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Physica A 320 (2003) 193–203www.elsevier.com/locate/physa

Onsager’s regression and the !eld theory ofparabolic transport processesKatalin Gamb&ara, Ferenc M&arkusb;∗

aDepartment of Atomic Physics, E�otv�os Roland University, P�azm�any P�eter s�et�any 1=A,H-1117 Budapest, Hungary

bInstitute of Physics, Budapest University of Technology and Economics, Budafoki �ut 8.,H-1521 Budapest, Hungary

Received 14 June 2002

Abstract

On the basis of a Hamiltonian and canonical formalism of simple parabolic transport processesthe action for short time intervals can be calculated. In the knowledge of action, we can examinethe stochastic behavior of these processes and we conclude that these are Markovian in the spaceof canonical variables. The constructed Poisson structure gives us an opportunity to calculateOnsager’s regression hypothesis.c© 2002 Elsevier Science B.V. All rights reserved.

PACS: 02.50.Ey

Keywords: Onsager’s regression; Field theory of thermodynamics; Hamilton formalism;Euler–Lagrange equation; Liouville equation; Chapman–Kolgomorov equation; Markov process;Fokker–Planck equation; Regression of <uctuations

1. Introduction

The Onsager principle [1–4], which de!nes the linear laws for irreversible processesin an equilibrium ensemble, states that the average time derivative of an extensivevariable is linearly related to the average deviations of the conjugate intensive variablesfrom their equilibrium values. This principle describes the linear relaxation of averagesof extensive variables to their equilibrium values. The hypothesis of regression of<uctuations, on which e.g. the proof of reciprocity relations is based, says that theaverage regression of <uctuations from a given nonequilibrium state obeys the same

∗ Corresponding author.E-mail addresses: gambar@ludens.elte.hu (K. Gamb&ar), markus@phy.bme.hu (F. M&arkus).

0378-4371/03/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.PII: S0378 -4371(02)01579 -0

194 K. Gamb�ar, F. M�arkus / Physica A 320 (2003) 193–203

laws as the corresponding irreversible processes, i.e., the <uctuating variables obey thephenomenological laws [3–7].One can introduce a Lagrangian O by

O(�; �) = 12 �R�+

12 �SLS� ; (1)

where � are the deviations from equilibrium of the extensive variables, � = d�=dt arecalled <uxes, S = (@2S=@�2)eq, the second derivatives of entropy with respect to the �parameter space, are evaluated at equilibrium; L and R are symmetric matrices. Wealso de!ne an action A

AO(�(t)) =∫ t2

t1O(�; �) dt : (2)

Given the state �(t1) we ask for the path �(t) (t1¡t¡ t2) that minimizes the functionalA. After the variation, the Euler–Lagrange equations are

ddt@O@�

− @O@�

= 0 (3)

and the condition(@O@�

)t=t2

= 0 (4)

holds. Substituting the de!nition of O into Eq. (3) gives

L�− (LS)2�= 0 (5)

ordinary second-order diMerential equations, as the Euler–Lagrange equations. Thesolutions are

�(t) = exp(LS)�+ + exp(−LS)�− : (6)

If we take (I) the initial condition �(0)=�0 and (II) the asymptotic condition �(∞)=0,corresponding to equilibrium, the solution of Eq. (6) close to equilibrium

�(t) = exp(LSt)�0 : (7)

Thus, the solution to Eq. (3) with the conditions (I) and (II) must satisfy

�− LS�= 0 : (8)

The Lagrange equations (3) with integration constants (I) and (II) imply the averagedequations of irreversible thermodynamics. The Lagrange equations give a global varia-tional principle (minimum loss principle): the average path between �0 and equilibriumis the path for which∫ ∞

0O(�; �) dt =min : (9)

The variational principles describe only the average path of motion in thermodynamicspace (�). The path of any particular system will deviate from the average path, due to

