Stochastic equations for thermodynamics

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However, there are many alternative forms of eq. (6) which provide ... ship between the rate change of Z and the gradient of the free energy. ( ) t. Z. dZ. M Z. F. = -.
Stochastic equations for thermodynamics Roumen Tsekov¤ Department of Physical Chemistry, University of SoÐa, 1126 SoÐa, Bulgaria

The applicability of stochastic di†erential equations to thermodynamics is considered and a new form, di†erent from the classical Itoü and Stratonovich forms, is introduced. It is shown that the new presentation is more appropriate for the description of thermodynamic Ñuctuations. The range of validity of the BoltzmannÈEinstein principle is also discussed and a generalised alternative is proposed. Both expressions coincide in the small-Ñuctuation limit, providing a normal distribution density.

The controversy concerning the proper meaning and the appropriate application in physics of stochastic di†erential equations is full of pitfalls. One is the unclear separation between the macroscopic value of a physical quantity and its Ñuctuations. The aim of the present paper is to clarify some problems that arise in the application of stochastic di†erential equations in thermodynamics. It is well known1 that many processes in nature can be described by the following stochastic equation dX \ A(X) dt ] B(X) dW (t) named after Paul Langevin. Here X is the quantity under observation, t is time, A and B are deterministic functions and W is a random Wiener process. The integral form of the Langevin equation is

P

dX \ A(X) dt ] dW

BDC A

B A BD

n]a n]1 n N~1 t W t [W t lim ; B X N N N N?= n/0 where a is a real number between 0 and 1. In contrast to the usual integrals, which are independent of the value of a, this expression depends substantially on the choice of the middle point in the time intervals. This is evident from the corresponding FokkerÈPlanck equation describing the evolution of the probability density P(x, t)

C

D

L B2a LB2(1~a)P LP \ [AP ] Lx Lx 2 Lt

(2)

which can be rigorously derived from eqn. (1).1 In the literature, there are two universally accepted choices for a leading to two di†erent forms of eqn. (2) : the Itoü form with a \ 0,2 and the Stratonovich form with a \ 1/2.3 From the mathematical point of view, both forms are correct but their application to physics and chemistry generates some problems.4 The main difficulty is related to the right attribution of physical meaning to the functions A and B.5 The latter are not independent and their relationship, being the subject of the ÑuctuationÈdissipation relation,6 is given by the equilibrium solution P1 (x) of eqn. (2) 1 [ a LB2 B2 L ln P1 ] A\ Lx 2 Lx 2 ¤ E-mail : tsekov=chem.uni-soÐa.bg

(3)

Regardless of the value of a, the corresponding FokkerÈ Planck equation is

C

D

L 1 LP LP \ [AP ] Lx 2 Lx Lt

and the drift term A is related to the equilibrium probability density as follows

P

t t (1) X(t) \ X(0) ] A(X) ds ] B(X)]a[ dW 0 0 where the notation ]a[ indicates a peculiar deÐnition of the integrals over a Wiener process. As was mentioned, the main problem in eqn. (1) is the interpretation of the last integral. Following standard mathematics, it can be expressed by a Riemann sum as

CA

As seen, the a-problem disappears if B is constant. For this reason, let us start with a pure random process obeying the following Langevin equation

A\

1 L ln P1 2 Lx

(4)

Let us now introduce a new random variable Y , being a deterministic function of X. The equilibrium probability density P1 (x) is connected to P1 (y) via the relation7 P1 (x) \ P1 (y)C(y)

(5)

where C(Y ) \ dY /dX. Multiplying eqn. (3) by C and using eqn. (4) and (5) one obtains a new stochastic di†erential equation for the random evolution of Y driven by a multiplicative white noise dY \

C2 d ln P1 C dt ] C]a[ dW dY 2

(6)

Eqn. (6) is a particular example of the Langevin equation with B \ C. Its corresponding FokkerÈPlanck equation is

