-diffusion process. The
particles perform random hoppings to nearest-neighbour
sites such that upon encounter of two particles, the arriving particle disappears. The stochastic reset is described
then by a given set of probabilities for having some consecutive empty sites.
A Markov monotonic continuous time random walk
model in the presence of a drift and Poisson resetting
events has been addressed in an elegant work by Montero and Villarroel [13], who derived general formulas for
the survival probability and the mean exit time. Simple
Brownian diffusion analogs in external potentials have
been further analyzed in a recent studies by Pal [14].
Many intriguing facets of the process in which a Brownian
particle is stochastically reset to its initial position with
a constant rate have been also investigated by Evans,
Majumdar and Kusmierz [15, 16]. The stationary state
of such process has been shown to be described by a
non-Gaussian distribution which, due to a non-vanishing
steady state current directed towards the resetting position, violates the detailed-balance condition. Moreover,
it has been proved [15, 16] that there exists an optimal
resetting rate that minimizes the average hitting time to
the target. Extensions to space depending rate, resetting
to a random position with a given distribution and to a
spatial distribution of the target have been also considered in [17].
In this work we concentrate on a variant of the model,
in which one-dimensional jump-like searching process and
resetting events are analyzed as renewal Markov model
with Lévy jumps. We assume that a random searcher
starts its motion at x0 = 0 and tries to find the object
located at some position x. The walker does not memorize its former locations, and the steps undertaken at any
instant of time are statistically independent and drawn
from symmetric stable distribution with a stability index
α ∈ (0, 2]. Furthermore, at random times following the
Poisson point process, the searcher decides to instantaneously reset to the initial position. We derive expression for the transition probability density of such process,
analyze existence and character of long-time stationary
distribution and discuss optimal conditions for the mean
first arrival time (MFAT).
The paper is organized as follows: Section II introduces
2
the model and discusses the structure and stationary solutions of evolution equations for corresponding probability distribution functions. The mean first arrival time
in the model is introduced in Section III and its optimization is further analyzed in Section IV which presents the
most important results of this project. We summarize
the paper with conclusions in Section V.
II.
TIME EVOLUTION AND TRANSITION
PROBABILITY
(1)
In course of time W (x, t|x0 , t0 ) is subject to possible reset events to x = 0 or jumps (Lévy flights). Resets are
independent from flights and occurring in time according
to Poisson statistics with an average expectation time
for the occurrence of the event given by r−1 . Note that
for the purpose of analysis, we have untied the initial
and resetting positions. We denote the former as x0 and
keep the latter at the origin. The overall process is time
homogeneous, i.e. W (x, t|x0 , t0 ) = W (x, t − t0 |x0 , 0) ≡
W (x, t − t0 |x0 ), so that the propagator satisfies the equation:
W (x, t|x0 ) = e−rt W0 (x, t|x0 ) +
Z t
+
dτ e−rτ rW0 (x, τ |0)
(2)
0
The first term on the RHS of the above renewal equation is survival of the probability mass without resetting
events, whereas the second term describes evolution after the last reset. Function W0 (x, τ |x0 , t0 ) denotes the
probability density function (PDF) of the process when
the resetting mechanism is switched off. In this case the
random walk propagator fulfills equation
Z t
W0 (x, t|x0 ) = δ(x − x0 ) 1 −
Φ(τ )dτ )
+
Z
0
t
Φ(t − t0 )
Z
0
+∞
−∞
(3)
with the stability index 0 < α ≤ 2. The resulting process is Markovian, with variance diverging for α < 2 and
fractional moments [18] scaling like:
(6)
where D = σ α /τ0 .
The asymptotic behavior of
W0 (k, s|x0 , 0) can be deduced by taking the limit k → 0
and s → 0 which implies:
Φ(s) ≈ 1 − sτ0 + ... p(k) ≈ 1 − D|k|α .
(7)
After proper rescaling of waiting times and jumps [18],
the diffusion limit of the integral equation (3) is obtained
in the form of a space fractional Fokker-Planck equation
(FFPE)
∂
∂α
W0 (x, t|x0 ) = D
W0 (x, t|x0 )
∂t
∂|x|α
(8)
α
∂
with ∂|x|
α denoting the symmetric Riesz space fractional
derivative which represents an integro-differential operator defined as:
∂α
−1
f (x) =
×
∂|x|α
2 cos(πα/2)Γ(2 − α)
Z ∞
∂2
f (x0 )
×
dx0
2
∂|x| −∞ |x − x0 |α−1
(9)
The total propagator of the process W (x, t|x0 ) can be
then obtained from Eq.(2). In the Laplace domain this
equation has the form
r
W (x, s|x0 ) = W0 (x, s + r|x0 ) + W0 (x, s + r|0) (10)
s
In the case of Lévy flights W0 (k, s|x0 ) =
W (k, s|x0 ) is given by
W (k, s|x0 ) =
eikx0
D|k|α +s ,
eikx0 + rs
D|k|α + s + r
sW (k, s|x0 ) − eikx0 =
r
= −D|k|α W (k, s|x0 ) − rW (k, s|x0 ) + .
s
hence
(11)
the integral Eq.(3) takes the form of
(4)
(12)
The inverse transformation yields to FFPE describing
evolution of the total probability distribution:
∂
∂α
W (x, t|x0 ) = D
W (x, t|x0 ) −
∂t
∂|x|α
−rW (x, t|x0 ) + rδ(x)
W (k, s) ≡ F[L[W (x, t); t → s]; x → k],
1
1 − Φ(s)
,
s
1 − Φ(s)p(k)
(5)
and obeys the differential equation
p(x − x0 )W0 (x0 , t0 |x0 )dx0 dt0
where Φ(t) is the waiting time PDF, independent from
the jump-length PDF p(x − x0 ). In the Fourier-Laplace
space
W0 (k, s|x0 ) =
F[p(x)] = exp [−σ α |k|α ]
h|x(t)|q i ∝ (Dt)q/α ,
We start with analysis of the integral equation that
governs evolution of the probability density function for
the process {X(t), t ≥ 0}:
W (x, t|x0 , t0 )dx ≡
Prob{x < X(t) ≤ x + dx|X(t0 ) = x0 }
R∞
where L[f (t)] ≡ 0 exp(−st)f (t)dt. We further assume
R∞
that Φ(t) has a well defined mean value, τ0 = 0 tΦ(t)dt,
and p(x) is PDF of the Lévy stable form, so that its
characteristic function reads
(13)
with initial condition W (x, 0|x0 ) = δ(x − x0 ). Eq.(13)
is analogous to the Fokker-Planck equation defining a
3
model of diffusion with stochastic resetting [15]. The
difference lies in the fact, that instead of a second order spatial derivative, characteristic of normal (Gaussian)
diffusion, we are dealing now with a non-local fractional
derivative, which describes Lévy flights. Note that the
model analyzed in this paper includes the other one as a
special case, for α = 2.
Having calculated the propagator in the FourierLaplace space, it is straightforward to obtain a characteristic function of the stationary distribution. For the sake
of simplicity, we also introduce a length scale λα ≡ D
r .
By definition, the stationary PDF can be then derived
from the relation
ps (k; λ, α) ≡ lim sW (k, s|x0 )
s→0
= rW0 (k, s = r|0) =
1
.
1 + |λk|α
(14)
W (x, t|0, 0) =
The resulting function Eq.(14) is known as the Linnik
distribution [19, 20], a special case of the family of geometric stable PDFs, approximating distribution of normalized sums of i.i.d random variables
SN =
N
X
xi
(15)
i
where the number of terms N is sampled from a geometric distribution, i.e. P (N = k) = (1 − p)k−1 p. Summation of that type has been used, among others, in modeling energy release of contributing earthquakes, water
discharge over a dam during the flood, or avalanche dynamics [21]. The Linnik PDF can be expressed in terms
of elementary functions only for α = 2, in which case it
becomes a well-known Laplace distribution:
|x|
1
ps (x; λ, 2) = √ e− λ
2 λ
(16)
with a zero mean and a variance V ar[x2 ] = λ2 . For α = 1
the closed-form expression for the corresponding Linnik
PDF can be obtained (cf. Appendix) in terms of special
Rx
R∞
functions Si(x) ≡ sinttdt and Ci(x) ≡ − costtdt , and in
0
THE PROBLEM OF THE FIRST ARRIVAL
TIME
For the stochastic process defined by Eqs.(1,2), a question of interest is estimation of the waiting time before
Zt
dτ pf a (τ, x)W (x, t|x, τ )
(18)
0
The above formula can be easily interpreted: it simply
states that the process which at time t finishes up at x,
has had to get to that point for the first time at some
time τ ∈ (0, t), then it could move freely until at time
t, it came back to the very same point. The assumption
of time-homogeneity (W (x, t|x, τ ) = W (x, t − τ |x, 0)) explains a convolution operator on the RHS of Eq.(18). The
density pf a is a function of the first argument. The second variable denotes that the first arrival to a position x
is evaluated. For readability, we skip D, r and α in the
parameter list. From now on, we also assume that the
initial and reset positions coincide.
By transforming Eq.(18) into the Laplace space a simple algebraic relation is obtained:
W (x, s|0) = pf a (s, x)W (x, s|x)
(19)
It is important to notice that W (x, s|x) 6= W (0, s|0), as
the resetting mechanism introduces space inhomogeneity.
Our aim is to derive a formula for the mean first arrival
time (MFAT) which can be obtained from the pf a (s, x)
as follows:
x
a scaled form reads:
1
1 Si(x)
λps (λx; 1, 1) =
−
sin |x| − Ci(|x|) cos x.
2
π
π
(17)
When passing to the analysis of the first arrival times
in a subsequent Section, we note here that the result
Eq.(14) has been obtained earlier in [22] for a discrete
time counterpart of the resetting model.
III.
the first event of magnitude greater than a given threshold is observed. However, as it has been discussed elsewhere [23–25], a super diffusive nature of Lévy flights
strongly influences the statistics of first passage times
over the threshold. In particular, due to the long-range
character of Lévy jumps occurring with an appreciable
probability, the trajectory of the process may cross the
threshold numerous times without actually hitting it. In
consequence, the statistics of first arrival times to a predefined barrier is different from the statistics of first passages over it.
Following [1, 23], we introduce the first arrival time
PDF pf a (t, x), which describes distribution of random
times Tf a in terms of the integral equation for the propagator W (x, t|0, 0):
hTf a (x)i = −
∂
1 − pf a (s, x)
pf a (s, x)|s=0 =
|s=0 (20)
∂s
s
We proceed by inserting the propagator, Eq.(10), and
the algebraic relation between propagator and pf a (s),
Eq.(19), into the formula for MFAT, Eq.(20). After
straightforward algebraic manipulations we arrive at:
1 W0 (x, s = r|x)
hTf a (x)i =
−1 =
r W0 (x, s = r|0)
1 ps (0; λ, α)
=
−1
(21)
r ps (x; λ, α)
Note that for simplicity we use a shortened notation
W0 (x, t) ≡ W0 (x, t|0, 0). Eq.(21) shows that analyzed
MFAT can be expressed either in terms of the Laplace
4
transform of the propagator of the standard Lévy αstable process without resetting, or in terms of the stationary PDF of the process with resetting mechanism
switched on. This result is very general, since in the
derivation no particular form of W0 (x, t|x0 ) has been assumed.
We further focus on the special case of Lévy flights,
α ≤ 1. In general, the propagator of a Lévy stable process
cannot be expressed in terms of elementary function as
a function of x. A representation in terms of the Fox
functions is known in literature, see e.g.[26], but it would
not be useful in our case. We can, however, calculate
W0 (x, s = r|x) and deduce from its form the range of the
stability parameter α which guarantees finiteness of the
evaluated MFAT:
1
W0 (x, s|x) =
2π
=
Z
dk
Z∞
dte
0
R
1
1
Γ( α )Γ(1 − α )
=
1
1
παD α s1− α
−st −D|k|α t
e
1
1
π
α sin α
D α s1− α
√s
1
W0 (x, s|0, 0) = √ e−|x| D
2 Ds
We may also expand MFAT around x = 0 using known
expansion of Linnik distribution [27, 28], leading to:
π
α sin α
IV.