K. Gamb�ar, F. M�arkus / Physica A 320 (2003) 193–203 195

the molecular randomness. The dynamic <uctuations are contained in the conditionalprobabilities. The averaged path of relaxation is determined by Eq. (8). We need moreinformation to describe the dynamical <uctuations. Onsager and Machlup proposedthat the spontaneous deviations from the average path were stationary, Gaussian andMarkovian [1–9]. This assumption implies the Langevin-type equations

�− LS�= f ; (10)

where f is a stationary white noise. The average of this equation is Eq. (8). Onsagerand Machlup proved that a knowledge of the equilibrium density

w(�)˙ exp(�2S(�)=2kB) (11)

is enough to provide a complete description of nonequilibrium thermodynamics of asystem in a neighborhood of equilibrium. They showed that the theory could be reducedto a variational principle. The conditional probability for short time

P(�0 | �(t))˙ exp[− 14kB

∫ t

0M (�; �) dt

]min

; (12)

where

M (�; �) = (�− LS�)R(�− LS�) (13)

is the Lagrangian and

AM (�) =∫ t

0M (�; �) dt (14)

is the action. The conditional probability P(�0 | �(t)) and the minimal increase of actionassociated with a change from the initial state �0 to the !nal state �(t)=�′ are related.The minimum path from �0 to �(t) = �′ obeys the Euler–Lagrange equations

ddt@M@�

− @M@�

= 0 : (15)

In other words, of all the path which take a time t between �0 and �′, the one whichminimizes the time integral of M determines the dynamic <uctuations. For a Gaussianprocess, the average and the most probable path are the same. We can conclude thatthe Onsager–Machlup principle governs the statistical behavior of irreversible systemsclose to equilibrium. Following Onsager and Machlup’s work Graham [10], Grabert andGreen [11] and Grabert et al. [12] made a hypothesis about the conditional probabilityfor a small time interval. This hypothesis is a certain extension to nonlinear cases

P�(�′=�) d�′ =exp[− (1=2kB)A�(�′=�)] d�∫

d� exp[− (1=2kB)A�]; (16)

where �= �(�; �′; �) are random forces. The action for the minimal path is

A� =∫ t1+�

t1dt 1

2 M (�; �)min : (17)

196 K. Gamb�ar, F. M�arkus / Physica A 320 (2003) 193–203

They showed that the conditional probability ful!lls the Chapman–Kolgomorovequation—for small time intervals—and the Fokker–Planck equation, thus it speci!esthe dynamics of <uctuations [10–12]. We can summarize that the irreversible process ofa discrete system is described by the <uctuations of the state variables, so the stochastictheory of it is developed [13].

2. Field theory of nonequilibrium thermodynamics

The most elegant ways of expressing the condition that determines the particularpath out of all possible path is the principle of least action. The classical path is thatfor which the action S is minimum. However, in the sense of Feynman’s method,the diMerent trajectories contribute to the total probability to go from a to b. Theycontribute equal amounts, but at diMerent phases. The contribution of a path has aphase proportional to the action [14].

The classical irreversible thermodynamics deals with such a large and continuousmedia which are described by !eld quantities: intensive �(r; t) and speci!c extensiveg(r; t) quantities. It was our purpose to apply the principle of least action for simpledissipative processes. We managed to give a suitable Lagrange density function—introduced certain potential functions—by which the complete Hamiltonian formalismcan be used [15].Now, we would like to show how the regression hypothesis can be deduced for con-

tinuum systems from the concepts of !eld theory of nonequilibrium thermodynamics.The method—what we apply now to calculate the regression—was originally used forthe examination of Markovian behavior of hyperbolic equations developed by V&azquezet al. [16]. This elegant formulation needed some modi!cation because they applieda not strictly applicable canonical calculation. With the Hamiltonian formalism fordissipative hyperbolic processes [17] we could describe the regression of dissipativehyperbolic processes.In the present paper, we examine the stochastic [13] properties of dissipative pro-

cesses [7,18] given by parabolic diMerential equations, for which the linear constitutiveequations hold (e.g. in the case of Fourier heat conduction the Fourier’s law). For thesake of simplicity, we consider only one process in our examination. We would like toemphasize that the Hamiltonian formulation of parabolic processes is not a simpli!ca-tion of the hyperbolic calculations, i.e., it is not enough to drop the second-order timederivatives in the telegraph equation. Now, we shortly summarize the canonical formal-ism for this kind of equations, containing nonself-adjoint operators, here the !rst-ordertime derivative. Then the Lagrangian L [15] is

