Linear Response Theory for Non Equilibrium Stationary Systems

PHYSICA ELSEVIER Physica A 246 (1997) 78-96

Linear response theory for nonequilibrium stationary systems

Masakazu Ichiyanagi *

Department of Mathematical Science, Gifu University of Economics, Ogaki, Gifu 503, Japan

Received 18 November 1996

Abstract

A nonequilibrium statistical mechanics is developed by extending the Kubo-Nakano theory of linear response to spatially inhomogeneous systems under the action of external fields. It is argued that the standard linear response theory bases upon the fluctuation-dissipation theorem. The stationary density matrix of the nonequilibrium system is given by applying Zubarev's method. It is found that Onsager's reciprocity relations require a redefinition of the responses to external fields, reflecting the fact that the stationary density matrix is not of the standard canonical form with the unperturbed Hamiltonian. Following Nakano, a variational principle is developed for nonequilibrium stationary systems. A link of the variational principle with Machlup-Onsager theory is explored.

1. Introduction

One of the comerstones o f nonequilibrium statistical mechanics is the fluctuation-

dissipation theorem which represents a generalization o f Nyquist 's theorem to near-

equilibrium thermodynamic processes. From the current point o f view, the theorem

is interpreted as a very general relationship between response and fluctuations. We recall that Nyquist 's theorem, which bases upon the second law of thermodynamics, establishes a power balance between the rate at which energy is absorbed by the system

and the rate at which energy is dissipated in the system. This assumes that the system

cannot be able to store energy so that there do not occur noninstantaneous responses to external fields. In physical terms, this is guaranteed by the postulate that time scales of

extemal perturbations and of excitations are well separated. It is important to examine the close relationship between a dynamical change and a notion o f "adibatic switching procedure".

* E-mail: [email protected].

0378--4371/97/$17.00 Copyright (~ 1997 Elsevier Science B.V. All rights reserved PH S0378-437 1 (97)00339-7

M. lehiyanaoilPhysica A 246 (1997) 78-96 79

We preface our remarks by clearly stating the basis of the standard linear response theory [1,2]. In contrast with isolated systems, which can be described in terms of Hamiltonians, open dissipative systems are described by phenomenological equations of one sort or another. While a Hamiltonian description is based on first principles of physics, the use of phenomenological equations always involves some additional as- sumptions, whose validity is limited. For instance, linear response theory employs the linear approximation, where by "linear" we mean that only terms linear in the depar- ture from equilibrium are retained. It is clear that such linear approximations are based upon the fluctuation~lissipation theorem [3,4]. This theorem is proved for a Hamilto- nian system but can be applied to non-Hamlitonian systems with dissipation under the

assumption that the irreversible processes occurring in the system can be analyzed by making use of the fluctuation~lissipation theorem. Accordingly, it is easy to see that linear response theory can simply be justified on the basis that the Hamiltonian of the system determines not only the dynamics of fluctuations around a reference state but also the ensemble to be used [5].

The problem considered in this paper is how the Kubo-Nakano theory of linear response for systems near equilibrium is extended with respect to stationary reference states of an open system under the action of time-dependent external forces. It would be possible to develop a dynamical response theory for stationary systems which are described by invariant solutions of the Liouville-von Neumann equation. Hence, the first thing to be established must be the definition of stationary state to be considered. In this paper, it is assumed that a stationary density matrix is of the type considered by Zubarev [6]. The density matrix, in fact, is of a generalized canonical one. We use his method to define a stationary density matrix of spatially inhomogeneous systems in question. Then, we try to formulate a linear response theory for stationary systems, under the assumption of the fluctuation--dissipation theorem.