C

L C2P d ln P1 C C2a LC2(1~a)P LP \ [ ] dy Ly Ly 2 2 Lt

D

which is correct in the equilibrium state (P \ P1 ) only if a \ 1/2. Hence,

C

D

L C2P L ln P/P1 LP \ 2 Ly Ly Lt

(7)

and the correct form of eqn. (6) is the Stratonovich one dY \

D C

1 C2 d ln P1 C dt ] C dY 2 2

dW

(8)

The fact that the Stratonovich form corresponds to the usual rules of mathematics is proven by the WongÈZakai theorem.8 However, there are many alternative forms of eqn. (8) which provide the same FokkerÈPlanck equation. Two J. Chem. Soc., Faraday T rans., 1997, 93(9), 1751È1753

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examples are dY \

C2 d ln P1 C2 dt ] C]0[ dW dY 2

dY \

C2 d ln P1 dt ] C]1[ dW 2 dY

(9)

(10)

Eqn. (9) is the Itoü form of the Langevin equation. Its advantage is that the equilibrium average value of the drift term is zero. For this reason it is suitable for juxtaposition with the corresponding macroscopic equations.7 Eqn. (10) is a new one. Its convenience is related to the proportionality of the drift term to the gradient of the equilibrium distribution density. In this sense, it could be appropriate for non-equilibrium thermodynamics. Eqn. (8)È(10) are exact. There is no physics there and using any one of them is a matter of convenience. In general, the transition to another a, without change in the corresponding FokkerÈPlanck equation, is ruled by L(B]a[ dW )/ La \ (dB/dX)B dt.1 According to thermodynamics, the characteristic function of a closed system at constant temperature T is the free energy F. Any spontaneous process in the system leads to a decrease in F to its minimal value corresponding to the equilibrium state. Hence, if Z is a variable parameter of the system, the rate of free energy decrease can be presented as dF LF dZ \ O0 LZ dt dt

(11)

On the other hand, according to non-equilibrium thermodynamics, there is a linear relationship between the rate change of Z and the gradient of the free energy LF dZ \ [M(Z) LZ dt

(12)

where M~1 is the resistance of the system. Owing to the positive deÐnition of M, inequality (11) is always fulÐlled. Eqn. (12) is not stochastic and describes only the irreversible evolution toward equilibrium without accounting for the Ñuctuations. The latter are not subject to the second law of thermodynamics. The di†erence between the equilibrium free energy and the conditional free energy for a given value of Z is proportional to the logarithm of the equilibrium probability density to observe this Ñuctuation value6 F(Z1 ) [ F(Z) P kT ln P1 (Z)

(13)

Introducing this expression in eqn. (12) one can easily obtain that the di†erential stochastic equation for the Z Ñuctuations should be written in the form dZ \ kT M

d ln P1 dt ] )(2kT M)]1[ dW dZ

(14)

Eqn. (14) is a particular example of eqn. (10) and the corresponding FokkerÈPlanck equation is

C

D

L L ln P/P1 LP \ kT MP Lz Lz Lt

As is seen, the most appropriate value of a in the description of thermodynamic Ñuctuations is a \ 1, which is not surprising. In physics, the causalty principle is very important. The only value of a which accounts for the whole inÑuence of the Wiener process to the contemporary evolution of the variable under observation is 1. Other a values correspond to a physically unacceptable inÑuence of Ñuctuations taking place after the current time to the moment state of the considered process. 1752 J. Chem. Soc., Faraday T rans., 1997, V ol. 93

A common mistake here is to treat eqn. (12) as the average product of eqn. (14). This is only true if M is constant. Eqn. (12) is not exact. It is a result of the Second Law and does not take into account the thermodynamic Ñuctuations which lead to the free energy increase. The exact equation is eqn. (14), which reduces to eqn. (12) in the absence of Ñuctuations. Finally, we shall pay attention to a problem of range of validity of the BoltzmannÈEinstein principle (13). It is obvious that the BoltzmannÈEinstein principle is not general because it is not invariant to non-linear change of the variable by the law (5). The only case for which it is always satisÐed is the small-Ñuctuation limit corresponding to a Gaussian distribution function. Hereafter, an alternative point of view is presented which seems to be more general. According to statistical mechanics, the equilibrium probability density for Ñuctuations of Z(X) can be calculated by the canonical Gibbs distribution P1 (z) \