L=12

(�S−1 @’

@t− LR’

)2(18)

and the connection between the measurable quantity � and the introduced potential ’can be expressed by

� = �S−1 @’@t

− LR’ ; (19)

K. Gamb�ar, F. M�arkus / Physica A 320 (2003) 193–203 197

where � is the mass density, S−1 is the generalized capacity, L is the conductivity(Onsager) coeScient and � is the Laplace operator. The introduction of a potentialis a consequence of nonself-adjoint operators in the !eld equations, similar to theHamilton formalism of Maxwell’s equations of electrodynamics. The Euler–Lagrangeequation (the well-known transport equation) can be written for the quantity �

�S−1 @�@t

+ LR� = 0 : (20)

The conjugated momentum p to ’ is

p=@L@’;t

= �S−1� = (�S−1)2’;t − �S−1L(R’) ; (21)

where we denote ;t = (@=@t). The Hamilton density function is

H = p’;t − L ; (22)

where we substitute the term containing the time derivative of ’;t and we obtain

H (p;R’) = 12 (�

−1S)2p2 + �−1SLpR’ : (23)

The canonical equations are

’;t =@H@p

(24)

and

p;t =−� @H@R’

: (25)

These equations can be written by the Poisson bracket expressions

’;t = [’;H ] = (�−1S)2p+ �−1SLR’ (26)

and

p;t = [p;H ] =−�−1SLRp : (27)

In order to examine Eqs. (26) and (27) from the stochastic viewpoint, we considerthe quantities ’ and p as the <uctuating variables of the system. This assumptionfollows from the fact that the system is out of equilibrium and uncontrolled innerprocesses may exist which can cause random microscopic changes. Therefore, the aimis to de!ne a probability !eld associated with the paths of the system in the conjugatedspace.

3. Liouville and Chapman–Kolgomorov equation

Let us consider a collection of replicas of the original system with the same initialcondition in the space of conjugated variables. We would like to describe this physical

198 K. Gamb�ar, F. M�arkus / Physica A 320 (2003) 193–203

ensemble in the phase space (’;p). The de!nition of probability !eld of paths mustbe in accordance with the fact that the action S will be minimum for path obtained bythe solution of Eqs. (26) and (27), because it represents the most probable behaviorof the system. The Poisson structure gives the possibility to describe the process as afunction of time in the new space of canonically conjugated quantities (’;p). On thebasis of Eqs. (26) and (27) it can be seen that the Liouville’s theorem is obeyed, since

��’

@’@t

+��p

@p@t

= 0 : (28)

This shows that the phase volume does not change in the space of conjugated variables.After this, it can be seen that the Liouville equation is valid for the probability densityW (’;p; t), i.e.,

@W@t

= [W;H ] : (29)

Here, we need to mention that in the framework of nonequilibrium statistical mechanicsearly studies starting from Liouville’s theorem were pioneered by Kirkwood [19] andGreen [8], the statistical theories were elaborated by Penrose [20], van Kampen [21]and many others.TheW (’;p; t) represents the set of probability densities: W1(’1; p1; t1); W2(’1; p1; t1;

’2; p2; t2), etc. If we know a single probability W1(’1; p1; t1) and the transition prob-ability P(’N−1; pN−1; tN−1=’N ; pN ; tN ) we can obtain the complete set through therelation

WN (’1; p1; t1; : : : ;’N ; pN ; tN )

=P(’N−1; pN−1; tN−1=’N ; pN ; tN )

