Download Nano Mechanics and Materials: Theory, Multiscale Methods

Nano Mechanics and Materials:
Theory, Multiscale Methods and Applications
by
Wing Kam Liu, Eduard G. Karpov, Harold S. Park
4. Methods of Thermodynamics and Statistical Mechanics
Definition and Features the Thermodynamic Method
Thermodynamics is a macroscopic, phenomenological theory of heat.
Basic features of the thermodynamic method:
• Multi-particle physical systems is described by means of a small number of
macroscopically measurable parameters, the thermodynamic parameters: V, P, T, S (volume,
pressure, temperature, entropy), and others.
Note: macroscopic objects contain ~ 1023…1024 atoms (Avogadro’s number ~ 6x1023mol–1).
• The connections between thermodynamic parameters are found from the general laws of
thermodynamics.
• The laws of thermodynamics are regarded as experimental facts.
Therefore, thermodynamics is a phenomenological theory.
• Thermodynamics is in fact a theory of equilibrium states, i.e. the states with timeindependent (relaxed) V, P, T and S. Term “dynamics” is understood only in the sense “how
one thermodynamic parameters varies with a change of another parameter in two successive
equilibrium states of the system”.
Classification of Thermodynamic Parameters
Internal and external parameters:
• External parameters can be prescribed by means of external influences on the system by
specifying external boundaries and fields.
• Internal parameters are determined by the state of the system itself for given values of the
external parameters.
Note: the same parameter may appear as external in one system, and as internal in another
system.
Intensive and extensive parameters:
• Intensive parameters are independent of the number of particles in the system, and they serve
as general characteristics of the thermal atomic motion (temperature, chemical potential).
• Extensive parameters are proportional to the total mass or the number of particles in the
system (internal energy, entropy).
Note: this classification is invariant with respect to the choice of a system.
Internal and External Parameters: Examples
A same parameter may appear both as external and internal in various systems:
System A
T  Const
System B
M
P
V = Const
External parameter: V
Internal parameter: P
P = Const
V
External parameter: P, P = Mg/A
Internal parameter: V, V = Ah
State Vector and State Equation
Application of the thermodynamic method implies that the system if found in the state of
thermodynamic equilibrium, denoted X, which is defined by time-invariant state parameters, such
as volume, temperature and pressure:
X  (V , T , P)
The parameters (V,T,P) are macroscopically measurable.
One or two of them may be replaced by non-measurable
parameters, such internal energy or entropy.
Note that only the mean quantity of a state parameter A
is time-invariant, see the plot.
A mathematical relationship that involve a complete set of measurable parameters (V,T,P) is
called the thermodynamic state equation
f (V , T , P,  )  0
Here, ξ is the vector of system parameters
Analysis of the State Equation: System Parameters
The knowledge of an equation of state allows evaluation of a group of the microscopic system
parameters, such as the compressibility, expansion and pressure coefficients:
f (V , T , P,  )  0 
1) Isothermal compressibility coefficient
T  
2) Isobaric thermal expansion coefficient
P 
3) Isochoric pressure coefficient
1  V 


V  P T
1  V 


V  T  P
1  P 
V   
P  T V
Examples of the State Equation
Standard forms of the state equations are readily available for ideal gases, real (Van der Waals)
gases, homogeneous liquid and homogeneous isotropic solids.
Ideal gas
PV  NkBT
Van der Waals gas
(pair wise interaction V of gas molecules is taken into account)

2 a 
P

n
V  nb   Nk BT

2 
V 

n

N
NA
2
a  2 N A2  W2 (r )r 2 dr , b  N A d 3
3
d
d – effective diameter of
the molecules
W – pairwise potential
Homogeneous isotropic liquid or solid
V  V0 1   (T  T0 )   ( P  P0 ) 
1  V 
1  V 
 