Asymptotic behavior
The average hTf a (x)i cannot be expressed in terms of
elementary functions for arbitrary α. Nevertheless, we
can learn something about its behavior for large and
small distances x to a target. By taking a well-known
expression for the asymptotic expansion of α-stable distributions [26] and transforming it to the Laplace space,
or otherwise, directly expanding
(23)
and transforming it back from the Fourier space term by
term, we obtain an asymptotic expansion of the propagator W0 in the Laplace space (see also [27, 28] for more
formal derivations):
∞
1X
π
Dn Γ(nα + 1)
(−1)n+1 sin( nα) n+1 nα+1
π n=1
2
s
x
(24)
This expression is correct for α ∈ (1, 2). For α = 2 we
don’t need the asymptotic expansion since in this case
W0 (x, s|0, 0) =
1
1
1
2 sin π(α−1)
Γ(α) r α D1− α
2
xα−1 + O(x2α−2 ).
(27)
OPTIMIZATION OF MFAT
Given a distance to a target x, one could be tempted
to determine the optimal search kinetics of this location.
We choose MFAT as an objective function, and minimize it in the space of parameters (r, α). We will denote derived parameters of the efficient strategy as r∗ (x),
α∗ (x), respectively and the corresponding optimal MFAT
as T ∗ (x).
A.
∞
X
1
(−D|k|α )n
=
D|k|α + s n=0
sn+1
(25)
(22)
For any x 6= 0, the propagator W0 (x, r|0) is finite, since
it is an integral of an oscillating function with an amplitude decreasing to zero, and can be rewritten as an
alternating series. We therefore conclude from Eq.(22)
that the MFAT diverges for α ≤ 1 and remains finite
for 1 < α ≤ 2. That apparent finiteness of the MFAT
in case of Lévy flights is rather surprising, taking into
account discontinuous character of superdiffusive trajectories and thus possibility of overshooting (i.e. jumping
over the target).
A.
(for α = 2)
One can easily verify that Eqs.(21,22) together with
Eq.(25) give the same result as the one derived in [15].
We truncate the series at the first term and so obtain
large x behavior of the MFAT:
(
xα+1
√ r ;1 < α < 2
hTf a (x)i ∝
(26)
x D
e
;α = 2
hTf a (x)i ≈
=
1
we have a closed-form expression:
Fair comparison
Since we want to compare Lévy flights with different
stability indices α, it is important to carefully choose the
parametrization of the family of
jump distributions. One
α
commonly used is φ(k) = e−|k| which in our case means
fixing D = 1 for every α. Alas, this choice is very arbitrary and based on simplicity of a characteristic function
for symmetric stable distributions. As an alternative option, we propose here a straightforward and consistent
approach based on fractional moments. Let us define a
random variable ξα to be a position of the process without resetting at time t = 1 (this fixes the time unit). The
p-th fractional moment may be expressed as
p
λp0 = h|ξα |p i = D α f (α, p)
(28)
where condition p < α has to be satisfied in order for
fractional moment to be finite. Function f (α, p) is known
and reads [29]
f (α, p) =
2p+1 Γ( p+1
)Γ(− αp )
√ 2
α πΓ(− p2 )
(29)
We want to keep λ0 constant (e.g. λ = 1) so our D will
depend on α and p. The most natural choice of p in our
5
case is p = 1 since it does not exclude any solution (in
line with findings of Section III, we refer to cases with
α > 1 assuring finiteness of MFAT) and it induces L1
norm quite commonly used in many applications. This
choice leads to the expression:
α
π
(30)
D(α) =
2Γ(1 − α1 )
In the following we will refer to this method of comparison based on the choice p = 1 as “fair comparison”,
in contrast to “naive comparison” based on simplicity of
characteristic function (D = 1).
B.
Asymptotic analysis
From the asymptotic behavior of the MFAT several
conclusions may be drawn: The prefactor in Eq.(27) is
bounded for α ∈ (1, 2]. Consequently, for given non-zero
r and D it is always possible to find x small enough, so
that α = 2 minimizes MFAT. In other words, Brownian
motion is expected to be optimal strategy at small distances to the target. In contrast, as it can be inferred
from the asymptotic behavior Eq.(26), for large enough
distances MFAT increases with x much faster for α = 2
than for α < 2. In this case, the Lévy motion with α < 2
minimizes MFAT, thus indicating a more efficient kinetics of space exploration to detect a target.
C.
Random distribution of target sites
In many natural scenarios, living organisms navigate to
unpredictable, or randomly distributed resources where
positions of the ”target” are not precisely known. How
is kinetics of random search with resetting affected by
location of targets in an unknown environment? In order
to address this point, we further explore MFAT under
the constraint that the searcher knows only the mean
(expected) distance to the target. Accordingly, instead
of a fixed x in evaluation of MFAT, we use the PDF
which satisfies the maximum entropy principle, i.e. a
Laplace distribution p(x) of target positions is assumed.
The MFAT in this more general setting can be calculated
by averaging over possible distances:
hTf a (λt )i =
Z∞
hTf a (λt )i =
dxhTf a (x)ip(x) =
Z∞
|x|
dxhTf a (x)ie− λt
(31)
−∞
Even though hTf a (λt )i is a different function from
hTf a (x)i, for readability we keep the same symbol for the
MFAT averaged over distribution of targets and denote
that by use of a different argument, only.
1
r
λ
λt
1
,
−1
(32)
q
where λ = D
r . Clearly, the MFAT is finite for λ ≥ λt
and optimization of Eq.(32) yields the value of resetting
D
frequency r2∗ (λt ) = 4λ
If a searcher does not know
2.
t
the distribution of target locations but was able to estimate via several measurements the mean distance to
the target, h|x|i ≈ λt , he might be prompted to use
that fixed position for further optimization of MFAT,
hTf a (x = λt )i. Derived optimal r∗ , see Eq.(B1), when
applied in the system with Laplace distributed distanceto-target, would then lead to infinite MFAT. This apparent inconsistency demonstrates that for the proper minimization of arrival times, the actual form of distance-totarget distribution p(x) is indispensable.
It can be easily shown that for heavy-tailed distanceto-target distributions, the Brownian strategy always
gives infinite MFAT. In contrast, strategies with Lévydistributed jumps (α < 2) may provide efficient algorithms for searching, for which MFAT remains finite, as
long as the p(x) distribution is characterized by a finite
variance. A simple example illustrating this case is optimization of MFAT given by Eq.(31) with the Student’s
t-distribution of distances to the target:
− ν+1
2
Γ ν+1
x2
2 p(x) = √
.
1+ 2
ν
νλt
νπΓ 2 λt
(33)
In this case the integral in Eq.(31) is convergent iff condition α < ν − 1 holds. Numerical integration of Eq.(31)
for ν = 2.7 and ν = 4 leads to MFAT functions displayed
in Figs.5,6.
D.
Scaling
Optimal parameters r∗ (x), α∗ (x) and optimal MFAT
T (x) depend on x and D. For the sake of simplicity,
from now on we fix D. It will be useful to take advantage of dimensional analysis to calculate scaling behavior
of optimal r∗ and MFAT for a given α. Let rα∗ (x) and
Tα∗ (x) be optimal r and optimal MFAT for fixed x, α
and D. Up to an arbitrary multiplicative constant, the
only combination
of x and D that has an unit of time is
α
t = xD . This leads to the following scaling equations:
(
Tα∗ (x) = Tα∗ (1)xα
(34)
r ∗ (1)
rα∗ (x) = αxα
∗
−∞
1
=
2λt
As explained in the following example, such averaging
over random distances to a target leads to modification of
the MFAT and becomes crucial for the optimal strategy
planning. Let us assume Brownian diffusion α = 2 with
the Laplace PDF of target positions characterized by the
mean distance to the target h|x|i = λt . In that case
MFAT is given by the formula:
6
One can easily verify that these equations hold, by calculating a derivative of MFAT (Eq.21) with respect to r,
comparing it to 0, and rewriting a corresponding equation so that it contains only function of rxα . Scaling
equations (34) imply also similar relations to be fulfilled
by T ∗ (λt ):
(35)
t
4.5
5
4
4
3.5
3
log <T>
(
Tα∗ (λt ) = Tα∗ (1)λα
t
r ∗ (1)
rα∗ (λt ) = αλα
6
3
2
2.5
1
Analytical formula
The above relations are used in a numerical algorithm of
optimization, as explained in details in the Appendix.
Monte Carlo δt=0.01
−1
1
0.2 0.6 1
Results
A comparison between analytical prediction, Eq.(21),
and numerical stochastic simulations has been performed
and the results are displayed in Fig.1 demonstrating a
perfect agreement between both approaches. Additionally, Fig.2 presents analytically derived MFAT functions
in 2-dim (α, r) parameter space.
MFAT diverges as r → 0 and r → ∞ (cf. Figs. 1,2).
Accordingly, a minimum of MFAT with respect to r can
be found in the interval [0, ∞) and its position depends on
the stability index α characterizing underlying diffusive
process.
For small x MFAT values are systematically higher for
non-Gaussian diffusion (α < 2) than for the Gaussian
case and the same resetting rates. Also, as displayed in
Fig.1, MFAT has a more pronounced, deeper minimum
in function of r for Lévy diffusion with heavier tails (i.e.
lower α’s), which suggests that the Gaussian strategy is
more robust to variations of r. This is, however, no longer
true for large x (cf. Fig.2). In that case MFAT values for
α = 2 are higher than than for α < 2 and the same r, at
least in the vicinity of the optimal rα∗ . Moreover, in this
limit Lévy flights become more resilient to changes in r,
especially in the range r ≥ rα∗ .
Results displayed in Fig.2 have been further analyzed
to derive minimal values of MFAT with respect to a pair
of parameters (α, r) for different values of a distance to
a target, x. Consecutive Fig.3 and Fig.4 show outcomes
of the optimization procedure described in Appendix B
for the cases of the immobile target located at a distance
x, and the target with position described by Laplace distribution with an average distance to a target λt , respectively.
No qualitative difference in derived optimal MFAT values has been found between the naive and fair comparison. We therefore hereafter present results of numerical
optimization of hTf a (x)i and hTf a (λt )i for fair comparison only.
As expected, for small x (or, equivalently for small λt )
Gaussian diffusive motion (α∗ = 2) is an optimal searching strategy. With growing distance to a target x (or
λt ) the minimum of hTf a i becomes shallower, up to some
1.5
Monte Carlo δt=0.001
−2
E.
2
Monte Carlo δt=0.03
0
1.5
2
2.5
3
3.5
4
0.5
0.11
3
5
x
7
9
11
13
r
FIG. 1. Comparison between MFATs obtained by numerical integration of the analytical formula (21) (lines) and by
averaging over N = 105 realizations of a simulated process.
Different lines (from the top to the bottom) correspond to
α = (1.4, 1.6, 1.8, 2). For the sake of simulation not only time
has to be discretized (δt), but also a finite target size is needed.
For each α the target size is chosen separately to match the
analytical result at x = 1, r = 1. The same target size is further used across different values of x and r. Estimated error
bars are smaller than markers used in plots and hence have
not been displayed.
point x∗ ≈ 10.8 (λ∗t ≈ 3.25), beyond which Gaussian diffusion is not efficient anymore and the optimal stability
index switches to values α∗ < 2. Corresponding values of bifurcation points x∗ and λ∗t have been obtained
by means of numerical optimization procedure and are
marked in Figs.3,4 with a cross sign.
We have also investigated an interplay of intermittent
Lévy diffusion, random resetting with rate r and an impact of heavy-tailed distribution of distances to the target
on the efficiency of the searching. MFAT analysis performed in this case is illustrated in Figs.5,6. Presented
plots indicate that heavy-tailed distribution of distanceto-target excludes Gaussian diffusion α = 2 from the set
of possible optimal search strategies. Moreover, in line
with the analysis of Section IVC, for Lévy flights a condition α < ν − 1 has to be met in order to perform a
successful search with a finite MFAT.
V.