×WN−1(’1; p1; t1; : : : ;’N−1; pN−1; tN−1) : (30)

If we know W , we are able to calculate the average of an arbitrary quantity F(’;p)according to

〈F(’;p)〉=∫F(’;p)W (’;p) d%∫

W (’;p) d%; (31)

where d%= d’ dp is the elementary volume for what we need to know the transitionprobability. Let us consider a system with the state (’;p) and this tends to the state(’′; p′). Now, we calculate the probability density between two states.The transition probability can be expressed as the condition probability of the tran-

sition from the state (’;p) at t = 0 to (’′; p′) at t = �, i.e.,

(’;p)t=0 → (’′; p′)t=� ; (32)

P�(’′=’) =exp[− (1=k)S�(’′=’)]@p=@’′∫

dp exp[− (1=k)S�(’′=p)]: (33)

K. Gamb�ar, F. M�arkus / Physica A 320 (2003) 193–203 199

This is a modi!ed version of the conditional probability introduced by Grabert andGreen [11]. The action S� for the time interval (0; �) is

S�(’′=’) =∫ �

0(p’;t − H) dt ;

=∫ �

0

(p’;t − 1

2(�S−1)2p2 − �−1SLpR’

)dt ; (34)

where we assume that � is a small !nite quantity [16,11]. � is smaller than the usualhydrodynamic time scale, but greater than the kinetic scale. So

’(�) = ’+ �’;t + · · · ⇒ ’;t =’′ − ’�

(35)

and

p= (�S−1)2’′ − ’�

− �S−1LR’ (36)

can be given in these forms. Then

@p@’′ =

(�S−1)2

�(37)

is completed. By the help of Eqs. (35)–(37) the action S� and the transition probabilityP� can be calculated:

S�(’′=’) =12(�S−1)2

(’′ − ’)2

�− �S−1L(’′ − ’)R’+

12�L2(R’)2 (38)

and

P�(’′=’) =

√(�S−1)2

2k&�exp(−12(�S−1)2

k�(’′ − ’)2

+�S−1Lk

(’′ − ’)R’− 12�L2

k(R’)2

): (39)

It is assumed that � is in!nitesimal, and now all �′; �′′ are of the same order. Thenotion R’ denotes the value of R’ in the point 1=2(’′ + ’). Let us calculate thefollowing integral:∫

d’′P�′′(’′′=’′)P�′(’′=’)

=∫

d’′ ×√

(�S−1)2

2k&�′′exp(−12(�S−1)2

k�′′(’′′ − ’′)2 +

�S−1Lk

(’′′ − ’′)R’

−12�′′L2

k(R’)2

)√(�S−1)2

2k&�′exp(−12(�S−1)2

k�′(’′ − ’)2

+�S−1Lk

(’′ − ’)R’− 12�′L2

k(R’)2

): (40)

200 K. Gamb�ar, F. M�arkus / Physica A 320 (2003) 193–203

Here, we used that∫ ∞

−∞exp(−ax2 + bx) dx =

√&aexp

b2

4a: (41)

After the integration we obtain√(�S−1)2

2k&(�′ + �′′)exp(−12(�S−1)2

k�′′’′′2 − 1

2(�S−1)2

k�′’2 +

12(�S−1)2

k

×(’′′

�′′+’�′

)2 �′�′′

�′ + �′′+�S−1Lk

(’′′ − ’)R’− 12(�′ + �′′)L2

k(R’)2

)

=

√(�S−1)2

2k&�exp(−12(�S−1)2

k�(’′′ − ’)2

+�S−1Lk

(’′′ − ’)R’− 12�L2

k(R’)2

)

=P�(’′′=’) ; (42)

which is precisely the Chapman–Kolmogorov equation. It can be seen that the parabolicprocesses are stochastic processes and these are Markov processes (see an original workabout the Markov processes [8]) on the new phase space.