,   


V0  T  P  P0
V0  P T T0
V0  V (T0 , P0 ) 
arbitrary initial state
The Postulate on Existence of Temperature
The temperature is introduced as a parameter, which:
1) serves as an intrinsic characteristic of any equilibrium system (similar to V and P)
2) determines thermodynamic equilibrium between two systems in thermal contact
Thus, it is postulated that:
If two adiabatically isolated systems in equilibrium are brought into thermal
contact with each other, their states of equilibrium will not be altered and the
total system will be in equilibrium, only if initial systems have the same
temperature.
(Also known as the zeroth law of thermodynamics)
Any state of thermodynamic equilibrium of an arbitrary system in entirely
determined by the set of external parameters and temperature.
Consequently,
All internal parameters of an equilibrium system are functions of the
external parameters and temperature.
The First Law of Thermodynamics
The first law of thermodynamics is essentially a form of the energy conservation law,
written in relation to thermodynamic systems:
 Q  dU   W
An amount of heat absorbed by the system is equal to the summary change of its
internal energy and the work done by the system over external bodies.
Note: The internal energy U is defined solely by the state of system, while the external
thermal energy Q and the mechanical work W may depend both on the internal state of
the system and other factors.
The Second Law of Thermodynamics
The second law of thermodynamics specifies the direction of thermodynamic
processes.
The simplest form of the second law is given by Clausius’ postulate:
Heat cannot flow spontaneously from a colder to a hotter system.
This is equivalent to the following (Kelvin’s postulate): It is impossible to devise an
engine (a perpetuum mobile of the second kind) which, working in a cycle, would
produce no other effect than the transformation of heat extracted from a reservoir
completely into work.
Entropy
The most important consequence of the second law of thermodynamics is that it asserts
the existence of a new function of state, namely the entropy, S.
The concept of entropy plays a crucial role in statistical mechanics.
For quasistatic processes, entropy is an extensive state function, which is defined by the
relation
 Q  TdS
Based on the first law of thermodynamics we obtain the fundamental differential
equation
dU  TdS  W  TdS  PdV
Alternative form of the second law of thermodynamics: All spontaneous processes in
adiabatically isolated systems, δQ = 0, occur at a constant or growing entropy:
S  0
The Third Law of Thermodynamics
The second law and the original definition of entropy does not specify the absolute
value of entropy, whose value is provided up to a constant value. The third law
claims that
All thermodynamic processes at T = 0 occur without a change of the entropy.
This allows establishing an absolute scale for measuring the entropy; so that
T  0: S  0
Also,
T  0 : C   P  V  0
Thermodynamic Potentials
The method of thermodynamic potentials is a powerful tool of thermodynamics.
Thermodynamic potentials are functions that uniquely describe the state of the system.
The relevant thermodynamic parameters are found as derivatives of the potentials.
The simplest thermodynamic potential is the internal energy, given by the first law,
U ( S ,V ) :
dU   Q   W  TdS  PdV
Other thermodynamic potentials for systems with constant number of particles:
free energy
Gibbs potential
enthalpy
F (T ,V )  U  ST :
dF   SdT  PdV
G(T , P)  F  PV :
I ( S , P)  U  PV :
dG   SdT  VdP
dI  TdS  VdP
Exercise 3-1 (can be done in class): derive the differentials of the free energy, Gibbs potential and enthalpy, by
utilizing the definitions of these potentials and the first law of thermodynamics.
4.1 Basic Results of the Thermodynamic Method
Differentiation of the thermodynamic potentials gives the thermodynamic parameters:
 U 
T 


S

V
 G 
S  

 T  P
 U 
P  


V

S
 G 
V 

 P T
 I 
T  
 S  P
 F 
S  

 T V
 I 
V 


P

S
 F 
P  

 V T
Also, system parameters can be evaluated using their relationship with the above state
parameters (see reading assignment [1], Section 4.1.5)
Hamiltonian Mechanics
Equations of motion:
H
q 
,
p
H
p  
q
Hamiltonian of the system and the generalized momentum:
H ( p, q, t )   p q  L,

L
p 
q
Cartesian coordinates:
p2
H (p, r, t )  T  U 
 U (r, t ), p 2  m2 r 2  m2 ( x 2  y 2  z 2 )
2m
Micromodel vs. Macromodel
In classical (deterministic) mechanics, the state of the system is completely
described by the phase vector Q. Such a state is called a microscopic state, and
the corresponding model of matter is called a micromodel.
The microstate is completely defined by specifying values of all canonical
variables, the components of the phase vector,
Q  ( p1 , p2 ,..., ps , q1 , q2 ,..., qs )
This approach is not tractable in modeling macroscopic objects.
Thermodynamics provides a macromodel; the state of the a system is
determined by a very limited number of thermodynamic parameters, which are
sufficient for macroscopic characterization of the system.
The prescription of these parameters, measured in a macroscopic experiment,
determines the macroscopic state the system.
A key point is that a single macroscopic state of the system corresponds to a
great number of different microscopic states.
4.2 Statistics of Multiparticle Systems in Thermodynamic Equilibrium
The macroscopic thermodynamic parameters, X = (V,P,T,…), are
macroscopically observable quantities that are, in principle, functions of the
canonical variables, i.e.
fi  fi ( p1 , p2 ,..., ps , q1 , q2 ,..., qs ), i  1, 2,..., n, n
s
( f1 , f 2 ,..., f n )  (V , P, T ,...)  X

X  X ( p1 , p2 ,..., ps , q1 , q2 ,..., qs )
However, the specification of all the macroparameters X does not determine a
unique microstate,
pi  pi ( X ), qi  qi ( X )
• Consequently, on the basis of macroscopic measurements, one can make
only statistical statements about the values of the microscopic variables.
Statistical Description of Mechanical Systems
Statistical description of mechanical systems is utilized for multi-particle
problems, where individual solutions for all the constitutive atoms are not
affordable, or necessary. Statistical description can be used to reproduce
averaged macroscopic parameters and properties of the system.
Comparison of objectives of the deterministic and statistical approaches:
Deterministic particle dynamics Statistical mechanics
Provides the phase vector, as
a function of time Q(t), based
on the vector of initial
conditions Q(0)
Provides the time-dependent
probability density to observe
the phase vector Q, w(Q,t),
based on the initial value
w(Q,0)
Statistical Description of Mechanical Systems
From the contemporary point of view, statistical mechanics can be regarded as a
hierarchical multiscale method, which eliminates the atomistic degrees of
freedom, while establishing a deterministic mapping from the atomic to
macroscale variables, and a probabilistic mapping from the macroscale to the
atomic variables:
Microstates
Macrostates
Xk
(p,q)
k
deterministic
conformity
probabilistic
conformity
Distribution Function
Though the specification of a macrostate Xi cannot determine the microstate
(p,q)i = (p1,p2,…,ps; q1,q2,…,qs)i, a probability density w of all the microstates
can be found,
w  p1 , p2 ,..., ps ; q1 , q2 ,..., qs ; t 
or abbreviated:
w( p, q, t )
The probability of finding the system in a given phase volume G:
W (G, t )   w( p, q, t )dpdq
G
The normalization condition:

( p ,q )
w( p, q, t )dpdq  1
Statistical Ensemble
Within the statistical description, the motion of one single system with given
initial conditions is not considered; thus, p(t), q(t) are not sought.
Instead, the motion of a whole set of phase points, representing the collection of
possible states of the given system.
Such a set of phase points is called a phase space ensemble.
If each point in the phase space is considered as a random quantity with
a particular probability ascribed to every possible state (i.e. a probability density
w(p,q,t) is introduced in the phase space), the relevant phase space ensemble is
called a statistical ensemble.
p
t = t2: G2
G – volume in the phase
space, occupied by the
statistical ensemble.
t = t1: G1
q
Statistical Averaging
Statistical average (expectation) of an arbitrary physical quantity F(p,q), is given
most generally by the ensemble average,
F (t ) 

F ( p, q) w( p, q, t )dpdq
( p ,q )
The root-mean-square fluctuation (standard deviation):
( F ) 
F F
2
The curve representing the real motion (the experimental curve) will mostly
proceed within the band of width 2Δ(F)
F
True value
F  ( F )
F
t
For some standard equilibrium systems, thermodynamic parameters can be
obtained, using a single phase space integral. This approach is discussed below.
Ergodic Hypothesis and the Time Average
Evaluation of the ensemble average (previous slide) requires the knowledge of the
distribution function w for a system of interest.
Alternatively, the statistical average can be obtained by utilizing the ergodic
hypothesis in the form,
F F
Here, the right-hand side is the time average
(in practice, time t is chosen finite, though as large as possible)
1 t
F   F  p( ), q( )  d , t  
t 0
This approach requires F as a function of the generalized coordinates.
Some examples
Internal energy:
U  H ( p, q )
Temperature:
T
2 Ek
kB
T2 C
dU
Change of entropy: S  
  V dT
U1 T
T1 T
vl
Gas diffusion constant: D 
3
U2
Here,
Ek  mean kinetic energy per degree of freedom
v  mean velocity of molecules
l  mean free path of gas particles
More examples are given in Ref. [1].
Law of Motion of a Statistical Ensemble
A statistical ensemble is described by the probability density in phase space,
w(p,q,t). It is important to know how to find w(p,q,t) at an arbitrary time t, when the
initial function w(p,q,0) at the time t = 0 is given.
In other words, the equation of motion satisfied by the function w(p,q,t) is needed.
p
w(p,q,0)
w(p,q,t1)
w(p,q,t2)
Γ1
Γ0
Γ2
q
The motion of of an ensemble in phase space may be considered as the motion of a
phase space fluid in analogy to the motion of an ordinary fluid in a 3D space.
Liouville’s theorem claims that
0  1   2  ...
Due to Liouville’s theorem, the following equation of motion holds
2s
 H w H w 
w
 [ H , w],
[ H , w]   

 (Poisson bracket)
t

q

p

p

q
i 1 
i
i
i
i 
Equilibrium Statistical Ensemble: Ergodic Hypothesis
For a system in a state of thermodynamic equilibrium the probability density in
phase space must not depend explicitly on time,
w
0
t
Thus, the equation of motion for an equilibrium statistical ensemble reads
[ H , w]  0
A direct solution of this equation is not tractable.
Therefore, the ergodic hypothesis (in a more general form) is utilized: the
probability density in phase space at equilibrium depends only on the total
energy:
w( p, q)    H ( p, q, a) 
Notes: the Hamiltonian gives the total energy required; the Hamiltonian may depend
on the values of external parameters a = (a1, a2,…), besides the phase vector X.
This distribution function satisfies the equilibrium equation of motion, because
Exercise: Check the above equality.
[ H ,  ( H )]  0
Canonical Ensembles
After adoption of the ergodic hypothesis, it then remains to determine the actual
form of the function φ(H). This function depends on the type of the
thermodynamic system under consideration, i.e. on the character of the interaction
between the system and the external bodies.
We will consider canonical ensembles of two types of systems:
1) Adiabatically isolated systems that have no contact with the surroundings and
have a specified energy E.
• The corresponding statistical ensemble is referred to as the microcanonical
ensemble, and the distribution function – microcanonical distribution.
2) Closed isothermal systems that are in contact and thermal equilibrium with an
external thermostat of a given temperature T.
• The corresponding statistical ensemble is referred to as the canonical
ensemble, and the distribution function – Gibbs’ canonical distribution.
Both systems do not exchange particles with the environment.
Microcanonical Distribution
For an adiabatically isolated system with constant external parameters, a, the total
energy cannot vary. Therefore, only such microstates X can occur, for which
H ( p, q, a)  E  constant
This implies (δ – Dirac’s delta function)
w( p, q)   E  H ( p, q, a) 
and finally:
w( p, q) 
1
  E  H ( p, q, a ) 
( E, a)
E, a
(P,T,V,…)
where Ω is the normalization factor,
( E , a ) 
   E  H ( p, q, a)  dpdq
( p ,q )
Within the microcanonical ensemble, all the energetically allowed microstates have
an equal probability to occur.
Microcanonical Distribution: Integral Over States
The normalization factor Ω is given by
 ( E, a) 
( E, a)  