CONCLUSIONS
Not only animals foraging patterns, but also memory
retrievals of humans [30] and fluctuations of their spontaneous activity [31] exhibit scaling statistics. The problem
devised in this paper models mechanism of stochastic resetting, or relaxation of a diffusive searching process to a
predefined threshold, and as such can be well adapted to
many natural scenarios of exploration processes such as,
7
x=10
20
15000
4
log 10 (r * )
x=1
1000
10
1.85
1.7
2
α
500
0
2
8
5
1.8
α
r
1.6
4
x=20
0.02
0.06
0
0.1
<T>
1000
1.75
α
0.01
1.5
0
2
0.02 0.03
40
60
80
100
-1
0.5 1
1.4
r
1.5
−3
x 10
1.2
0
20
40
60
40
60
80
100
100
FIG. 4. Optimal parameters (α∗ , r∗ ) and MFAT in function
of λt . Target position is a random variable with the Laplace
PDF of distances and the average distance-to-target λt .
4
log 10 (r * )
T*
10000
2
T*
log 10 (r * )
20
80
6t
5000
0
-2
2
0
-2
0
0
-4
0
2
1.4
1.7
15000
0
1
1.6
4
5000
0
log 10 (6 t )
2
FIG. 2. MFAT in function of parameters (α, r) for different
values of distance to an immobile target x = (1, 10, 20, 100).
Contour plots beneath surfaces help to guide an eye towards
the minimum.
10000
-2
1.8
α
r
0
6t
5
0
2
20
r
10
2000
2
-4
0
x=100
x 10
3000
<T>
5000
,*
0
2
T*
<T>
<T>
10000
-2
-1
0
1
log 10 (x)
x
20
40
60
80
100
-3
-2
-1
0
1
2
log 10 (6 t )
6t
2
2
2
,*
1.8
,*
1.8
1.6
1.4
1.6
1.2
0
1.4
0
20
40
60
80
100
x
20
40
60
80
100
6t
FIG. 3. Optimal parameters (α , r ) and MFAT in function
of a distance to a target x.
FIG. 5. Optimal parameters (α∗ , r∗ ) and MFAT in function
of λt . Target position is given by Student’s t-distribution
Eq.(33) with ν = 4 and the average distance-to-target λt .
e.g. quests for food in a given territory [32], translocation
and recruitment of repair proteins seeking for a disrupted
DNA strand to be repaired [33], optimal computer-aided
web search [11] or statistics of recall periods in retrospective memory [30]. The efficiency of a search may be
defined and analyzed by use of different measures, like
e.g. the number of encounters of searchers and targets
per unit of time or the exploration range of space per
unit of time. Here, we have focused on efficiency measure expressed by the mean time to reach an immobile
target, MFAT.
The first arrival time statistics has been analyzed for
the one-dimensional problem with a constant resetting
rate r. The acts of trajectory relocation have been
assumed independent from the intermittent free diffusive motion described by Lévy jumps with the exponent
1 < α ≤ 2. Despite discontinuity of trajectories, typical
for Lévy motion, the MFAT remains finite in this case
with a rich characteristics of optimal (minimal) times
T ∗ (x). By use of the designed optimization method (Section IV), we have been able to derive optimal parameters
r∗ (x) and α∗ (x) for the range of target positions x. We
have shown that randomized distribution of targets with
some average distance to a target results in severe reduction of distances, for which Gaussian search remains
an optimal strategy. Moreover, our analysis of optimal
searching times for exponential distribution of distances
to a target (Section IV C) clearly indicates that not only
∗
∗
8
can assume that x ≥ 0.
#10 4
3
log 10 (r * )
2
T*
2
1
f (x) = πps (x, 1, 1) =
0
-2
0
0
20
40
60
80
100
-3
-2
-1
0
1
2
=
dk
=
Z∞
,*
1.6
1.5
se−s
ds 2
=
x + s2
−
1.2
0
20
40
60
80
100
6t
FIG. 6. Optimal parameters (α∗ , r∗ ) and MFAT in function
of λt . Target position is given by Student’s t-distribution
Eq.(33) with ν = 2.7 and the average distance-to-target λt .
first moment of that distribution but rather its actual
form is desired for a proper optimization planning: an
optimization procedure based solely on the information
about average distance to a target would result in an
optimal r∗ leading to infinite searching times MFAT.
Altogether, the proposed optimization scheme and
scaling analysis can be further exploited , e.g. for twodimensional searching scenarios. A plausible modification of the proposed procedure could be also implementation of Lévy walks, with coupled space-time distributions, or truncated Lévy flights, both penalizing very long
steps in motion.
VI.
ACKNOWLEDGMENTS
This project has been supported in part (EGN) by
National Science Center (ncn.gov.pl), a grant no. DEC2014/13/B/ST2/020140. Authors acknowledge many
valuable discussions with Martin Bier.
Appendix A: Linnik distribution
Derivation of the Linnik PDF, expressed in terms of
ps (x, λ, 1) in Eq.(17) proceeds as follows. Since ps (x, λ, 1)
as a function of x is even, without loss of generality, we
e−ikx
1 + |k|
ds cos (kx)e−s(1+k) =
0
1.4
1.3
dk
0
0
1.7
Z
R
Z∞
log 10 (6 t )
6t
Z∞
1
2
Z∞
dt
te−tx
=
1 + t2
0
d
dx
Z∞
0
dt
−tx
e
d
≡ − g(x)
2
1+t
dx
(A1)
One can verify by inspection that g(x) is a solution of an
equation
g 00 (x) + g(x) =
1
x
(A2)
which is a second order inhomogeneous linear differential
equation with constant coefficients. We can easily solve
it by using a variation of constants. Two constants in
the general solution are calculated from the boundary
conditions g(0) = π2 and lim g(x) = 0. The solution
x→∞
reads:
g(x) = (
π
− Si(x)) cos x + Ci(x) sin x
2
(A3)
which, after differentiation, leads to formula (17).
Appendix B: Numerical scheme
The optimization problem at hand could not be solved
analytically, thus we have solved it numerically. Scaling
formulas Eq.(34) allow very fast numerical optimization,
by reducing numerical calculation of MFAT to one value
of x for each α and r. The algorithm then proceeds as
follows: For each α we perform numerical integration by
use of the reverse Fourier transform of the Linnik distribution Eq.(14) for a given value x, e.g. x = 1, and a
few values of r. We fit quadratic function to calculated
points and find its minimum. Then we refine interval of
r values centering it at estimated minimum and, consecutively, we reduce its length. This procedure is repeated
until desired accuracy is achieved. We end up with a
quadratic function which, by means of its vertex coordinates, defines our Tα∗ (1) and rα∗ (1). Scaling equations
Eq.(34) allow to extend these results for arbitrary x.
When we start the calculation for a new value of α we
face a problem of choosing a proper interval of values of
r. Since we fit a quadratic function, it is important that
the interval contains the optimal r. For this reason, we
∗
can make use of the optimal rprev
calculated in a previous
step, for a value of α close to a new one. Accordingly, we
∗
choose an interval of r which contains rprev
. The formula
9
for optimal resetting frequency for the Brownian motion
is known [15] and reads:
r2∗ (x) =
Dz 2
x2
(B1)
[1] S. Redner, A guide to first-passage processes, Cambridge University Press, 2001.
[2] D. S. Novikov, E. Fieremans, J. H. Jensen, and J. A.
Helpern, Nature Phys. 7, 508 (2011).
[3] D. W. Sims, A. M. Reynolds, N. E. Humphries,
E. J. Southall, V. J. Wearmouth, B. Metcalfe,
and R. J. Twitchett, Proc. Natl. Acad. Sci. USA 111,
11073 (2014).
[4] G. Harik, C.-P. Erick, D. E. Goldberg, and B. L.
Miller, Evol. Comput. 7, 231 (1999).
[5] J. Zhu and J. C. Rasaiah, J. Chem. Phys. 101, 9966
(1994).
[6] O. Bénichou, C. Loverdo, M. Moreau, and R. Voituriez, Rev. Mod. Phys. 83, 81 (2011).
[7] V. V. Palyulin, A. V. Chechkin, and R. Metzler,
Proc. Natl. Acad. Sci. USA 111, 2931 (2014).
[8] D. Galbraith, Steven, Mathematics of public key
cryptography, Cambridge University, 2012.
[9] P. DeScioli, R. Kurzban, E. N. Koch, and D. LibenNowell, Perspect. Psychol. Sci. 6, 6 (2011).
[10] D. Liben-Nowell and J. Kleinberg, Proc. Natl. Acad.
Sci. USA 105, 4633 (2008).
[11] H. Tong, C. Faloutsos, and J.-Y. Pan, Fast Random
Walk with Restart and Its Applications, in Proceedings
of the Sixth International Conference on Data Mining,
pp. 613–622, IEEE Computer Society, 2006.
[12] X. Durang, M. Henkel, and H. Park, J. Phys. A:
Math. Theor 47, 045002 (2014).
[13] M. Montero and J. Villarroel, Phys. Rev. E 87,
012116 (2013).
[14] A. Pal, Phys. Rev. E 91, 012113 (2015).
[15] M. R. Evans and S. N. Majumdar, Phys. Rev. Lett.
106, 160601 (2011).
[16] L. Kusmierz, S. N. Majumdar, S. Sabhapandit, and
G. Schehr, Phys. Rev. Lett. 113, 220602 (2014).
[17] M. R. Evans and S. N. Majumdar, J. Phys. A: Math.
Theor. 44, 435001 (2011).
with z ≈ 1.5936. Therefore, when performing numerical
analysis, we have started our calculations from α = 2.
In the very last step we find, for each x, an α-parameter
for which the smallest optimal MFAT is obtained. This
is our global minimum. A numerical scheme used for
optimization of hTf a (λt )i is analogous.
[18] R. Klages, G. Radons, and I. M. Sokolov Eds.,
Anomalous diffusion, Wiley VCH, 2008.
[19] Y. V. Linnik, Selected Transl. Math. Statist. and Prob.
3, 41 (1953).
[20] T. J. Kozubowski and S. T. Rachev, J. Comput. Anal.
Appli. 1, 177 (1999).
[21] V. Pisarenko and M. Rodkin, Heavy Tailed Distributions in Disaster Analysis, Springer Verlag, 2010.
[22] D. N. Anderson and B. C. Arnold, J. Appl. Prob. ,
330 (1993).
[23] A. V. Chechkin, R. Metzler, V. Y. Gonchar,
J. Klafter, and L. V. Tanatarov, J. Phys. A: Math.
Theor. 36, L537 (2003).
[24] B. Dybiec, E. Gudowska-Nowak, and P. Hänggi,
Phys. Rev. E 73, 046104 (2006).
[25] B. Dybiec and E. Gudowska-Nowak, J. Stat. Mech.
Theor. and Exp. , P05004 (2009).
[26] R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000).
[27] S. Kotz, I. Ostrovskii, and A. Hayfavi, J. Math.
Anal. Appli. 193, 353 (1995).
[28] S. Kotz, T. Kozubowski, and K. Podgorski, The
Laplace distribution and generalizations: a revisit with
applications to communications, economics, engineering,
and finance, Number 183, Springer Science & Business
Media, 2001.
[29] M. Shao and C. Nikias, Proc. IEEE 81, 986 (1993).
[30] C. T. Kello, G. D. Brown, R. F. i Cancho, J. G.
Holden, K. Linkenkaer-Hansen, T. Rhodes, and
G. C. V. Orden, Trends Cogn. Sci. 14, 223 (2010).
[31] J. K. Ochab, J. Tyburczyk, E. Beldzik,
D. R. Chialvo, A. Domagalik, M. Fafrowicz,
E. Gudowska-Nowak, T. Marek, M. A. Nowak,
H. Oginska, and J. Szwed, PLoS One , e107542
(2014).
[32] A. Reynolds, Phys. Life Rev. , (2015).
[33] A. Badrinarayanan, T. B. Le, and M. T. Laub, J.
Cell Biol. 210, 385 (2015).