We develop our theory by following the old idea proposed by Takahasi [7]. It is based on the Gibbs formula relating fluctuating quantities with experimentally observable average quantities. A conceivable way to establish between the dynamics and the ensemble to be used is capitalized on the ergodicity property. The essential point here is that the dependence of this formula upon extemal fields is twofold; the dynamics of the system in question as well as the ensemble to be used depend upon exter- nally applied fields. Almost of the models used in this field, like those defined by the Liouville-von Neumann equation with time-dependent Hamiltonian, are not of this type. Here, we first solve the time-dependent Liouville-von Neumann equation in such a way as to reformulate the Gibbs formula. The aim of the present paper is to clarify the thermodynamic nature of the difference between the generalized canonical ensemble described by the time-independent Hamiltonian and the ensemble which is the solution to the Liouville-von Neumann equation.

It is important to show that such approaches are really a theory of irreversible processes and have a possible link with irreversible thermodynamics. A key to realize this can be found in Nakano's variational principle [8], where it is shown that the linear response theory is reformulated in terms of a variational functional whose stationary

80 M. lchiyanagi/ Physica A 246 (1997.) 78-96

value is equal to the rate of entropy production. Nakano's variational principle, which has an interesting connection with the classical Onsager theory of linear irreversible processes, is concerned with the time-dependent yon Neumann equations in the respec- tive cases that some incoming and outgoing external fields are applied to a macroscopic system which was in absolute equilibrium at remote past. This feature is quite similar to the occurrence of incoming and outgoing waves in the scattering theory developed by Lippmann and Schwinger [9]. It has been pointed out that the contraction of microscopic information is essential to reconcile the reversibility of the von Neumann equation with the irreversibility of macroscopic behavior [3,4,10].

In Section 2, by adopting the method by Zubarev [6] we define the stationary density matrix which is a solution of the time-dependent Liouville-von Neumann equation of an open system subjected to time-dependent external fields. In order to study the properties of fluctuations around the steady state described by the stationary density matrix obtained, a generalization of the response theory is given in Section 3. We find that the difference between the dynamical and the effective Hamiltonians destroys Onsager's reciprocity theorem for transport coefficients. Section 4 is devoted to develop a Variational principle similar to Nakano's. The final section is for the concluding remarks. In order to point out the importance of the fluctuation--dissipation theorem for the response theory, some remarks are added in the appendix.

2. A stationary density matrix

When the state of a system, surrounded by and interacting with large systems, does not vary in the course of time, the density matrix is stationary. The expectation value of any dynamical variable with respect to the stationary density matrix will also be stationary. Then, this leads to define an open system, subject to weakly time-dependent external fields Fi(t). For the sake of simplicity, let us suppose that

Fi(t) = Fiexp(st) (e ~ 0) for t < 0,

=F; , for t>~0. (2.1)

The total Hamiltonian of the system is given by

H(t) = H - ~ AiFi(t) , (2.2)

where H denotes the unperturbed Hamiltonian of the system and Ai are the physical quantities coupled to the external fields Fi. The problem, when the external fields do not explicitly depend on time, has been treated by Takahasi [7] by applying the Gibbs algorithm of equilibrium statistical mechanics.

Let us assume that the von Neumann equation for the density matrix p(t) is given by

~ p ( t ) + i [H - ~ AiFi(t),p(t)] = 0 , (2.3)

M. IchiyanagilPhysica A 246 (1997) 78 96 81

with the initial condition at t ~ -cx~: p ( - ~ ) = e x p ( F - f i l l ) , F denoting the

normalization constant so that Tr p ( - ~ ) = 1 and fl = (kT) -l , T being the temperature of heat bath. Eq. (2.3) is equivalent to

~ l n p ( t ) + i [ H - Z A i F i ( t ) , l n p ( t ) ] = 0 . (2.4)

We are interested in a steady-state density matrix Pst(t) which is the solution of Eq. (2.4);

[ln pst(t), H ( t ) ] -- 0 (2.5)

for t~<0. It is supposed in Eq. (2.5) that the total Hamiltonian H ( 0 ) ( = H - ~ A i F i ) is

invariant under the reversal o f time: H ( 0 ) -- TH(O)T-1 = H(0 ) , where T denotes the time-reversal operation. That is, the parities o f A i and Fi with respect to the reversal

o f time are same:A--i = TAiT -1 = ~iAi, and Fi = eiFi. Here, ei is +1 or - 1 according to the operator Ai is even or odd with respect to the reversal of time. In what follows,

the bar on the upper of any operator indicates the time-reversal o f it.