 E a
where Γ is the integral over states,
or phase integral:
( E , a ) 

Phase volume
1
dpdq
(2 ) f N N ! H ( p ,q,a ) E
1
  E  H (p, q)  dp1...dp N dq1...dq N
(2 ) f N N ! (p,q )
0, x  0
 ( x)  
,
1,
x

0

- Planck's constant,
j - number of DOF per particle
Γ(E,a) represents the normalized phase volume, enclosed within the
hypersurface of given energy determined by the equation H(X,a) = E.
Phase integral Γ is a dimensionless quantity.
Thus the normalization factor Ω shows the rate at which the phase volume
varies due to a change of total energy at fixed external parameters.
Microcanonical Distribution: Integral Over States
The integral over states is a major calculation characteristic of the microcanonical
ensemble. The knowledge of Γ allows computing thermodynamic parameters of the
closed adiabatic system:
1
S  k ln ,
 S 
T 
 ,
 E 
P
1
  


( E,V )  V  E
(These are the major results in terms of practical calculations over microcanonical ensembles.)
Microcanonical Ensemble: Illustrative Examples
We will consider one-dimensional illustrative examples of computing the phase
integral, entropy and temperature for microcanonical ensembles:
Spring-mass harmonic oscillator
Pendulum (non-harmonic oscillator)
We will use the Hamiltonian equations of motion to get the phase space trajectory,
and then evaluate the phase integral.
Harmonic Oscillator: Hamiltonian
Hamiltonian: general form
H  T ( p)  U ( x)
Kinetic energy
p2
T
2m
Potential energy
k x2
U
2
k
m
x
Potential energy is a quadratic function of the coordinate (displacement form the
equilibrium position)
The total Hamiltonian
p2 k x2
H ( p, x ) 

2m
2
Parameters:
m  1021 kg, k  25 1021 N/m
Harmonic Oscillator: Equations of Motion and Solution
Hamiltonian and equations motion:
p2 k x2
H
H
H ( x, p ) 

 x
, p

2m
2
p
x
Initial conditions (m, m/s):
x(0)  0.2, x(0)  0
x(0)  0.4, x(0)  0
x(0)  0.6, x(0)  0
Parameters:
m  1021 kg, k  25 1021 N/m
x
p
, p  k x
m
Harmonic Oscillator: Total Energy
Total energy:
E  H ( x, p)  Const (at any x(t ), p(t ))
E1
E2
x(0)  0.2
x(0)  0
x(0)  0.4
x(0)  0
E3
x(0)  0.6
x(0)  0
E1  0.5 1021 J, E2  2.0 1021 J, E3  4.5 1021 J
Harmonic Oscillator: Phase Integral
Phase integral:
( E ) 
1
2
A( E ),
A( E ) 
   E  H ( x, p)  dxdp
( x, p )
 x p
2
  E  H (i
N
N
i 0 j 0
x
, j p ) 
 x  2 xmax / N  step for x,  p  2 pmax / N  step for p, N  number of integration steps
A1
A2
A3
1  0.95  1012
 2  3.80 1012
3  8.54  1012
For the harmonic oscillator, phase
volume grows linearly with the
increase of total energy.
Harmonic Oscillator: Entropy and Temperature
S  k ln , k  1.38 1023 J/K
Entropy:
S1  3.811022 J/K
S2  4.00 1022 J/K
S3  4.111022 J/K
Temperature:
 S 
T 

 E 
1
We perturb the initial conditions (on 0.1% or less) and compute new values E and S.
The temperature is computed then, as
T
 S S 


E

E


1
(benchmark: T 
T1  36.3 K
T2  145.1 K
T3  326.5 K
2 kin
E )
k
Pendulum: Total Energy
Total energy:
p2
E  H ( , p) 
 mgl cos   Const (at any  (t ), p(t ))
2
2ml
E1
 (0)  0.3
 (0)  0
E2
 (0)  1.8
 (0)  0
E3
 (0)  3.12
 (0)  0
p   l - angular momentum
E1  1.87 1021 J, E2  0.45 1021 J, E3  1.96 1021 J
Pendulum: Phase Integral
Phase integral:
( E ) 
1
2
A( E ),
A( E ) 
  E  H ( , p)  d dp


( , p)
  p
2
  E  H (i , j ) 
N
N
i 0 j 0
p
  2 xmax / N  step for  ,  p  2 pmax / N  step for p, N  number of steps
A1
A2
1  0.4 1012
 2  11.2 1012
3  21.4 1012
For the pendulum, phase volume
grows NON-linearly with the increase
of total energy at large amplitudes.
A3
Pendulum: Entropy and Temperature
Entropy:
S  k ln , k  1.38 1023 J/K
S1  3.68 1022 J/K
S2  4.15 1022 J/K
S3  4.24 1022 J/K
 S 
T 