P

A

d(z [ Z)exp

B

F[H dX kT

where H(X) is Hamilton function of the system and the free energy F is given by

A B P A B

exp [

F \ kT

exp [

H kT

dX

(15)

A more appropriate statistical quantity for description of the Ñuctuations is the charcteristic function H which is deÐned as the Fourier transform of the probability density H(s) 4

P

exp(isz)P1 (z) dz

\ exp

A BP A F kT

exp [

B

H [ iskT Z dX kT

(16)

According to thermodynamics, the average value Z1 of the Ñuctuating thermodynamic parameter is equal to the Ðrst derivative of the free energy F with respect to its thermodynamically conjugated quantity m, i.e. Z1 \ LF/Lf. For instance, in simple systems f could be temperature, volume or the number of particles and the corresponding Z quantities are entropy, pressure or chemical potential. From this point of view and eqn. (15), the characteristic function (16) can be expressed as

C

H(s) \ exp

F(f) [ F(f [ iskT ) kT

D

and an alternative to the BoltzmannÈEinstein principle (13) is F(f) [ F(f [ iskT ) \ kT ln H(s), which is invariant to any change in the variable. It is clear that, for Gaussian Ñuctuations, both the expressions coincide. Owing to the minimal value of the free energy at equilibrium, the necessary condition 0 O H O 1 follows from the equation above. The logarithm of the characteristic function is the so-called cumulant generating function6 using which a number of correlation characteristics of the system Ñuctuations can be calculated. The results obtained here are methodological rather than being concrete new knowledge for nature. The foregoing discussion aims to acquaint the readers with the common difficulty arising from the use of stochastic di†erential equations and to prevent undesirable mistakes. Since the theory of the equilibrium state is much more developed, the basic checkpoint for any kinetic theory is the reproduction of equilibrium theory results. The reader can note that this is a good tool for discrimination between the di†erent interpretations of the stochastic equations. Of course, the latter can be obtained from Ðrst principles. However, owing to the complexity of the systems this is only possible in very simple cases, e.g. harmonic oscillators. In a previous paper,9 starting from classical mechanics, we have demonstrated that the dynamics of a Brownian particle

in solids obeys the following equation

References

dR \ [M(R)(LU/LR) dt ] )[2kT M(R)]]1[ dW

1

where R is the particle coordinate, U(R) is potential and M(R) is position-dependent mobility. As is seen, this equation is a particular example of eqn. (14). Any other choice of a will reÑect in a physically undue dependence of the equilibrium distribution on the particle friction.

2 3 4 5 6 7 8 9

The author acknowledges Ðnancial support by the Bulgarian National ScientiÐc Foundation under grant X-508.

C. W. Gardiner, Handbook of Stochastic Methods, Springer, Berlin, 1985. K. Itoü, Nagoya Math. J., 1951, 3, 55. R. L. Stratonovich, SIAM J. Control, 1966, 4, 362. N. G. van Kampen, J. Stat. Phys., 1981, 24, 175. R. Tsekov, Ann. Univ. SoÐa, Fac. Chim., 1995, 88, 57. R. L. Stratonovich, Nonlinear Nonequilibrium T hermodynamics, Nauka, Moscow, 1985. D. Ryter, Z. Phys. B, 1978, 30, 219 ; P. Mazur and D. Bedeaux, L angmuir, 1992, 8, 2947. E. Wong and M. Zakai, Ann. Math. Stat., 1965, 36, 1560. R. Tsekov and E. Ruckenstein, J. Chem. Phys., 1994, 100, 1450.

Paper 6/07594K ; Received 7th November, 1996

J. Chem. Soc., Faraday T rans., 1997, V ol. 93

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