Following Zubarev [6], let us introduce the operators of the form

Gi(r, t) = U-l(t, -oo)Gi(r)U(t, -cx:~), (2.6)

in the Heisenberg representation. Here, the evolution operator U(t,s) is a solution to the equation of motion

(H Z AiFi(t)) U(t,s) (2.7) ~tU(t,s) = - i -

with the initial condition: U(t = s,s) = 1, at t = s. It follows that U-~(t,s) =- U(s,t). The operator G( r , t ) satisfies the equation of motion

~G(r , t )0 = i [H-~-~AiFi(t),G(r,t)] . (2.8)

From Eq. (2.8) it follows that

t

- - c )C~

for t ~< 0. Then, it is readily seen that

[H(t) , G(r, t)l = 0 . (2.10)

We have for all s

G ( r , t ; s ) = U-l(s ,-c~)G(r, t )U(s,-c~)

= U-l(s,-c~)G(r,-cxD)U(s,-cx~)

t

82 M. IchiyanaoilPhysiea A 246 (1997) 78-96

from which follows that

O ~s G(r, t; s) = i[H(s), G(r, t; s)] = 0.

Hence, G(r, t) = G(r, t; s -- 0) commutes with H(t). Consequently, we see that the steady-state density matrix Pst(t) (for t ~ 0 ) has the

form

In Pst(t) = Fst - Z / fli(r)Gi(r, t) dr . (2.11)

Fst is the normalization factor so that Trpst(t) = 1 and fli(r) are c-number functions to be specified.

We could formally solve Eq. (2.5) in the following form:

In p(t) = U(t, -oo) ln p(-cx3)u- l ( t , -oo ) = F - f l fI(t) . (2.12)

Here,

i¢-I(t) = U ( t , - c c ) t t U - l ( t , - c ~ ) . (2.13)

Hence, we postulate that the fli(r) are determined by the conditions

Tr Pst(t)Gi(r, t) -- Tr p(t)Gi(r) (2.14)

for t ~< 0. This procedure to define the stationary density matrix bases on the assumption that the surroundings are very large compared to the system in question so that an approximate stationary state can be attained in which macroscopic variables referring to the system change slowly enough in time. Taking the derivatives of both the sides of Eq. (2.14) with respect to t yields

Tr Pst(t)[iH(t), Gi(r, t)] = Tr p(t)[iH(t), Gi(r)]. (2.14a)

Eq. (2.14a) implies that the observation level t2 = {Gi; i = 1,2 . . . . . f } is sufficient in view of the fluxes, [ill, G/(r)], when

Tr Pst(0)[iH, Gi(r, 0) ~ Tr p(0)[iH, Gi(r)] (2.14b)

holds to a good approximation [11]. It is easy to see from Eq. (2.5) that for a statistical entropy for t~>0

S(t) = - k Tr Pst(t)ln Pst(t) (k; Boltzmann's constant)

-- - k Tr p(0)ln Pst(0) • (2.15)

There is no change of the statistical entropy. It should be so, because we are considering a stationary process. However, there are constant dissipative fluxes flowing in the open system in question and they generate heat.

Let us examine this point a little closer. The heat generated by the dissipative fluxes is considered to transfer from the system to its surroundings. It is here assumed that disturbances of the states of the surroundings equalize practically in zero time and

M. Ichiyanagi/ Physica A 246 (1997) 78-96 83

the surroundings have a well-defined temperature which is taken as the temperature of the system. Hence, the second law of thermodynamics tells us that, as far as the assumption holds, the irreversible change of entropy only occurs in the surroundings for stationary processes.