E


Temperature:
1
We perturb the initial conditions (on 0.1% or less) and compute new values E and S.
The temperature is computed then, as
T
 S S 


E

E


1
(benchmark: T 
T1  6.3 K
T2  153.8 K
T3  96.0 K
2 kin
E )
k
Summary of the Statistical Method: Microcanonical Distribution
1.
Analyze the physical model; justify applicability of the microcanonical
distribution.
2.
Model individual particles and boundaries.
3.
Model interaction between particles and between particles and boundaries.
4.
Set up initial conditions and solve for the deterministic trajectories (MD).
5.
Compute two values of the total energy and the phase integral – for the
original and perturbed initial conditions.
6.
Using the method of thermodynamic parameters, compute entropy,
temperature and other thermodynamic parameters. If possible compare the
obtained value of temperature with benchmark values.
1
S  k ln ,
7.
 S 
T 
 ,
 E 
P
1
  


( E,V )  V  E
If required, accomplish an extended analysis of macroscopic properties (e.g.
functions T(E), S(E), S(T), etc.) by repeating the steps 4-7.
Canonical Distribution: Preliminary Issues
One important preliminary issue related to the use of Gibbs’ canonical distribution is
the additivity of the Hamiltonian of a mechanical system.
Structure of the Hamiltonian of an atomic system:
H  E kin (p1 , p 2 ,..., p N )  U (r1 , r2 ,..., rN )
pi2

 W1 (ri )   W2 (ri , r j )   W3 (ri , r j , rk )  ...
i 2m
i
i , j i
i , j i , k  j
Here, kinetic energy and the one-body potential are additive, i.e. they can be
expanded into the components, each corresponding to one particle in the system:
E kin (p1 , p 2 ,..., p N )  E kin (p1 )  E kin (p 2 )  ...
W (r )  W (r )  W (r )  ...
1
i
1
1
1
2
i
Two-body and higher order potentials are non-additive (function Q2 does not exist),
W2 (rij )  W | ri  rj | 
 W (r , r )  ...  Q (r )  ...  Q (r )  ...,
i , j i
2
i
j
2
i
2
j
Canonical Distribution: Preliminary Issues
Thus, if the inter-particle interaction is negligible,
W2  W3  ...  E kin  W1
the system is described by an additive Hamiltonian,
H   hi ,
i
pi2
hi 
 W1 (ri )
2mi
Here, H is the total Hamiltonian, and hi is the one-particle Hamiltonian.
• For the statistical description, it is sufficient that this requirement holds for the averaged
quantities only.
• The multi-body components, W>1, cannot be completely excluded from the physical
consideration, as they are responsible for heat transfer and establishing the thermodynamic
equilibrium between constitutive parts of the total system.
• A micromodel with small averaged contributions to the total energy due to particle-particle
interactions is called the ideal gas.
Example: particles in a circular cavity. Statistically averaged value W2 is small:
3 particles:
W2
0.049 E tot
5 particles:
W2
0.026 E tot
Canonical Distribution
Suppose that system under investigation Σ1 is in
thermal contact and thermal equilibrium with a
much larger system Σ2 that serve as the
thermostat, or “heat bath” at the temperature T.
From the microscopic point of view, both Σ1 and
Σ2 are mechanical systems whose states are
described by the phase vectors (sets of canonical
variables X1 and X2). The entire system Σ1+Σ2 is
adiabatically isolated, and therefore the
microcanonical distribution is applicable to Σ1+Σ2,
w( p1 , q1 ; p2 , q2 ) 
Thermostat T
N1
Σ1
1
  E  H ( p1 , q1; p2 , q2 ) 
( E )
Assume N1 and N2 are number of particles in Σ1
and Σ2 respectively. Provided that N1 << N2, the
Gibbs’ canonical distribution applies to Σ1:
1 
w( p, q)  e
Z
H ( p,q )
kT
N2
Σ2
( p, q)  ( p1 , q1 )
Canonical Distribution: Partition Function
Thermostat T
The normalization factor Z for the canonical
distribution called the integral over states or
partition function is computed as
N1
Σ1
H ( p ,q ,a )

1
kT
Z
e
dpdq
3N

(2 ) N ! ( p ,q )
N2
Σ2
H ( p ,r )

1
kT

e
dp1...dp N dr1...drN
3N

(2 ) N ! (p,r )
Before the normalization, this integral represents the statistically averaged phase
volume occupied by the canonical ensemble.
The total energy for the canonical ensemble is not fixed, and, in principle, it may
occur arbitrary in the range from – to  (for the infinitely large thermostat,
N2  ).
Partition Function and Thermodynamic Properties
The partition function Z is the major computational characteristic of the
canonical ensemble. The knowledge of Z allows computing thermodynamic
parameters of the closed isothermal system (a  V, external parameter):
Free energy:
(relates to mechanical work)
Entropy
(variety of microstates)
Pressure
Internal energy
F (T ,V )  kT ln Z