B. MARCUS CANONICAL EQUATION
B
Marcus canonical equation
B.1
Recipe
As discussed in Section 2.1.2, the Itô-Stratonovich dilemma arises in the context of SDEs
with multiplicative white Gaussian noise. The Stratonovich prescription is usually used
in the physical literature in favor of Itô’s approach since it a) preserves standard rules of
calculus and b) appears as the proper limit of colored noise (the Wong-Zakai theorem).
The first feature makes it convenient to use the Stratonovich integral in stochastic differential equations, allowing to solve them with a standard change of variables. When
the noise-generating process has jumps, neither Stratonovich nor Itô integrals preserve
the chain rule, thus some additional care has to be taken to find the solutions of the
corresponding SDEs. For systems driven by a compound Poisson process or Lévy flights
these two features can be retained by use of the Marcus integral.
Let (X )t≥0 be a stochastic process which solves the SDE:
Xt = X0 +
Zt
a(Xs , s)ds +
0
Zt
b(Xs , s) dZs ,
(1)
0
where denotes a stochastic integral which preserves the chain rule. The proper definition
of such stochastic integral has been proposed by Marcus [Mar78; Mar81], and the resulting
equation is called a Marcus canonical equation (or Marcus SDE). Here we show a recipe
how to use it in practice and give some examples. For a formal and complete treatment
consult [App09]. A less formal derivation is presented in [CP14].
Let us first assume that (Zt )t≥0 = (Pt )t≥0 , i.e. the noise-generating process is a pure
jump compound Poisson process. It turns out that the following prescription leads to the
desired solution
Xt
=
X0
+
Zt
0
a(Xs , s)ds
+
X k:Tk ≤t
φ(1; XT − , Jk , Tk )
k
−
XT −
k
,
(2)
where Jk and Tk are jumps and arrival times of the noise-generating process, whereas
Xt− = limε→0+ Xt−ε . The function φ is the solution of the following ordinary differential
124
B. MARCUS CANONICAL EQUATION
equation


d
φ(u; x, z, t)
du
= b(φ, t)z
(3)
φ(0; x, z, t) = x.
The noise-generating process (Zt )t≥0 in the Marcus canonical equation can be replaced
by any Lévy process (or even any semimartingale) [App09]. In the case of more general
noise-generating processes, e.g. with both pure-jump and continuous components or with
an infinite number of jumps in finite time, the sum in (2) is performed over every step of
the integration ∆t, and the limit ∆t → 0 is taken. In practice an approximate numerical
integration is performed with finite ∆t. The Euler scheme can be written as
Xt+∆t
= a(Xt , t)∆t + φ(1, Xt , ∆Zt , t).
B.2
(4)
Examples
Stochastic exponential
5
15
0.05<q<0.95
0.25<q<0.75
q=0.5
quantiles of ln Xt
quantiles of ln Xt
0
0.05<q<0.95
0.25<q<0.75
q=0.5
10
−5
−10
5
0
−5
−10
−15
−15
−20
0
5
10
−20
0
15
t
5
10
15
t
Figure 1: Stochastic exponentials of the Wiener process (left) and the Cauchy process (right).
Along with median (red) and quantile areas (gray and violet), ten sample paths are presented.
The simplest example of a multiplicative stochastic differential equation is
dXt = Xt dZt ,
(5)
where (Zt )t≥0 is a given noise-generating stochastic process. Its solution for X0 = 1 is
called a stochastic exponential (or Doléans-Dade exponential) of (Zt )t≥0 [Pro04]. If one
naively used the standard rules of differential calculus, he would arrive at the following
125
B. MARCUS CANONICAL EQUATION
solution
?
Xt = X0 eZt ,
(6)
which for nondifferentiable processes (e.g. Wiener process) is wrong due to the nonvanishing quadratic variation. The correct solution for the Wiener process (Zt = Wt ) reads
1
Xt = X0 eWt − 2 t ,
(7)
which can be shown by applying Itô’s lemma [Pro04; App09].
−5
0
−10
−2
5
10
15
−10
−4
0
5
−20
−20
15
Marcus
0
−2
−2
−10
−4
0
−4
0
−20
−20
5
10
15
0
Wt
10
−20
−20
20
10
0
15
−10
10
2
10
−10
20
2
t
0
20
4
lnXc2
t
quantiles of ln Xt
Colored
10
4
5
10
0
lnXst
2
20
lnXm
t
0
−15
0
10
Stratonovich
4
lnXit
quantiles of ln Xt
Ito
5
0
−10
0
Wt
10
20
−10
0
Wt
10
20
0
−10
−10
t
0
c2
Wt
10
−20
−20
20
(b)
(a)
Figure 2: Visualizations of the solutions of Eq. (5) with white Gaussian noise for four different
types of integrals. (a) Medians (red) and quantile areas q ∈ (0.25; 0.75) (violet) of ln Xt . (b)
ln Xt and corresponding Wt at t = 15. All integrals lead to an exponential dependence of Xt on
Wt . Also, as expected from Eq. (7), Itô integral gives the line shifted by − 2t .
By direct summation we can compute the stochastic exponential of the Poisson counting process (Nt )t≥0 :
dXt = Xt d(zNt ) =⇒ Xt = (1 + z)Nt .
(8)
If we change the integral in Eq. (5) to the Stratonovich integral,
dXto = Xto ◦ dZt ,
(9)
the solution is in general different from (Xt )t≥0 . In the case of Wiener process standard
rules of differential calculus hold and we arrive at the expected solution
Xto = X0o eWt .
126
(10)
B. MARCUS CANONICAL EQUATION
If we use either colored noise or Marcus prescription we arrive at the same result, which
can be seen in Fig. 2. Note that this is not the case for discontinuous processes, e.g. for
the Poisson counting process:
dXto
=
Xto
◦ d(zNt ) =⇒
Xto
=
2+z
2−z
Nt
.
(11)
For stochastic processes with discontinuous sample trajectories both Itó and Stratonovich
stochastic integrals does not preserve standard rules of differential calculus. This is in
particular true for α-stable processes with α < 2, as shown in Fig. 3 for α = 1, i.e. for the
Cauchy process. Interestingly, the logarithm of the stochastic exponential of the Cauchy
√
process does not scale with time as the Cauchy process. Instead, it scales as t, similarly
to the Wiener process (see Fig. 1). Also, there is no drift term as in the case of the
stochastic exponential of the Wiener process. These features are related to the fact, that
the quadratic variation of the Cauchy process is a random variable correlated with the its
value Ct . This is also the reason why the stochastic exponential of the Cauchy process is
tempered, i.e. its jumps are much smaller than these of the noise itself (cf. Fig. 2.1).
Ito
Stratonovich
200
100
100
0
lnXst
0
0
−100
5
10
15
−5
0
5
−200
−200
15
Marcus
20
20
10
10
0
0
lnXc2
t
Colored
10
100
100
0
−10
−20
0
−20
0
−200
−200
10
15
5
10
15
t
−200
−200
200
200
−10
t
0
Ct
200
−100
5
0
−100
lnXm
t
−5
0
quantiles of ln Xt
200
5
lnXit
quantiles of ln Xt
5
0
Ct
200
0
Ct
200
0
−100
0
Cc2
t
−200
−200
200
(b)
(a)
Figure 3: Visualizations of the solutions of Eq. (5) with white Cauchy noise for four different
types of integrals. (a) Medians (red) and quantile areas q ∈ (0.25; 0.75) (violet) of ln Xt . (b)
ln Xt and corresponding Ct at t = 15 (ordering of figures as in the left panel). Clearly, Itô and
Stratonovich integrals lead to a nonfunctional dependence of Xt on Ct , i.e. the solution depends
on the whole history of the noise-generating process (Cs )0≤s≤t . Due to a relatively long time
constant (τ = 0.2) colored noise also gives a nonfunctional dependence. The Marcus integral
gives the expected exponential dependence.
127
B. MARCUS CANONICAL EQUATION
Stochastic Verhulst equation
Ito
5
Stratonovich
1.2
1
0.8
0.6
0.4
0.2
0.6
0.4
0.2
3
0
2
4
6
2
1
0
0
2
4
t
6
8
1.2
1
0.8
0.6
0.4
0.2
0
2
2
4
t
6
0
Ito
2
4
t
6
quantiles of X
t
15
Stratonovich
1.5
t
0.05<q<0.95
0.25<q<0.75
q=0.5
mean
0.15
0.1
1
0.05
0.5
0
10
2
4
6
0
2
quantiles of X
t
Colored
5
2
6
(b)
20
0
0
4
Marcus
1.2
1
0.8
0.6
0.4
0.2
(a)
quantiles of X
0
Colored
quantiles of Xt
quantiles of Xt
4
quantiles of Xt
0.05<q<0.95
0.25<q<0.75
q=0.5
mean
4
t
6
1.2
1
0.8
0.6
0.4
0.2
0
8
4
6
Marcus
1
0.8
0.6
0.4
0.2
2
4
t
(c)
6
0
2
4
t
6
(d)
Figure 4: Exact (a,c) and approximate (b,d) solutions of the stochastic Verhulst equation (12)
with white Gaussian (a,b) and shot (c,d) noise. Mean (green), median (red) and quantile areas
(gray and violet) have been obtained by randomly sampling many (M = 105 ) noise trajectories,
from which trajectories of the process have been obtained according to the exact solution (13)
in (a,c) and four integration schemes in (b,d) with ∆t = 0.01. Additionally, five exemplary
trajectories have been plotted in (a,c).
As another example let us take a generalized stochastic Verhulst (logistic) equation:
dXt = Xt (1 − Xtµ )dt + Xt dZt ,
(12)
with an unique solution
Xt = X0 et+Zt
1 + µX0
Rt
0
128
es+Zs ds
µ1 ,
(13)
B. MARCUS CANONICAL EQUATION
which is obtained assuming that the standard chain rule holds. Many analytical results
regarding moments of that process with jump noise Zt have been obtained by Zygadło
[Zyg93]. Different variants of the stochastic Verhulst equation with Lévy α-stable noise
have been analyzed in [DS08; Dub12], where rules of ordinary calculus have been also
implicitly assumed. Here we focus on the µ = 1 case and solve it numerically with
different methods for white Gaussian noise and for white shot noise, i.e. generated by the
compensated Poisson process.
Results are presented in Fig. 4, for a process starting from x0 = 0.1. Plots on the
left are shown mainly to present exemplary sample trajectories. As expected, in both
cases the Marcus integral gives the same plots as predicted from the exact solution (13),
therefore it can be used as a reference for the comparison. Quantiles obtained with colored
noise are close to the exact result, with a visible difference due to a relatively long time
constant (τ = 0.2). The Stratonovich integral works fine for Gaussian noise, but leads to
a rescaled process (see axis scales) for shot noise. The Itó integral for both noises gives
completely different results, which is especially visible for shot noise, where the resulting
process has highly skewed PDF with mean and median going with time to 0.
129
C. LÉVY FLIGHTS AND DETAILED BALANCE
C
Lévy flights and detailed balance
Let (Xt )t≥0 be a stochastic process solving the following SDE.
(α)
dXt = −Xt dt + dLt .
(14)
α
∂t f (x, t|x0 ) = ∂|x|
f (x, t|x0 ) + ∂x (xf (x, t|x0 )) ,
(15)
The corresponding FFPE reads
where we condition on the value of the stochastic process at time t0 = 0 w.l.o.g. (the
process is time-homogeneous). The solution of this FFPE is
1
Lα
f (x, t|x0 ) =
γ(t)
x − µ(t)
γ(t)
,
(16)
where Lα (·) is the standard symmetric α-stable distribution
F{Lα }(k) = e−|k|
α
(17)
and where the time-dependent parameters are given by
µ(t) = x0 e−t ,
γ(t) =
1 − e−αt
α
1/α
(18)
.
(19)
The unique stationary state is denoted as fs (x) = limt→∞ fs (x, t|x0 ).
Definition 9. We say that there is a detailed balance iff the following condition is met
∀x,y∈R,t>0 : f (x, t|y)fs (y) = f (y, t|x)fs (x).
(20)
Theorem 3. Detailed balance holds for the solution of (14) iff α = 2.