In this paper, we wish to evaluate the "dynamical" entropy production in terms of a relative entropy [12], which measures an entropic distance between two states. The word "dynamical" is in quotation marks because the entropy production is only positive in the mean [13]. Let us introduce the relative entropy of the form

S[p(O)lp(-oo)] = k Tr p(0)(ln p(0) - lnp(-c~))]/> 0. (2.16)

The relative entropy is not negative by the Klein inequality. By introducing Eq. (2.12) with Eq. (2.13) into Eq. (2.16), one gets at once

0 1 / S[p(O)IP(-°°)] = ~ Z Ji(t)Fi(t)dt. (2.17)

--00

Here, the macroscopic fluxes Ji(t) are defined in terms of the exact density matrix p(t) a s

Ji(t) = Tr p(t)[iH, Ai] . (2.18)

As can be seen from the integrand of the right-hand side of Eq. (2.17), the relative entropy is equal to the time integral of the total amount of the Joule heat, ~ Ji(t)Fi(t), generated by the extemal fields.

It is convenient to introduce the operator

with

fl / t( t) = flH - ~ ( t ) , (2.19)

!

~(t) = fl f U(t,s) [ill, Z AiFi(s)] U-l(t,s)ds. (2.20)

Making use of Eq. (2.19), together with Eq. (2.20), into Eq. (2.18) yields

J~(t = 0) = Tr p(0)[i@(0), Ai]/fl 0

i Tr / dt Z [ i H , dk] U-l(0, t)[Ai, p] U(O, t)Fk(t). (2.21 )

This result, a generalized response, shows clearly that the macroscopic fluxes depend nonlinearly upon the external fields Fi(t) [12].

Before concluding this section, it is pointed out that the solution to the strong stationarity condition, Eq. (2.5), could be given by

Pst(0) : exp[f - f i l l (0)] , (2.22)

84 M. Ichiyanagi/Physica A 246 (1997) 78-96

where e x p ( - f ) = Trexp(-f lH(0)) . This is the ensemble employed by Takahasi [7]. However, this is one of the (possibly many) stationary density matrixes characterizing a dissipation of the open system in question. The use of it does not destroy the principle of microscopic reversibility (see the appendix). We conclude that this density matrix is uninteresting since there are qualitative differences of fluctuations occurring from equilibrium or a nonequilibrium stationary state.

3. A response theory

The linear response theory is based upon the following idealized experiment [7]. A dynamical system, canonically distributed in the infinite past, is adiabatically removed from an absolute equilibrium by the time-dependent external fields, Fi(t) which are coupled to the dynamical variables Ai. At t = 0 these fields are suddenly decreased from their initial values by the amounts f and the relaxation back to the stationary state begins. In this event, we can treat the external fields, f , as a small perturbation. Here, we assume that the unperturbed stationary state is described by the density matrix p obtained in the last section.

Let us consider the von Neumann equation

O ,

and its equivalence

-~lnp'(t) + i [ H ( t ) - Z Aif(t),lnp'(t)] = 0 , (3.1a)

where

f =fexp(st) (s + 0) for t~<0,

= 0 for t > 0. (3.2)

We set the initial condition of Eq. (3.1a) as

p'(t) ~ p -- pst(t), ast --* - c ~ , (3.3)

where Pst(t) is given by Eq. (2.11). We shall solve Eq. (3.1a) in the form

In p'(t) = In p + ¢( t ) . (3.4)

Then, we shall rewrite Eq. (3.1) in the following form:

~t¢(t ) + i[H(t), ¢(t)] = - iOn p, Z Air(t)] (3.5)

up to the first order o f f ( t ) . This is an analogy with the linear response theory for the near equilibrium systems.