 F 


S      k  ln Z  T
ln Z 
T
 T V



 F 
P      kT
ln Z
V
 V T
U  F  TS  kT 2

ln Z
T
These are the major results in terms of practical calculations over canonical ensembles.
Class exercise: check the last three above formulas with the the method of thermodynamic potentials,
using the first formula for the free energy.
Free Energy and Isothermal Processes
Free energy, also Helmholtz potential is of importance for the description of
isothermal processes. It is defined as the difference between internal energy and
the product of temperature and entropy.
F  U  TS
Since free energy is a thermodynamic potential, the function F(T,V,N,…)
guarantees the full knowledge of all thermodynamic quantities.
Physical content of free energy: the change of the free energy dF of a system at
constant temperature, represents the work accomplished by, or over, the system.
Indeed,
dF  dU  TdS  SdT
 dU  TdS   W 
  SdT   W
dF   W
Isothermal processes tend to a minimum of free energy, i.e. due to the definition,
simultaneously to a minimum of internal energy and maximum of entropy.
Canonical vs. Microcanonical: Factorization of the Partition Function
In terms of practical calculations, there exists one major
difference between the canonical and microcanonical
distributions:
Thermostat T
6
N1
Σ1
For additive Hamiltonians, the canonical distribution
factorizes,
1
w( p, q)  e
Z
N
H ( p ,q )

kT
1 
w( p, q)   e
i 1 zi
1 i
 e
Z

hi ( pi , qi )
kT
hi ( pi , qi )
kT
Σ2

zi is the one-particle partition function
Note that this property does not hold for the
microcanonical distribution,
w( p, q) 
N2
1
  E  H ( p, q ) 
( E )
E, a
Factorization of the Partition Function: Computational Issues
Factorization of the canonical distribution is crucial in terms of practical
calculations over real-life systems.
In computing the partition function Z, this property reduces the calculation of a 6Ndimensional phase integral to a product of N 6-dimensional integrals:
1 N
Z
zi ,

N ! i 1
zi 
1
(2 )
3

e

hi ( pi ,ri )
kT
dpi dri
( p ,r )
Here, zi is the one-particle partition function, and hi is the one-particle Hamiltonian,
pi2
hi 
 W1 (ri )
2mi
W
2
 E kin  W1  0 
In case that the system is comprised of identical particles, calculation of Z
requires evaluation of a single 6-dimensional integral z:
h1  h2  ...  hN
 z1  z2  ...  z N  z 
zN
Z
N!
The factorized canonical distribution
can be very effective computationally.
Analytical Example: Non-Interactive Ideal Gas
The canonical distribution allows an exact solution for the non-interactive ideal gas:
pi2
H 
i 1 2mi
N
W1 ,W2
0
Analytical results for this system are useful, because they provide acceptable “first
guess” assessments for a wide class of systems.
One-particle partition function:
zi 

1
(2 )
V
3

e

pi2
2 mkT
dpi dri
p
2
i
 pi2 , dpi  pi2 sin i di di 
( p ,r )

4  e

pi2
2 mkT
2
pi2 dpi
(2 )3
0
V
3/ 2

(2

mkT
)
(2 )3
Partition function (total system):
   pi 2


3/ 2
2 mkT
pi dpi 
(mkT ) 
e
0

2


zN
VN
3N / 2
Z

(2

mkT
)
N ! (2 )3 N N !
Non-Interactive Ideal Gas: Thermodynamic Parameters
Partition function:
Free energy
(recall the method of
thermodynamic potentials)
Entropy
Pressure
Total internal energy
(differs form the earlier
MD definition)
VN
3N / 2
Z
(2

mkT
)
(2 )3 N N !
 N (2 )3 
F (T ,V )  kT ln Z  NkT ln 
 NkT
3/ 2 
 V (2 mkT ) 
 V (2 mkT )3/ 2  5
 F 
S  
  Nk
  Nk ln 
3
 T V
 N (2 )
 2
NkT
 F 
P  



V
V

T
3
E  F  TS  NkT
2
f
(generally: E  NkT ,
2
f - number of DOF per atom)
Numerical Example: Interactive Gas
Repulsive interaction between the particles and the
wall is described by the “wall function”, a one-body
potential that depends on ri – distance between the
particle i and the chamber’s center):
Wwl (ri )   e
y
  ( R ri )
Interaction between particles is modeled with the
two-body Lennard-Jones potential (rij – distance
between particles i and j):
ri
rij
R
x
rj
  12  6 
WLJ (rij )  4  12  6 
r
rij 
 ij
The Hamiltonian:
pi2
H (p, r )  
 Wwl (| ri |)  WLJ (| ri  r j |),
2
m
i
i
i j i
One particle is initially at rest. This illustrates the concept of heat exchange between the
smaller subsystem, for which the canonical distribution holds, and the external thermostat.
Interactive Gas: Equations of Motion and Solution
The total potential:
U  Wwl (r1 )  Wwl (r2 )  Wwl (r3 )  Wwl (r4 )  Wwl (r5 )
 WLJ (r12 )  WLJ (r13 )  WLJ (r14 )  WLJ (r15 )
 WLJ (r23 )  WLJ (r24 )  WLJ (r25 )
 WLJ (r35 )  WLJ (r35 )
 WLJ (r45 )
Equations of motion:
U
U
mi xi  
, mi yi  
, i  1, 2,3, 4,5
xi
yi
Parameters:
m  1023 kg,   1023 J,   2.7m,
  1023 J,   4m 1 , R  10m
Initial conditions (nm, nm/s):
x1 (0)  0, x1 (0)  25, y1 (0)  5.8, y1 (0)  0
x2 (0)  0, x2 (0)  30, y2 (0)  3.1, y2 (0)  0
x3 (0)  0, x3 (0)  0, y3 (0)  0.1, y3 (0)  0
x4 (0)  0, x4 (0)  24, y4 (0)  2.9, y4 (0)  0
x5 (0)  0, x5 (0)  22, y5 (0)  5.7, y5 (0)  0
Interactive Gas: Temperature
Time averaged kinetic energy vs. time (five particles)
Time averaged kinetic energy
of particles is approaching the
value
E kin  2.438 1021 J
which corresponds to
temperature
1 kin
E
k
2.438 1021