4. Fokker–Planck equation

The Kramers–Moyal expansion [18] can be written for the probability density

�@W (’; t)

@t=

∞∑n=1

(− @@’

)n(Mn

n!W); (43)

where Mn means the nth momentum. The !rst and second momenta are

M1 =∫

d’′(’′ − ’)P�(’′=’) (44)

and

M2 =∫

d’′(’′ − ’)2P�(’′=’) : (45)

We can calculate these by the help of Eq. (39) and we obtain

M1 =L��S−1R’ (46)

and

M2 =k�

(�S−1)2+

L2�2

(�S−1)2(R’)2 : (47)

K. Gamb�ar, F. M�arkus / Physica A 320 (2003) 193–203 201

We neglect the second- and higher-order terms of � because it is a small !nite param-eter, and therefore we neglect the terms higher than second-order momenta. Now, wesubstitute both of the momenta M1 and M2 into the Kramers–Moyal expansion (seeEq. (43)). In consequence of the above results, we obtain a Fokker–Planck equation

@W (’; t)@t

=− L�S−1 R’

@W@’

+12

k(�S−1)2

@2W@’2 ; (48)

which is generally valid for parabolic diMerential equations. The solution of this partialdiMerential equation can be calculated

W (’; t) =(

2&kt(�S−1)2

)−1=2

exp

(− (’− (LR’=�S−1)t)2

(2k=(�S−1)2)t

); (49)

which is a Gaussian distribution in the space of the potential ’.

5. Regression of &uctuations

In this section, we examine the <uctuation of the measurable quantity �(x; t) =�(’;p). The <uctuation is the deviation from the steady-state �s, i.e.,

�� = � − �s : (50)

We express the time evolution of the <uctuation by the Poisson bracket

@��@t

= [��;H ] : (51)

We obtain the diMerential equation for �� by the help of H (p; �’); �(’;p) = �−1Spand Eqs. (26) and (27)

@��@t

=−�−1SLR(��) ; (52)

which has the same form as that of Eq. (20). After these, we can deal with theaverage regression of <uctuation of measurable quantity. We de!ne the conditionalaverage (similar to Refs. [3,22])

〈��=��′〉� =∫��P�(��1=��′) d(��1) ; (53)

where the <uctuation �� evolves from the <uctuation ��′ after the time �. Here, ��1

is an intermediate value of the <uctuation and P� is the transition probability. TheOnsager’s regression hypothesis asserts that Eq. (53) obeys the same time evolutionas the corresponding irreversible process. We would like to show that this conditionalaverage obeys a parabolic diMerential equation as Eq. (20) with the same coeScients.Now, we calculate the time derivative of the right-hand side of Eq. (53)∫

@��@t

P�(��1=��′) d(��1) ; (54)

202 K. Gamb�ar, F. M�arkus / Physica A 320 (2003) 193–203

where P� does not change during the time interval �. To continue the calculationwe should express P�(��′=��). In the !rst step, we can obtain the P�(�′=�) usingEqs. (19), (35) and (39), i.e.,

P�(�′=�) =

√(�S−1)2

2k&�exp(− �2k

(�′ − �)2): (55)

We take into account that

��′ = �′ − �s

and

�� = � − �s ;

then P�(��′=��) can be obtained

P�(��′=��) =

√(�S−1)2

2k&�exp(− �2k

(��′ − ��)2): (56)

After these we can express the time derivative by the help of Eqs. (52) and (56)@〈��=��′〉�

@t= �−1SLR〈��=��′〉 : (57)

If we compare this equation with Eq. (20) we recognize that the average regression of<uctuation of the measurable quantity � obeys the regression hypothesis.

6. Conclusion

We can conclude that in the description of stochastic behavior of parabolic transportprocesses the principle of least action and the Hamilton–Lagrange formalism may havecentral roles. We could calculate the transition probability, so we could deduce thestatistical properties of processes. The knowledge of Hamilton function of the systemenables us to calculate the diMerential equation for the time evolution of probabilitydensity function and to study the average regression of <uctuations.

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