M. Ichiyanagi/ Physica A 246 (1997) 78-96 85

It is assumed that the two parameters, e and s, satisfy the condition; e <is. This means, in practice, that the variations of fi(t ) in time are much faster than those of Fi(t). We also assume that the magnitudes of fi are much smaller than those of Fi. Accordingly, we can replace Eq. (3.5) by

~ q~(t) [ln p, ~-'~ Aifi(t)] (3.5a) + i [H(O) , q~(t)] = - i

in a good approximation, since the time variation of qS(t) is much faster than that of H(t) .

Let us define the relative entropy in this case as

S[p'(t)[Pst(t)] = k Tr p'(t){ln p'(t) - In pst(t)} (3.6)

which is equal to

= kTrp'(t)4)(t)>~O. (3.7)

Then, for t ~< 0

d s [ p ' ( t ) p ] = ( - ik ) y ~ Tr p'(t)[ lnp, Ai]f i ( t ) . (3.8)

This can be interpreted as an excess entropy production in the system due to the perturbation f i( t) . Eq. (3.7) suggests that the effective dissipative fluxes in the stationary state are required to identify as

Bi = i1,Ai, (3.9)

where

~ - k T [ l n p ] . (3.10)

The time derivative of the entorpic distance from the nonequilibrium state characterized by p'(t) from the steady state described by Pst(t) is expressed in terms of the effective dissipative fluxes.

We shall put

~(t) = qt(t) + ~ ( t ) , (3.11)

where

while

~(t) is given by a solution of the equation of motion

~ ( t ) + i[H(0), ~(t)] = - i [ k T l n p + H(O), qt(t)],

qt(t) satisfies

+ i / .o ( t ) = - Z n , , ( t ) .

The initial conditions for the two equations, Eqs. (3.12) and (3.13), are

~(t --+ -oc ) = 0 and qt(t ---+ - o c ) = 0.

In this way, we can replace Eq. (3.5) in the set of Eqs. (3.12) and (3.13).

(3.12)

(3.13)

(3.14)

86 M. Ichiyanaoi/Physica A 246 (1997) 78-96

Note that the ~( t ) would vanish if the canonical ensemble to be used is completely determined by the Hamiltonian H(0) as was the case of the linear response theory for near equilibrium systems. Hence, we are interested in the observable effect due to the difference between the dynamical Hamiltonian H(0) and the effective Hamiltonian kT In p, p being the stationary density matrix obtained by applying Zubarev's method [6].

A formal solution of Eq. (3.12) is given by

t

- i / dt'V(t,t ')[krlnp + H(t ') ,~(t ')]V-l(t , t ' ) , (3.15) ~(t) - o o

where [6]

V(t, t ') -- exp{iH(0)(t - t ' )}. (3.15a)

We shall write a formal solution of Eq. (3.13) in the following form:

t

fl Z / dt' exp{-iL(t' - t)}Bif(t ') (3.16) ~(t) - - 0 0

for t~>0. The expression (3.4) for the nonequilibrium density matrix is rewritten in the form

( J ) p'(t) = 0 1 + dxp-X~(t)p x (3.17)

o

up to the lowest order o f f . We are interested in the quantities defined by

Aei(t) = Tr p'(t)Ai - TroAi

= ((~k(t),Ai)) + ((~(t) ,Ai)) , (3.18)

where we have defined the inner product as

1

C)) = Tr / dxpl-xC+pX.4 (3.19) ((A, J 0

for any operators .4 and C of the system. C + denotes the Hermite conjugate of C. The first term of the last equation of Eq. (3.18) defines the time-correlation function expression for the response of the quantity A; of the stationary system to the perturbation

In order to see this, let us introduce Eq. (3.14) into Eq. (3.16). Then it is readily seen that

0

= - i f l ' f f~ . / dfl((/lk, .4i(t -- d)))fk exp(st') + ((~(t) ,Ai)) , (3.20) A~i(t)

- o o

M. IchiyanagiIPhysica A 246 (1997) 78-96 87

where the effective Heisenberg operator is defined by

,'ti(t) = exp(iLt)Ai . (3.2l)

It is noted here that the operator I, has the property

((&, LAk )) = - ( (LBi , & )) . (3.22)

The first term on the right-hand side of Eq. (3.20) shows the dynamical response of the system to the external perturbation, which is exactly analogous to the concept treated by the linear response theory, while the second and third members represent the kinematical response, we shall call it.