1.38 1023
T
176.68K
Information on temperature allows computing the partition function (integral over
states), using canonical distribution. Subsequently, the partition function, computed at
various temperatures, can provide all the remaining thermodynamic parameters.
For sufficiently long simulations, the value of temperature does not depend on the
choice of a subsystem (particle).
Interactive Gas: Partition Function and Free Energy
H   hi ,
Hamiltonian
(can be viewed as additive in
the statistical sense, due to
smallness of the time averaged
pair-wise interaction)
One-particle partition
function
(value at given
T and V = πR2)
5 particles:
i
pi2
hi 
  e  ( R |ri |)
2mi
z
1
(2 ) 2

e
Free energy
p2 / 2 m  e  ( R |r|)
kT
dpdr
 p2  p2 , r  r,

 dpdr  pdpd p rdrd r   polar coordinates



(total system)
0.026 E tot
( p ,r )

4 2

e
(2 ) 2 0
Partition function

W2
zN
Z
N!
p 2 / 2 m  e  ( Rr )
2 mkT
pdp rdr
1.383 1026
(1.383 1026 )5
 4.223 10128
5!
F (T ,V )  kT ln Z  7.22 1019 J
Interactive Gas: Thermodynamic Parameters
In order to compute the thermodynamic quantities, it necessary to evaluate 2 values of the
partition function:
1) Z – for the initially computed temperature T,
2)
– for a perturbed temperature T +ΔT (ΔT/T < 0.1%).
Z
Note: the simulation needs to be run once only (not two times).
Entropy:

Z Z 

 
21
S  k  ln Z  T
ln Z  k  ln Z  T
  4.16 10 J/K
T
T 

 
Pressure:
P  kT
Internal energy:
U  kT 2
Ideal gas benchmark:

ln Z
T
kT
Z Z
 7.211023 Pa
T

Z Z
ln Z kT 2
 1.27 1020 J
T
T
f
U (i.g.)  NkT  1.22 1020 J  f  2, N  5 
2
Other parameters can be computed using the method of thermodynamic potentials
Phase Integral, Free Energy and Entropy vs. Temperature
Assume that we observe the same isothermal system at various
temperatures of thermostat. The following trends are available:
Partition function
Free energy
Entropy
Partition function, and therefore the phase volume occupied by this canonical ensemble, grows
exponentially vs. temperature.
Free energy decreases linearly; the work done by the system does not depend on temperature.
Entropy decays vs. temperature. Physical implication (according to the second law):
temperature cannot grow spontaneously in an isothermal system, once thermal equilibrium
with the thermostat is established.
Specifics of Calculations for Liquids and Solids
In liquids, the energy due to pair-wise interaction between
particles is close to the kinetic energy (per particle).
However, interaction, wij, between separate constitutive
parts (subdomains) i and j is still weak, if compared with
the total kinetic energy of the smaller domain. Indeed, the
kinetic energy depends on the subdomain volume, while wij
depends on the surface area.
Therefore, the Hamiltonian
can be expanded into hi – Hamiltonians
of the sufficiently large subdomains.
Partition function
(z – partition functions for
N identical subdomains,
n – number of subdomain particles)
hi
H   hi   wij
i
i
j i
j
i
h
i
i
hi  Eikin  W1( i )  W2( i )
W
(i )
2
 wij 
h ( p ,r )