Proof. The following calculations show that the detailed balance condition (20) is fulfilled
130
C. LÉVY FLIGHTS AND DETAILED BALANCE
for α = 2:
∀x,y∈R,t>0 : ln
=−
f (x, t|y)fs (y)
f (y, t|x)fs (x)
2 
−t )2
exp − (x−ye
exp
− y2
−2t
2(1−e
)

= ln
=
(y−xe−t )2
x2
exp − 2(1−e−2t ) exp − 2

(x − ye−t )2 − (y − xe−t )2 + (1 − e−2t )y 2 − (1 − e−2t )x2
=0.
2(1 − e−2t )
For α < 2, from (20) follows that in particular
g(x, t) ≡
f (x, t|0)fs (0)
=1
f (0, t|x)fs (x)
(21)
and thus
h(x) ≡ lim g(x, t) = 1.
t→0
(22)
Conditions (21) and (22) are necessary but not sufficient for the detailed balance to hold.
For α < 2 and x > 0 we have (note that limt→0 γ(t) = 0)
x
1+α −t 1+α
L
α γ(t)
f (x, t|0)
γ(t)
xe
= lim
lim
= 1.
= lim
t→0 f (0, t|x)
t→0
t→0
x
γ(t)
Lα − µ(t)
γ(t)
(23)
Intuitively this means that for very short times the propagator does not depend on the
deterministic part of the SDE (force felt by the particle). Plugging this into (21) and (22)
we arrive at (x > 0):
h(x) =
Lα (0)
> 0,
Lα (x)
which shows that the detailed balance does not hold for α < 2.
131
(24)
D. FIRST ARRIVAL AND FIRST PASSAGE TIMES
D
First arrival and first passage times
Let (Xt )t≥0 be a real-valued stochastic process starting from X0 = x0 . Let xT ≤ x0 be a
position of an immobile target. We define a first arrival time (FAT):
Tf a = inf(t : Xt = xT ),
(25)
Tf p = inf(t : Xt < xT ).
(26)
and a first passage time (FPT):
Trajectories of the Wiener process are continuous thus Tf p ≥ Tf a .
Moreover,
not cross it later is equal to 0, i.e.
We there-
the probability that at some time t the process hits the origin (target) but do
p(Tf p > t|Tf a = t) = 0.
fore conclude that the FAT and the FPT of the Brownian motion are equal.
20
for α < 2 have discontinu-
15
15
ous trajectories. A discon-
10
10
tinuous process may jump
t
20
t
Lévy α-stable processes
5
over the target without ar-
0
−6
5
−4
−2
0
2
0
−6
4
−4
x
riving at it, i.e. Tf p < Tf a
0
2
4
x
(a)
is possible. As illustrated in
−2
(b)
20
20
15
15
10
10
Fig. 5, a reversed inequalt
t
ity is also possible for a discrete time process. Many aspects of first passage times
of the Brownian motion are
explained in [Red01].
A
5
0
−6
5
−4
−2
x
(c)
0
2
0
−6
−4
−2
x
0
2
(d)
discussion of differences be- Figure 5: An illustration of the FAT (red) and the FPT (green)
tween first passage and ar- on two exemplary sample paths of a 1-dimensional stochastic
rival times of Lévy flights process. The time (vertical axis) and the position (horizontal
axis) are discrete. The first trajectory (a and b) has longer
and some methods of finding FAT=17 than FPT=11, whereas the second trajectory (c and
their PDFs can be found in d) has shorter FAT=1 than FPT=3.
[Che+03b].
132
E. CONTINUOUS VERSUS DISCRETE JUMP DISTRIBUTION
E
Continuous versus discrete jump distribution
In order to check if it is possible to extend results from Paper III to discontinuous jump
length distributions, a simple case of purely discrete jump length distribution is analyzed:
1
1
f (η) = δ(η − 1) + δ(η + 1)
2
2
(27)
As before the probability of resetting is denoted by r. Hereafter, a random walk based
on Eq. (27) is called a lattice random walk. An elementary derivation of the MFPT is
presented here, although one could also adapt the technique used in Paper III.
Let Xn (n ∈ {0, 1, ...}) be a random variable describing position of the walker at
discrete time n (so that a sequence (Xn )∞
n=0 is a sample path of the process), ηn (n ∈
{1, ...}) be jump lengths drawn from distribution (27) and Rn - resetting random variable
- be 1 with probability r and 0 with probability 1 − r. With these definitions the process
of lattice random walk with resetting may be written in a recursive form:
Xn = (1 − Rn )(Xn−1 + ηn ).
(28)
Resetting takes the walker to the origin but for the moment initial position X0 is an
arbitrary constant. Additionally, let T = inf{n : Xn < 0} denote a first passage time, i.e.
time at which the walker crosses the origin for the first time. Clearly T is also a random
variable. In further calculations survival probability will be employed:
Q(t, x) = P (T > t|X0 = x)
(29)
where P (A|B) denotes conditional probability of A given B. Thanks to the Markov
property of the process and its time-homogeneity, the following recurrence relation holds:
Q(t + 1, x) = rQ(t, 0) +
1 − r
Q(t, x − 1) + Q(t, x + 1)
2
133
(30)
E. CONTINUOUS VERSUS DISCRETE JUMP DISTRIBUTION
Proof:
P (T > t|X0 = x) = P (T > t, R1 = 0|X0 = x) + P (T > t, R1 = 1|X0 = x) =
= P (T > t|R1 = 0, X0 = x)P (R1 = 0) + P (T > t|R1 = 1, X0 = x)P (R1 = 1) =
= (1 − r) P (T > t|X1 = x + 1)P (η1 = 1) + P (T > t|X1 = x − 1)P (η1 = −1) +
1 − r
P (T > t − 1|X0 = x + 1) + P (T > t − 1|X1 = x − 1) +
+ rP (T > t|X1 = 0) =
2
1 − r
+ rP (T > t − 1|X0 = 0) = rQ(t − 1, 0) +
Q(t − 1, x − 1) + Q(t − 1, x + 1)
2
Note that mutual independence of R1 , η1 and X0 have been assumed implicitly. In the
last step time is shifted (t → t + 1) and desired Eq. (30) is obtained.
The recurrent relation could be used to derive the full distribution of T . Here, however,
only the MFPT is of interest and so, for the sake of clarity, a summation over t is applied
to both sides of Eq. (30). Since the expected value of T is given by the sum of survival
probabilities,
T(x) := E[T |X0 = x] =
∞
X
Q(t, x),
(31)
t=0
the summation leads to a recurrence relation for function T(x) (not to be confused with
random variable T ):
1 − r
T(x) − 1 = rT(0) +
T(x − 1) + T(x + 1)
2
(32)
where an initial condition Q(0, x) = 1, valid for x ≥ 0, has been accounted for. This
recurrence can be solved with the following ansatz,
T(x) = C0 (r)a(r)x + C1 (r),
(33)
together with the boundary condition T(−1) = 0. The solution reads:
r
1
1
a= −
−1
r̄
r̄2
1 − ax+1
T(x) =
ar
(34)
where r̄ := 1 − r. To meet the assumption that resetting takes the walker back to the
134
E. CONTINUOUS VERSUS DISCRETE JUMP DISTRIBUTION
initial position, x must be set to 0, leading to the final result:
T(0) =
1−a
.
ar
(35)
Now it is possible to compare this result with the universal formula valid for continuous
jump length distributions (3.17):
p
Tlattice
r + r(2 − r)
√
=
> 1.
Tcontinuous
r+ r
(36)
The inequality holds for r ∈ (0, 1). When r = 0 (no resetting) or r = 1 (the walker stays
at the origin) both MFPTs diverge. It is thus evident that the lattice random walk admits
higher values of the MFPT than a random walk based on continuous jump distributions,
when starting from the origin.
In Fig. 6 both MFPTs
are plotted against r (lines),
8
with additional estimates
Discrete jump PDF − simulation
Continuous jump PDF − gaussian mixture
Discrete jump PDF − analytical prediction
Continuous jump PDF − analytical prediction
7.5
from stochastic simulations
(points).
7
For the sake of
6.5
jump length distribution has
been obtained from the lat-
<T>
simulation, the continuous
6
5.5
tice jumps, by simply adding
5
Gaussian noise with a stan-
4.5
dard deviation σ.
In that
4
way a mixture of Gaussians
3.5
is generated.
0
0.1
In the limit
σ → 0 the lattice random
0.2
0.3
0.4
0.5
0.6
0.7
r
Figure 6: A comparison between lattice random walk (red)
walk is recovered. Nonethe- and random walk based on continuous jump length distribuless, for any σ > 0, how- tion. Points and lines represent results of stochastic simulations
and analytical predictions, respectively. Additional crosses mark
ever tiny, the correspond- minima. The number of samples averaged over: M = 106 . The
ing distribution is continu- standard deviation of Gaussians in the mixture: σ = 10−10 .
ous. Therefore, although the
random walk generated by the mixture of Gaussians and the lattice random walk may
look quite alike, in the former case the MFPT is given by the universal formula (3.17),
135
E. CONTINUOUS VERSUS DISCRETE JUMP DISTRIBUTION
whereas in the latter case the MFPT is given by Eq. (34).
For small values of σ the motion of the walker may be decomposed into two distinct
scales: discrete jumps on the lattice and tiny jerks around positions of the lattice due to
Gaussian noise. If σ is assumed to be infinitesimal these two scales do not mix, i.e. jerks
will not bring the walker to a different lattice position. Then one might ask: what is the
difference between this particular continuous jump walker and the lattice random walker,
which leads to different values of the MFPT? The answer is quite simple: the difference
lies in the possibility of crossing the origin when the non-resetting lattice jump to the
origin is mixed with jerks. This also explains why values of the MFPT are consistently
larger in the case of the lattice random walk and leads to the the hypothesis that any
lattice random walk (i.e. with discrete jump length distribution) should bring about
slower passage times than random walks with continuous jump length distribution, when
starting from the origin.
136
F. DISCUSSION OF α ≤ 1 CASE
F
Discussion of α ≤ 1 case
Definition 10. Let Xt (α, r), with α ∈ (0, 2], r ∈ (0, ∞) and t ∈ [0, ∞), denote the
stochastic process defined in Paper IV and called Lévy flights (LFs) with resetting. The
initial condition is tied to the origin, i.e. X0 (α, r) = 0.
Definition 11. Let x ∈ R \ {0} be a fixed position of a target. Then a random variable
T (α, r, x), called a first arrival time (FAT), is defined as:
T (α, r, x) = inf{t : Xt (α, r) = x}.
(37)
Lemma 1. Let r > 0, α > 0, and x ∈ R. Then the following convergence criteria hold:

Z

−ikx
c:0<c<∞
e
dk
=
|k|α + r ∞
R
for x 6= 0 ∨ (x = 0 ∧ α > 1)
.
(38)
for (x = 0 ∧ α ≤ 1)
Proof sketch: The integral may be estimated from above and from below by series of the
form:
 ∞
P −α


k
C


C
for x = 0
k=0
∞
P
.
(−1)k k −α
k=0
(39)
for x 6= 0
The hyperharmonic series is convergent iff α > 1, whereas the alternating series is convergent for any α > 0, which can be checked with Leibniz’s convergence test.
Theorem 4. LFs with resetting cannot reach the point target if the first moment of the
corresponding stable distribution diverges, i.e.:
α ≤ 1 =⇒ ∀r : P (T (α, r, x) < ∞) = 0.
(40)
Proof. In Paper IV propagator of the process Xt (α, r),
P (x < Xt (α, r) < x + h|X0 (α, r) = x0 )
,
h→0
h
W (x, t|x0 ) := lim
(41)
has been calculated in Fourier-Laplace space (which for readability is denoted by different
137
F. DISCUSSION OF α ≤ 1 CASE
arguments only). The result, given by Eq. (11) in Paper IV, reads:
eikx0 + rs
W (k, s|x0 ) =
.