In order to see the implication of the kinematical response, let us use the general result, Eq. (3.18), and evaluate the response currents for t>~0:

0

ABi(t) = [3 Z f d t t ( (Bk,Bi( t - t')))fk exp(st t) - - 0 0

+ ( (~( t ) , / l i ) ) . (3.23)

The first term on the right-hand side of Eq. (3.23) provides a time-correlation function expression for the "transport" coefficients,

0

Lik = fl / p ,

dt((/lk, B, ( - t ) ) )exp(st )

= L,i • (3.24)

The "transport" coefficients, by the definition, enjoy a reciprocal relation similar to Onsager's theorem. We are interested in the quantity ((~(t) ,Bi)) involved in Eq. (3.23). By making use of Eq. (3.15) into it, we get

0

((~(0),/l~)) = dt((i[kTlnp + H(t),O(t)],B~(t))).

o

= - d t ( ( i [kT In pH(t), B~(t)l, ~ , ( t ) ) ) . (3.25)

Here, Bi(t) is

~i(t) = V -1 (0, t)l~i V(O, t). (3.26)

By making use of Eq. (3.16) in Eq. (3.25), we have

( ( ~ ( o ) , l L ) ) o t

=-flZ i dt / dt'((i[kTlnp+H(t),~k(t)],JBi(t'-t~(,'))). (3.25a)

- - O O - - D O

88 M. Iehiyanagi/ Physica A 246 (1997) 78-96

This is the second part of the response of Bk to the perturbation, ~ Air(t) , considered and thus suggests the definition

0

Mki[F] =- f l S - - 0 0

t

dt / dt'((i[kT In p ÷ H(t), ~k(t)], Bi(t' - t)exp(st'))). (3.27)

- - 0 0

Then, combining Eqs. (3.23), (3.24) and (3.26) yields

A l O e : Z(Zik ÷ Mzk[F]))~ • (3.28)

It is emphasized that the coefficients Mik go to zero as all the Fk tend to zero but are not symmetric with respect to the suffixes. We thus cannot have a reciprocity theorem for the transport coefficients defined by Eq. (3.28), like Onsager's, for nonequilibrium stationary processes in the open system subjected to the finite external perturbations. The origin of the failure of reciprocity relations for the transport coefficients defined by Eq. (3.28) stems from the fact that the Hamiltonian for the microscopic dynamics, H(t), in general, differs from the effective Hamiltonian, - kT lnp , which determines the stationary ensemble to be used. Note that the density matrix p is for the spatially inhomogeneous system in question and remember that the standard linear response theory bases upon the special situation in which the Hamiltonian plays the double role; that is, it determines both the dynamics of fluctuations and the equilibrium ensemble to be used. Incidentally, if one neglects the difference between the two Hamiltonians, H(0) and kT In p, the present formulation becomes identical to Takahasi's theory [7].

An alternative interpretation can be given for the responses given by Eq. (3.28). We could follow the analogy to the idea given by Onsager and Machlup [14] and rewrite it as

= ~ Lik~k, (3.29)

with the generalized forces defined by

Ck = fk + Z ( L - 1 M ) k j f j . (3.30)

Here, L and M denote the matrixes whose elements are given by Lik and Mik, respectively. The reciprocity theorem holds for the coefficients L~k. The second numbers on the right-hand side of Eq. (3.30) might be able to describe the relaxation phenomena towards the spatially homogeneous stationary state of the system. If one wishes to make further development in this line of thought, one will have to deal with them.