zN
1
kT
Z
, z
e
dpdr
3n

N!
(2 ) n! (p,r )
For reasonably small subdomains, numerical evaluation
of the liquid’s partition function can be effective.
A similar approach is also applicable
to solids. Example:
H
Note: In case of large wij, the microcanonical distribution should be utilized.
Summary of the Statistical Method: Canonical Distribution
1.
Analyze the physical model; justify applicability of the Gibbs’ canonical
distribution.
2.
Model individual particles and boundaries.
3.
Model interaction between particles and between particles and boundaries.
4.
Set up initial conditions and solve for the deterministic trajectories (MD).
5.
Compute the averaged kinetic energy and temperature, or assume T given.
6.
Based on the canonical distribution, compute two values of the partition
function: for the original and perturbed temperatures.
7.
Compute the free energy and other thermodynamic parameters, using the
method of thermodynamic potentials. If possible compare the obtained value
of internal energy with a benchmark value.
8.
If required, accomplish an extended analysis of macroscopic properties (e.g.
dependences P(T), P(N), S(T), etc.) by repeating the steps 4-7.
4.3 Numerical Heat Bath Techniques
•
•
•
•
•
Berendsen thermostat
Adelman-Doll thermostatting GLE
Phonon heat bath
Time-history kernel and transform techniques
Random force and lattice normal modes
Finite Temperatures
A heat bath technique is required to represent a peripheral region at finite temperatures
Berendsen thermostat for a standard Langevin equation
mi ui (t )  iU   ui (t )  R i (t )
R i (t ) R j (t   )  2mi i k BT0  ( ) ij
T0  target system temperature
Berendsen et al., JCP 81(8), 1984
T / T0
t
Finite Temperatures
Adelman-Doll’s thermostatting GLE for gas-solid interface
t
ui (t )  02ui (t )   (t   ) ui ( )d  R i (t )
0
R i (t ) R j (t   )  k BT0 ( ) ij
Adelman, Doll et al., JCP 64(6), 1976
Almost exactly what we seek, however, the update is needed:
gas/solid interface -> solid/solid interface
Phonon Heat Bath
Phonon heat bath represents energy exchange due to correlated motion of lattice atoms
along an imaginary atomic/continuum (solid-solid) interface
Phonon heat bath is a configurational method
atom next to
the interface
mui (t )  iU   (t   )  ui ( )  R i ( )  d  k R j (t )
t
0
R (t )   lattice normal mode (an , n )
n
 an (T0 ), n   sampled from the Gibbs distribution
 (t)  mechanical response if the thermostat
R(t)  random thermal motion of thermostat atoms
T / T0
k  lattice stiffness constant
Karpov, Liu, preprint.
t
Time History Kernel (THK)
The time history kernel shows the dependence of dynamics in two adjacent cells.
Any time history kernel is related to the response function.
u1 (t )   u0 (t ) ,   ?
fn (t )   n,0f (t )
f(t)
…
-2
-1
0
1
2
…
t
u n (t )   g n  n ' (t   ) f ( )d , U n ( s)  G n ( s) F( s), U1 ( s)  G1 ( s)G 01 ( s) U 0 ( s)
0
t
u1 (t )   (t   )u 0 ( )d , (t )  L 1 G1 ( s)G 01 ( s)
0
0.6
1
(t )  L 1 
4


 2
s 2  4  s   J 2 (2t )
 t
2
θ(t ) 
0.4
2
J 2 (2t )
t
0.2
0
-0.2
0
2
4
6
8
10
12
14
Bridging Scale at T = 0: Impedance Boundary Conditions
The MD domain is too large to solve, so that we eliminate the MD degrees of freedom
outside the localized domain of interest.
Collective atomic behavior of in the bulk material is represented by an impedance force
applied at the formal MD/continuum interface:

Md  N T f (u)
t
MD degrees of freedom outside the
localized domain are solved implicitly
M Aq  f (u)   Θ(t   )  q( )  u( )  d
FE + Reduced MD +
Impedance BC
0
MD
FE
+

Due to atomistic nature of the model, the structural impedance is evaluated computed at
the molecular scale.
Dynamic Response Function: 1D Illustration
Assume first neighbor interaction only:
Mu n (t ) 
n 1
K
n ' n 1
1
g n (t )  L F
1

nn '
un ' (t )   n ,0  (t ),
 s M  k (e
2
 ip

ip 1
2e
…
n-2
n-1
n
n+1
K 0  2k , K 1  k

K nn '

1
 L 
2 n
2
( M  k 1)
2 s s  4
1
Displacements

s 4s
2

2n
…
n+2
 2U

un un ' u  0



Velocities
1
0.7
0.8
0.6
g n (t )  J 2 n (2t )
0.6
0.5
0.4
0.4
0.2
0.3
t
g n (t )   J 2 n (2 )d
0.2
0.1
0
-0.2
0
0
-0.4
0
2
4
6
8
Illustration
Transfer of a unit pulse
due to collision (movie):
10
12
14
0
2
4
6
8
10
12
14
Discrete Fourier Transform (DFT)
Discrete functional sequences
Infinite:
un  f (nx / a), n  0,  1,  2, ...
Periodic:
un kN  un , N is integer, k  0,  1,  2, ...
DFT of infinite sequences


1
ipn
u ( p )   un e
un 
u
(
p
)
e
dp

2

n 

p – wavenumber, a real value between – and 
 ipn
DFT of periodic sequences
i 2 p n
1 N / 21
u ( p )   un e
un 
u ( p )e N

N p  N / 2
n  N / 2
Here, p – integer value between –N/2 and N/2
N / 2 1

i 2 p n
N
Discrete convolution
K  un   K n  n ' un '
n'
F  K  un   K ( p ) u ( p )
Numerical Laplace Transform Inversion
Most numerical algorithms for the Laplace transform inversion utilize series
decompositions of the sought originals f(t) in terms of functions whose Laplace
transform is tabulated. The expansion coefficients are found numerically from F(s).
Examples:
• Weeks algorithm
(J
Assoc Comp Machinery 13, 1966, p.419)
f t   e
 c T / 2t
S
a L  t / T 


0
L (t ) – Laguerre polynomials,
a – coefficients computed using F(s)
• Sin-series expansion (J Assoc Comp Machinery 23, 1976, p.89)
For an odd function f gives
 k i   k t 
f  t   2 Im F 
 sin 

T

  T 
k 1
N