D|k|α + s + r
(42)
Note that for the moment the initial position is arbitrary (X0 (α, r) = x0 ) but the reset
position is still at the origin. Also in Paper IV an algebraic relation between Laplace
transforms of the propagator and PDF (probability density function) of the FAT has
been obtained (cf. Eq. (19) in Paper IV):
W (x, s|0) = pT (s, x)W (x, s|x),
(43)
where pT (t, x) is a PDF of T (α, r, x) for the process with x0 = 0. With these preliminaries
the rest of the proof is straightforward:
lim P (T (α, r, x) < t) =
t→∞
R
= lim RR
s→0
R
Z∞
(43)
W (x, s|0)
=
s→0 W (x, s|x)
dtpT (t, x) = lim pT (s, x) = lim
s→0
0
dke−ikx W (k, s|0)
(42)
dke−ikx W (k, s|x)
R
= lim RR
s→0
R
138
−ikx
e
(s+r)
dk D|k|
α +s+r
−ikx
s+re
dk D|k|
α +s+r

1 for α > 1
(38)
=
0 for α ≤ 1
.
Bibliography
[Åbe13]
J. Åberg. “Truly work-like work extraction via a single-shot analysis.” In: Nature
Communications 4 (2013).
[An+15]
S. An et al. “Experimental test of the quantum Jarzynski equality with a trappedion system.” In: Nature Physics 11.2 (2015), pp. 193–199.
[And54]
E. S. Andersen. “On the fluctuations of sums of random variables II.” In: Mathematica Scandinavica 2.195-223 (1954), p. 3.
[App09]
D. Applebaum. Lévy processes and stochastic calculus. Cambridge university press,
2009.
[Atk+02]
R. Atkinson, C. Rhodes, D. Macdonald, and R. Anderson. “Scale-free dynamics in
the movement patterns of jackals.” In: Oikos 98.1 (2002), pp. 134–140.
[Bac00]
L. Bachelier. Théorie de la spéculation. Gauthier-Villars, 1900.
[Bar+02]
F. Bartumeus, J. Catalan, U. Fulco, M. Lyra, and G. Viswanathan. “Optimizing
the encounter rate in biological interactions: Lévy versus Brownian strategies.” In:
Physical Review Letters 88.9 (2002), p. 097901.
[Bar+05]
F. Bartumeus, M. G. E. da Luz, G. Viswanathan, and J. Catalan. “Animal
search strategies: a quantitative random-walk analysis.” In: Ecology 86.11 (2005),
pp. 3078–3087.
[Bat+14]
T. B. Batalhão et al. “Experimental reconstruction of work distribution and study
of fluctuation relations in a closed quantum system.” In: Physical Review Letters
113.14 (2014), p. 140601.
[Ben07]
S. Benhamou. “How many animals really do the Levy walk?” In: Ecology 88.8
(2007), pp. 1962–1969.
[Bén+05]
O. Bénichou, M. Coppey, M. Moreau, P. Suet, and R. Voituriez. “Optimal search
strategies for hidden targets.” In: Physical Review Letters 94.19 (2005), p. 198101.
[Bén+06]
O. Bénichou, C. Loverdo, M. Moreau, and R. Voituriez. “Two-dimensional intermittent search processes: An alternative to Lévy flight strategies.” In: Physical Review
E 74.2 (2006), p. 020102.
[Bén+11]
O. Bénichou, C. Loverdo, M. Moreau, and R. Voituriez. “Intermittent search strategies.” In: Review of Modern Physics 83.1 (2011), p. 81.
[Bla97]
P. Blackwell. “Random diffusion models for animal movement.” In: Ecological Modelling 100.1 (1997), pp. 87–102.
[Bot11]
D. Bothe. “On the Maxwell-Stefan approach to multicomponent diffusion.” In:
Parabolic Problems. Springer, 2011, pp. 81–93.
[BP03]
J.-P. Bouchaud and M. Potters. Theory of financial risk and derivative pricing:
from statistical physics to risk management. Cambridge university press, 2003.
139
BIBLIOGRAPHY
[BB88]
P. Bovet and S. Benhamou. “Spatial analysis of animals’ movements using a correlated random walk model.” In: Journal of Theoretical Biology 131.4 (1988), pp. 419–
433.
[Boy+06]
D. Boyer et al. “Scale-free foraging by primates emerges from their interaction
with a complex environment.” In: Proceedings of the Royal Society of London B:
Biological Sciences 273.1595 (2006), pp. 1743–1750.
[Bra+15]
F. Brandão, M. Horodecki, N. Ng, J. Oppenheim, and S. Wehner. “The second laws
of quantum thermodynamics.” In: Proceedings of the National Academy of Sciences
112.11 (2015), pp. 3275–3279.
[BHG06]
D. Brockmann, L. Hufnagel, and T. Geisel. “The scaling laws of human travel.” In:
Nature 439.7075 (2006), pp. 462–465.
[Bro28]
R. Brown. “A brief account of microscopical observations made in the months of
June, July and August 1827, on the particles contained in the pollen of plants; and
on the general existence of active molecules in organic and inorganic bodies.” In:
The Philosophical Magazine, or Annals of Chemistry, Mathematics, Astronomy,
Natural History and General Science 4.21 (1828), pp. 161–173.
[BF08]
M. Buchanan and A. Frood. “The mathematical mirror to animal nature.” In:
Nature 453.7196 (2008), pp. 714–716.
[BC04]
A. A. Budini and M. O. Cáceres. “Functional characterization of generalized
Langevin equations.” In: Journal of Physics A: Mathematical and General 37.23
(2004), p. 5959.
[BLR05]
C. Bustamante, J. Liphardt, and F. Ritort. “The nonequilibrium thermodynamics
of small systems.” In: Physics Today 58.7 (2005), p. 43.
[Các99]
M. O. Cáceres. “Lévy noise, Lévy flights, Lévy fluctuations.” In: Journal of Physics
A: Mathematical and General 32.33 (1999), p. 6009.
[CMS76]
J. M. Chambers, C. L. Mallows, and B. Stuck. “A method for simulating stable random variables.” In: Journal of the American Statistical Association 71.354 (1976),
pp. 340–344.
[Che+02]
A. Chechkin, V. Gonchar, J. Klafter, R. Metzler, and L. Tanatarov. “Stationary
states of non-linear oscillators driven by Lévy noise.” In: Chemical Physics 284.1
(2002), pp. 233–251.
[Che+04]
A. V. Chechkin, V. Y. Gonchar, J. Klafter, R. Metzler, and L. V. Tanatarov. “Lévy
flights in a steep potential well.” In: Journal of Statistical Physics 115.5-6 (2004),
pp. 1505–1535.
[Che+03a]
A. V. Chechkin, J. Klafter, V. Y. Gonchar, R. Metzler, and L. V. Tanatarov.
“Bifurcation, bimodality, and finite variance in confined Lévy flights.” In: Physical
Review E 67.1 (2003), p. 010102.
[Che+03b]
A. V. Chechkin, R. Metzler, V. Y. Gonchar, J. Klafter, and L. V. Tanatarov. “First
passage and arrival time densities for Lévy flights and the failure of the method
of images.” In: Journal of Physics A: Mathematical and Theoretical 36.41 (2003),
p. L537.
140
BIBLIOGRAPHY
[CP14]
A. Chechkin and I. Pavlyukevich. “Marcus versus Stratonovich for systems with
jump noise.” In: Journal of Physics A: Mathematical and Theoretical 47.34 (2014),
p. 342001.
[CS15]
C. Christou and A. Schadschneider. “Diffusion with resetting in bounded domains.”
In: Journal of Physics A: Mathematical and Theoretical 48.28 (2015), p. 285003.
[CL98]
S. Ciliberto and C. Laroche. “An experimental test of the Gallavotti-Cohen fluctuation theorem.” In: Le Journal de Physique IV 8.PR6 (1998), Pr6–215.
[Col+05]
D. Collin et al. “Verification of the Crooks fluctuation theorem and recovery of
RNA folding free energies.” In: Nature 437.7056 (2005), pp. 231–234.
[Cro99]
G. E. Crooks. “Entropy production fluctuation theorem and the nonequilibrium
work relation for free energy differences.” In: Physical Review E 60.3 (1999),
p. 2721.
[DHH08]
S. Denisov, W. Horsthemke, and P. Hänggi. “Steady-state Lévy flights in a confined
domain.” In: Physical Review E 77.6 (2008), p. 061112.
[DF93a]
M. Di Paola and G. Falsone. “Itô and Stratonovich integrals for delta-correlated
processes.” In: Probabilistic engineering mechanics 8.3 (1993), pp. 197–208.
[DF93b]
M. Di Paola and G. Falsone. “Stochastic dynamics of nonlinear systems driven
by non-normal delta-correlated processes.” In: Journal of Applied Mechanics 60.1
(1993), pp. 141–148.
[DV75]
M. D. Donsker and S. S. Varadhan. “Asymptotics for the Wiener sausage.” In:
Communications on Pure and Applied Mathematics 28.4 (1975), pp. 525–565.
[DS08]
A. Dubkov and B. Spagnolo. “Verhulst model with Lévy white noise excitation.”
In: The European Physical Journal B 65.3 (2008), pp. 361–367.
[Dub12]
A. Dubkov. “Transient Dynamics of Verhulst Model with Fluctuating Saturation
Parameter.” In: Acta Physica Polonica-Series B Elementary Particle Physics 43.5
(2012), p. 935.
[DSU08]
A. A. Dubkov, B. Spagnolo, and V. V. Uchaikin. “Lévy flight superdiffusion: an
introduction.” In: International Journal of Bifurcation and Chaos 18.09 (2008),
pp. 2649–2672.
[DGS07]
B. Dybiec, E. Gudowska-Nowak, and I. Sokolov. “Stationary states in Langevin
dynamics under asymmetric Lévy noises.” In: Physical Review E 76.4 (2007),
p. 041122.
[DGH06]
B. Dybiec, E. Gudowska-Nowak, and P. Hänggi. “Lévy-Brownian motion on finite intervals: Mean first passage time analysis.” In: Physical Review E 73 (2006),
p. 046104.
[DPG12]
B. Dybiec, J. M. Parrondo, and E. Gudowska-Nowak. “Fluctuation-dissipation relations under Lévy noises.” In: EPL (Europhysics Letters) 98.5 (2012), p. 50006.
[DS15]
B. Dybiec and K. Szczepaniec. “Escape from hypercube driven by multi-variate
α-stable noises: role of independence.” In: The European Physical Journal B 88.7
(2015), pp. 1–8.
141
BIBLIOGRAPHY
[Edw+07]
A. M. Edwards et al. “Revisiting Lévy flight search patterns of wandering albatrosses, bumblebees and deer.” In: Nature 449.7165 (2007), pp. 1044–1048.
[Ein05]
A. Einstein. “Über die von der molekularkinetischen Theorie der Wärme geforderte
Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen.” In: Annalen der
Physik 322.8 (1905), pp. 549–560. issn: 1521-3889.
[ECM93]
D. J. Evans, E. Cohen, and G. Morriss. “Probability of second law violations in
shearing steady states.” In: Physical Review Letters 71.15 (1993), p. 2401.
[ES94]
D. J. Evans and D. J. Searles. “Equilibrium microstates which generate second law
violating steady states.” In: Physical Review E 50.2 (1994), p. 1645.
[ES02]
D. J. Evans and D. J. Searles. “The fluctuation theorem.” In: Advances in Physics
51.7 (2002), pp. 1529–1585.
[EM11]
M. R. Evans and S. N. Majumdar. “Diffusion with stochastic resetting.” In: Physical
Review Letters 106.16 (2011), p. 160601.
[EMM13]
M. R. Evans, S. N. Majumdar, and K. Mallick. “Optimal diffusive search: nonequilibrium resetting versus equilibrium dynamics.” In: Journal of Physics A: Mathematical and Theoretical 46.18 (2013), p. 185001.
[Fel71]
W. Feller. “An introduction to probability and its applications, Vol. II.” In: Wiley,
New York (1971).
[Fic55]
A. Fick. “On liquid diffusion.” In: The London, Edinburgh, and Dublin Philosophical
Magazine and Journal of Science 10.63 (1855), pp. 30–39.
[FW12]
J. Findlay and R. Walker. “Human saccadic eye movements.” In: Scholarpedia 7.7
(2012), p. 5095.
[Fog94]
H. C. Fogedby. “Langevin equations for continuous time Lévy flights.” In: Physical
Review E 50.2 (1994), p. 1657.
[Fou78]
J. B. J. baron Fourier. The analytical theory of heat. The University Press, 1878.
[FF95]
U. Frisch and H. Frisch. “Universality of escape from a half-space for symmetrical
random walks.” In: Lévy Flights and Related Topics in Physics. Springer, 1995,
pp. 262–268.
[GC95a]
G. Gallavotti and E. Cohen. “Dynamical ensembles in nonequilibrium statistical
mechanics.” In: Physical Review Letters 74.14 (1995), p. 2694.
[GC95b]
G. Gallavotti and E. Cohen. “Dynamical ensembles in stationary states.” In: Journal of Statistical Physics 80.5-6 (1995), pp. 931–970.
[GEW15]
R. Gallego, J. Eisert, and H. Wilming. “Defining work from operational principles.”
In: arXiv preprint arXiv:1504.05056 (2015).
[Gar+85]
C. W. Gardiner et al. Handbook of stochastic methods. Vol. 4. Springer Berlin, 1985.
[GK54]
B. Gnedenko and A. Kolmogorov. “Limit distributions for sums of independent
random variables.” In: American Journal of Mathematics 105 (1954), pp. 28–35.
[Gre54]
M. S. Green. “Markoff Random Processes and the Statistical Mechanics of TimeDependent Phenomena. II. Irreversible Processes in Fluids.” In: The Journal of
Chemical Physics 22.3 (1954), pp. 398–413.
142
BIBLIOGRAPHY
[Gut01]
M. Gutowski. “L\’evy flights as an underlying mechanism for global optimization
algorithms.” In: arXiv preprint math-ph/0106003 (2001).
[HS01]
T. Hatano and S.-i. Sasa. “Steady-state thermodynamics of Langevin systems.” In:
Physical Review Letters 86.16 (2001), p. 3463.
[Her11]
R. Herrmann. Fractional calculus: An introduction for physicists. World Scientific,
2011.
[HL84]
W. W. Horsthemke and R. Lefever. Noise-induced transitions : theory and applications in physics, chemistry, and biology. Springer series in synergetics. Berlin, New
York: Springer-Verlag, 1984. isbn: 0-387-11359-2.
[Imp+07]
A. Imparato, L. Peliti, G. Pesce, G. Rusciano, and A. Sasso. “Work and heat
probability distribution of an optically driven Brownian particle: Theory and experiments.” In: Physical Review E 76.5 (2007), p. 050101.
[IM74]
K. Itô and H. P. McKean Jr. Diffusion Processes and their Sample Paths. Springer
Berlin Heidelberg, 1974.
[JW93]
A. Janicki and A. Weron. Simulation and chaotic behavior of alpha-stable stochastic
processes. Vol. 178. CRC Press, 1993.
[Jar97]
C. Jarzynski. “Nonequilibrium equality for free energy differences.” In: Physical
Review Letters 78.14 (1997), p. 2690.
[Jar11]
C. Jarzynski. “Equalities and inequalities: irreversibility and the second law of thermodynamics at the nanoscale.” In: Annual Review of Condensed Matter Physics
2.1 (2011), pp. 329–351.
[Jeo+11]
J.-H. Jeon et al. “In vivo anomalous diffusion and weak ergodicity breaking of lipid
granules.” In: Physical Review Letters 106.4 (2011), p. 048103.
[JMF99]
S. Jespersen, R. Metzler, and H. C. Fogedby. “Lévy flights in external force fields:
Langevin and fractional Fokker-Planck equations and their solutions.” In: Physical
Review E 59.3 (1999), p. 2736.
[KSH12]
K. Kanazawa, T. Sagawa, and H. Hayakawa. “Stochastic energetics for non-gaussian
processes.” In: Physical Review Letters 108.21 (2012), p. 210601.
[KT98]
S. Karlin and H. M. Taylor. An introduction to stochastic modeling; 3rd ed. New
York, NY: Harcourt, 1998.
[KSZ96]
J. Klafter, M. F. Shlesinger, and G. Zumofen. “Beyond brownian motion.” In:
Physics today 49.2 (1996), pp. 33–39.
[KS11]
J. Klafter and I. M. Sokolov. First steps in random walks: from tools to applications.
Oxford University Press, 2011.
[KRS08]
R. Klages, G. Radons, and I. M. Sokolov. Anomalous transport: foundations and
applications. John Wiley & Sons, 2008.
[Kop95]
I. Koponen. “Analytic approach to the problem of convergence of truncated Lévy
flights towards the Gaussian stochastic process.” In: Physical Review E 52.1 (1995),
p. 1197.
[Kre10]
G. M. Kremer. An introduction to the Boltzmann equation and transport processes
in gases. Springer Science & Business Media, 2010.
143
BIBLIOGRAPHY
[Kub66]
R. Kubo. “The fluctuation-dissipation theorem.” In: Reports on progress in physics
29.1 (1966), p. 255.
[Kub57]
R. Kubo. “Statistical-mechanical theory of irreversible processes. I. General theory
and simple applications to magnetic and conduction problems.” In: Journal of the
Physical Society of Japan 12.6 (1957), pp. 570–586.
[KTH12]
R. Kubo, M. Toda, and N. Hashitsume. Statistical physics II: nonequilibrium statistical mechanics. Vol. 31. Springer Science & Business Media, 2012.
[Kur98]
J. Kurchan. “Fluctuation theorem for stochastic dynamics.” In: Journal of Physics
A: Mathematical and General 31.16 (1998), p. 3719.
[KBD99]
D. Kusnezov, A. Bulgac, and G. Do Dang. “Quantum levy processes and fractional
kinetics.” In: Physical Review Letters 82.6 (1999), p. 1136.
[Lan08]
P. Langevin. “Sur la théorie du mouvement brownien.” In: CR Acad. Sci. Paris
146.530-533 (1908), p. 530.
[Las00]
N. Laskin. “Fractional quantum mechanics and Lévy path integrals.” In: Physics
Letters A 268.4 (2000), pp. 298–305.
[Lév25]
P. Lévy. Calcul des probabilités. Vol. 9. Gauthier-Villars Paris, 1925.
[LMW13]
T. Li, B. Min, and Z. Wang. “Marcus canonical integral for non-Gaussian processes and its computation: Pathwise simulation and tau-leaping algorithm.” In:
The Journal of chemical physics 138.10 (2013), p. 104118.
[LAM05]
M. A. Lomholt, T. Ambjörnsson, and R. Metzler. “Optimal target search on a fastfolding polymer chain with volume exchange.” In: Physical Review Letters 95.26
(2005), p. 260603.
[Lom+08]
M. A. Lomholt, K. Tal, R. Metzler, and K. Joseph. “Lévy strategies in intermittent
search processes are advantageous.” In: Proceedings of the National Academy of
Sciences of the United States of America 105.32 (2008), pp. 11055–11059.
[MWW07]
M. Magdziarz, A. Weron, and K. Weron. “Fractional Fokker-Planck dynamics:
Stochastic representation and computer simulation.” In: Physical Review E 75.1
(2007), p. 016708.
[Mai97]
F. Mainardi. “Fractional calculus.” In: Fractals and fractional calculus in continuum
mechanics. Springer Vienna, 1997, pp. 291–348.
[Maj10]
S. N. Majumdar. “Universal first-passage properties of discrete-time random walks
and Lévy flights on a line: Statistics of the global maximum and records.” In:
Physica A: Statistical Mechanics and its Applications 389.20 (2010), pp. 4299–
4316.
[Man60]
B. Mandelbrot. “The Pareto-Levy law and the distribution of income.” In: International Economic Review 1.2 (1960), pp. 79–106.
[Man83]
B. B. Mandelbrot. The fractal geometry of nature. Vol. 173. Macmillan, 1983.
[MV68]
B. B. Mandelbrot and J. W. Van Ness. “Fractional Brownian motions, fractional
noises and applications.” In: SIAM review 10.4 (1968), pp. 422–437.
144
BIBLIOGRAPHY
[MM11]
R. Mannella and P. V. E. McClintock. “Comment on "influence of noise on
force measurements".” In: Physical Review Letters 107.7 (2011), p. 170602. issn:
00319007.
[MS94]
R. N. Mantegna and H. E. Stanley. “Stochastic process with ultraslow convergence
to a Gaussian: the truncated Lévy flight.” In: Physical Review Letters 73.22 (1994),
p. 2946.
[Man94]
R. N. Mantegna. “Fast, accurate algorithm for numerical simulation of Lévy stable
stochastic processes.” In: Physical Review E 49 (5 May 1994), pp. 4677–4683.
[M+95]
R. N. Mantegna, H. E. Stanley, et al. “Scaling behaviour in the dynamics of an
economic index.” In: Nature 376.6535 (1995), pp. 46–49.
[Mar+08]
U. M. B. Marconi, A. Puglisi, L. Rondoni, and A. Vulpiani. “Fluctuation–
dissipation: response theory in statistical physics.” In: Physics reports 461.4 (2008),
pp. 111–195.
[Mar78]
S. I. Marcus. “Modeling and analysis of stochastic differential equations driven
by point processes.” In: Information Theory, IEEE Transactions on 24.2 (1978),
pp. 164–172.
[Mar81]
S. I. Marcus. “Modeling and approximation of stochastic differential equations
driven by semimartingales.” In: Stochastics: An International Journal of Probability
and Stochastic Processes 4.3 (1981), pp. 223–245.
[Max67]
J. C. Maxwell. “On the dynamical theory of gases.” In: Philosophical transactions
of the Royal Society of London 157 (1867), pp. 49–88.
[MK00]
R. Metzler and J. Klafter. “The random walk’s guide to anomalous diffusion: a
fractional dynamics approach.” In: Physics Reports 339.1 (2000), pp. 1–77.
[MK04]
R. Metzler and J. Klafter. “The restaurant at the end of the random walk: recent
developments in the description of anomalous transport by fractional dynamics.”
In: Journal of Physics A: Mathematical and General 37.31 (2004), R161.
[Mor65]
H. Mori. “Transport, collective motion, and Brownian motion.” In: Progress of
theoretical physics 33.3 (1965), pp. 423–455.
[ME07]
G. P. Morriss and D. J. Evans. Statistical Mechanics of Nonequilbrium Liquids.
ANU Press, 2007.
[Nee93]
T. Needham. “A visual explanation of Jensen’s inequality.” In: The American mathematical monthly 100.8 (1993), pp. 768–771.
[Obu03]
A. Obuchowicz. “Multidimensional mutations in evolutionary algorithms based
on real-valued representation.” In: International Journal of Systems Science 34.7
(2003), pp. 469–483.
[Ons31a]
L. Onsager. “Reciprocal relations in irreversible processes. I.” In: Physical Review
37.4 (1931), p. 405.
[Ons31b]
L. Onsager. “Reciprocal relations in irreversible processes. II.” In: Physical Review
38.12 (1931), p. 2265.
[PCM14]
V. V. Palyulin, A. V. Chechkin, and R. Metzler. “Lévy flights do not always optimize random blind search for sparse targets.” In: Proceedings of the National
Academy of Sciences 111.8 (2014), pp. 2931–2936.
145
BIBLIOGRAPHY
[Pav07]
I. Pavlyukevich. “Lévy flights, non-local search and simulated annealing.” In: Journal of Computational Physics 226.2 (2007), pp. 1830–1844.
[Pes+13]
G. Pesce, A. McDaniel, S. Hottovy, J. Wehr, and G. Volpe. “Stratonovich-to-Itô
transition in noisy systems with multiplicative feedback.” In: Nature Communications 4 (2013), p. 2733. issn: 2041-1723.
[PSW05]
A. Piryatinska, A. Saichev, and W. Woyczynski. “Models of anomalous diffusion:
the subdiffusive case.” In: Physica A: Statistical Mechanics and its Applications
349.3 (2005), pp. 375–420.
[Pol52]
F. Pollaczek. “Fonctions caractéristiques de certaines répartitions définies au moyen
de la notion d’ordre. Application à la théorie des attentes.” In: Comptes rendus
hebdomadaires des séances de l’Académie des Sciences 234.24 (1952), pp. 2334–
2336.
[PW90]
G. Poppe and C. Wijers. “Algorithm 680: evaluation of the complex error function.”
In: ACM Transactions on Mathematical Software (TOMS) 16.1 (1990), p. 47.
[Pro04]
P. E. Protter. Stochastic integration and differential equations. Applications of
mathematics. Berlin, Heidelberg, New York: Springer, 2004. isbn: 3-540-00313-4.
[Ram+04]
G. Ramos-Fernández et al. “Lévy walk patterns in the foraging movements of spider
monkeys (Ateles geoffroyi).” In: Behavioral Ecology and Sociobiology 55.3 (2004),
pp. 223–230.
[Rap+03]
E. Raposo et al. “Dynamical robustness of Lévy search strategies.” In: Physical
Review Letters 91.24 (2003), p. 240601.
[Ray97]
K. Rayner. “Eye movements in reading and information processing: 20 years of
research.” In: Psychological Bulletin 124.3 (1997), pp. 372–422. issn: 0033-2909.
doi: 10.1037/0033-2909.124.3.372.
[Red01]
S. Redner. A guide to first-passage processes. Cambridge University Press, 2001.
[Rey05]
A. Reynolds. “Scale-free movement patterns arising from olfactory-driven foraging.”
In: Physical Review E 72.4 (2005), p. 041928.
[Rey+07]
A. M. Reynolds et al. “Displaced honey bees perform optimal scale-free search
flights.” In: Ecology 88.8 (2007), pp. 1955–1961.
[RF07]
A. M. Reynolds and M. A. Frye. “Free-flight odor tracking in Drosophila is consistent with an optimal intermittent scale-free search.” In: PloS one 2.4 (2007),
e354–e354.
[Rhe+11]
I. Rhee et al. “On the levy-walk nature of human mobility.” In: IEEE/ACM transactions on networking (TON) 19.3 (2011), pp. 630–643.
[Ros06]
J. S. Rosenthal. A first look at rigorous probability theory. World Scientific, 2006.
[SU10]
T. Sagawa and M. Ueda. “Generalized Jarzynski equality under nonequilibrium
feedback control.” In: Physical Review Letters 104.9 (2010), p. 090602.
[ST94]
G. Samoradnitsky and M. S. Taqqu. Stable non-Gaussian random processes:
stochastic models with infinite variance. Vol. 1. CRC Press, 1994.
146
BIBLIOGRAPHY
[Sch+05]
S. Schuler, T. Speck, C. Tietz, J. Wrachtrup, and U. Seifert. “Experimental test of
the fluctuation theorem for a driven two-level system with time-dependent rates.”
In: Physical Review Letters 94.18 (2005), p. 180602.
[Sch+13]
H. G. Schuster, R. Klages, W. Just, and C. Jarzynski. Nonequilibrium Statistical
Physics of Small Systems: Fluctuation Relations and Beyond. John Wiley & Sons,
2013.
[Sei05]
U. Seifert. “Entropy production along a stochastic trajectory and an integral fluctuation theorem.” In: Physical Review Letters 95.4 (2005), p. 040602.
[Sei12]
U. Seifert. “Stochastic thermodynamics, fluctuation theorems and molecular machines.” In: Reports on Progress in Physics 75.12 (2012), p. 126001.
[Sek10]
K. Sekimoto. Stochastic energetics. Vol. 799. Springer, 2010.
[SWK87]
M. Shlesinger, B. West, and J. Klafter. “Lévy dynamics of enhanced diffusion:
Application to turbulence.” In: Physical Review Letters 58.11 (1987), p. 1100.
[SK86]
M. F. Shlesinger and J. Klafter. “Lévy walks versus Lévy flights.” In: On growth
and form. Springer, 1986, pp. 279–283.
[Sim+08]
D. W. Sims et al. “Scaling laws of marine predator search behaviour.” In: Nature
451.7182 (2008), pp. 1098–1102.
[Sim+14]
D. W. Sims et al. “Hierarchical random walks in trace fossils and the origin of
optimal search behavior.” In: Proceedings of the National Academy of Sciences of
the United States of America 111.30 (2014), pp. 11073–11078.
[Smo06]
M. Smoluchowski. “Zur kinetischen Theorie der Brownschen Molekularbewegung
und der Suspensionen.” In: Annalen der physik 326.14 (1906), pp. 756–780.
[SED11]
I. M. Sokolov, W. Ebeling, and B. Dybiec. “Harmonic oscillator under Lévy noise:
Unexpected properties in the phase space.” In: Physical Review E 83.4 (2011),
p. 041118.
[Spi56]
F. Spitzer. “A combinatorial lemma and its application to probability theory.” In:
Transactions of the American Mathematical Society 82.2 (1956), pp. 323–339.
[Sro09]
T. Srokowski. “Multiplicative Lévy processes: Itô versus Stratonovich interpretation.” In: Physical Review E 80.5 (2009), p. 051113.
[Sro10]
T. Srokowski. “Nonlinear stochastic equations with multiplicative Lévy noise.” In:
Physical Review E 81.5 (2010), p. 051110.
[Ste71]
J. Stefan. “Über das Gleichgewicht und die Bewegung, insbesondere die Diffusion
von Gasgemengen.” In: Sitzber. Akad. Wiss. Wien 63 (1871), pp. 63–124.
[Str94]
S. H. Strogatz. Nonlinear dynamics and chaos : with applications to physics, biology, chemistry, and engineering. Studies in nonlinearity. Cambridge (Mass.): Westview Press, 1994. isbn: 978-0-7382-0453-6.
[SD14a]
K. Szczepaniec and B. Dybiec. “Resonant activation in 2D and 3D systems driven
by multi-variate Lévy noise.” In: Journal of Statistical Mechanics: Theory and Experiment 2014.9 (2014), P09022.
[SD14b]
K. Szczepaniec and B. Dybiec. “Stationary states in two-dimensional systems
driven by bivariate Lévy noises.” In: Physical Review E 90.3 (2014), p. 032128.
147
BIBLIOGRAPHY
[TFP06]
H. Tong, C. Faloutsos, and J.-Y. Pan. “Fast Random Walk with Restart and Its Applications.” In: Proceedings of the Sixth International Conference on Data Mining.
IEEE Computer Society. 2006, pp. 613–622.
[TC07]
H. Touchette and E. Cohen. “Fluctuation relation for a Lévy particle.” In: Physical
Review E 76.2 (2007), p. 020101.
[Tou09]
H. Touchette. “The large deviation approach to statistical mechanics.” In: Physics
Reports 478.1 (2009), pp. 1–69.
[Vah+13]
M. Vahabi, J. H. Schulz, B. Shokri, and R. Metzler. “Area coverage of radial Lévy
flights with periodic boundary conditions.” In: Physical Review E 87.4 (2013),
p. 042136.
[Van+13]
C. Van den Broeck et al. “Stochastic thermodynamics: a brief introduction.” In:
Physics of Complex Colloids 184 (2013), pp. 155–193.
[Van92]
N. G. Van Kampen. Stochastic processes in physics and chemistry. Vol. 1. Elsevier,
1992.
[VC03b]
R. Van Zon and E. Cohen. “Extension of the fluctuation theorem.” In: Physical
Review Letters 91.11 (2003), p. 110601.
[VC03a]
R. Van Zon and E. Cohen. “Stationary and transient work-fluctuation theorems
for a dragged Brownian particle.” In: Physical Review E 67.4 (2003), p. 046102.
[VC04]
R. Van Zon and E. Cohen. “Extended heat-fluctuation theorems for a system with
deterministic and stochastic forces.” In: Physical Review E 69.5 (2004), p. 056121.
[VR08]
J. M. Vilar and J. M. Rubi. “Failure of the work-Hamiltonian connection for freeenergy calculations.” In: Physical Review Letters 100.2 (2008), p. 020601.
[Vis+96]
G. M. Viswanathan et al. “Lévy flight search patterns of wandering albatrosses.”
In: Nature 381.6581 (1996), pp. 413–415.
[Vis+99]
G. Viswanathan et al. “Optimizing the success of random searches.” In: Nature
401.6756 (1999), pp. 911–914.
[Vol+10]
G. Volpe, L. Helden, T. Brettschneider, J. Wehr, and C. Bechinger. “Influence of
noise on force measurements.” In: Physical Review Letters 104.17 (2010), pp. 1–4.
issn: 00319007.
[Vol+11]
G. Volpe, L. Helden, T. Brettschneider, J. Wehr, and C. Bechinger. “Volpe et al.
Reply:” in: Physical Review Letters 107.7 (2011), p. 078902. issn: 0031-9007.
[Wan+13]
G. Wang et al. “Lévy-flight krill herd algorithm.” In: Mathematical Problems in
Engineering 2013 (2013).
[Wan+02]
G. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J. Evans. “Experimental
demonstration of violations of the second law of thermodynamics for small systems
and short time scales.” In: Physical Review Letters 89.5 (2002), p. 050601.
[Wan+05]
G. Wang et al. “Experimental study of the fluctuation theorem in a nonequilibrium
steady state.” In: Physical Review E 71.4 (2005), p. 046142.
[Wei+04]
M. Weiss, M. Elsner, F. Kartberg, and T. Nilsson. “Anomalous subdiffusion is
a measure for cytoplasmic crowding in living cells.” In: Biophysical journal 87.5
(2004), pp. 3518–3524.
148
BIBLIOGRAPHY
[Wer96]
R. Weron. “On the Chambers-Mallows-Stuck method for simulating skewed stable
random variables.” In: Statistics & Probability Letters 28.2 (1996), pp. 165–171.
[WZ65]
E. Wong and M. Zakai. “On the convergence of ordinary integrals to stochastic
integrals.” In: The Annals of Mathematical Statistics (1965), pp. 1560–1564.
[Yan10]
X.-S. Yang. “Firefly algorithm, Levy flights and global optimization.” In: Research
and development in intelligent systems XXVI. Springer, 2010, pp. 209–218.
[YD09]
X.-S. Yang and S. Deb. “Cuckoo search via Lévy flights.” In: Nature & Biologically
Inspired Computing, 2009. NaBIC 2009. World Congress on. IEEE. 2009, pp. 210–
214.
[YD10]
X.-S. Yang and S. Deb. “Eagle strategy using Lévy walk and firefly algorithms for
stochastic optimization.” In: Nature Inspired Cooperative Strategies for Optimization (NICSO 2010). Springer, 2010, pp. 101–111.
[YH13]
X.-S. Yang and X. He. “Bat algorithm: literature review and applications.” In:
International Journal of Bio-Inspired Computation 5.3 (2013), pp. 141–149.
[Yan+00]
V. Yanovsky, A. Chechkin, D. Schertzer, and A. Tur. “Lévy anomalous diffusion
and fractional Fokker–Planck equation.” In: Physica A: Statistical Mechanics and
its Applications 282.1 (2000), pp. 13–34.
[Zas94]
G. Zaslavsky. “Fractional kinetic equation for Hamiltonian chaos.” In: Physica D:
Nonlinear Phenomena 76.1 (1994), pp. 110–122.
[Zha+15]
K. Zhao et al. “Optimal Lévy-flight foraging in a finite landscape.” In: Journal of
The Royal Society Interface 12.104 (2015), p. 20141158.
[Zwa60]
R. Zwanzig. “Ensemble method in the theory of irreversibility.” In: The Journal of
Chemical Physics 33.5 (1960), pp. 1338–1341.
[Zwa61]
R. Zwanzig. “Memory effects in irreversible thermodynamics.” In: Physical Review
124.4 (1961), p. 983.
[Zwa65]
R. Zwanzig. “Time-correlation functions and transport coefficients in statistical
mechanics.” In: Annual Review of Physical Chemistry 16.1 (1965), pp. 67–102.
[Zyg93]
R. Zygadło. “Verhulst-type kinetics driven by white shot noise: Exact solution by
direct averaging.” In: Physical Review E 47.1 (1993), p. 106.
149