4. The variational principle

In order to explore the link of the developed response theory with irreversible thermodynamics, we have to calculate irreversible production of entropy. In this section, we will formulate a quantum variational principle for the von Neumann Eq. (3.1).

M. IchiyanagilPhysica A 246 (1997) 78-96 89

Following Nakano [8] we consider two boundary conditions of incoming and outgoing perturbations for Eq. (3.1):

(1) The time dependence of the perturbation is

f/(+)(t) =fexp(st), t~<0 (s J, 0) (4.1a)

and the density matrix is required to satisfy the condition

lira p'(t) = p. (4.2a) l----~ - - O O

(2)

f ( - ) ( t ) =fiexp(-st) , t > 0, (s ~ 0) (4.1b)

and

lira p'(t) =/5. (4.2b) t ----* ~

Here, in Eqs. (4.2a) and (4.2b), p is the steady-state density matrix defined by Eq. (3.3) and the bar above the operator denotes its time-reversal operation.

The von Neumann equations for the two cases are

, ~ p ( t ) + i [ H ( t ) - Z Aifi(+)exp(±st)'p'(t)] = 0 . (4.3)

or equivalently

~lnp ' ( t )+i[H( t ) -~Ai f i (+)exp(+s t ) , lnp ' ( t ) ] = 0 . (4.4)

Again, as in the previous section, we shall write

In p'(t) = In p + {~(+) + @(i)}exp(±st), (4.5)

for the two cases (1) and (2), respectively. Inserting Eq. (4.5) into Eq. (4.4) yields

:t: e/tP (±) + i[H(0), ~P(+)] = i[kTlnp + H(0), @ (+)] (4.6)

and

( i s + iL)@ ( i ) : - i [ln p , ~ Aif i] , (4.7)

up to the approximations considered in the previous section. Here, in Eq. (4.7), /~ is defined by Eq. (3.10). The formal solution of Eq. (4.7) is given by

, , i , = - i (+s + iL) -1 [lnp, Z Air] . (4.8)

By making use of Eq. (4.8) in Eq. (4.6), we get

{+s + iL(0)}'P (+)

. + iL)- . (49)

90 M. Ichiyanagil Physica A 246 (1997) 78-96

where

iL(0)~P (+) = i[H(0), ~(+)]. (4.10)

Following Nakano [8], the variational principle can be formulated as follows. Find the operators q)(-), q~(+),8 (-) and 8 (+) which make the variational functional

W[~(+), (p(-); 8(+), 8(-)] _= ((q¢-), iLl,o(+))) + ((8 (-), iL~(0)8(+)))

- y ] ( ( i [ ln p, Ai], ~o (+) - ~o(-))~

A - - 1

+ ~ ( ( i [ k T In p + H(0), -L~ [In p, Ai]], 8 ( - ) ) ) f

Z ( ( i [ k T l n p + ^-l - H(O),L~ [lnp, Ai]],8(+)))f (4.11)

stationary. Here, the inner product is defined by Eq. (3.19), and we set

iL,(0) = s + iL(0) and i/~,~ = s + iL. (4.12)

It is stressed here that, to define the variational functional, we have to use the two operators, L~(0) and i/~, since the Hamiltonian H(0) only determines the motion of fluctuations around the stationary state described by the p. As has been pointed out in the previous section, this implies that the usual fluctuation-dissipation theorem for near equilibrium systems, which stems from the fact that the unperturbed Hamiltonian H determines not only the motion out from equilibrium but also the equilibrium ensemble to be used, cannot be extended to the nonequilibrium steady state considered. The second point is that the variational principle defined by Eq. (4. l 1) involves the operator L~ which may be evaluated by using the ordinary variational principle [15]. These internal variational calculations should be carried out to the same order of approximation as the external variational calculations.

The operators satisfying this stationarity condition are equal to the solutions, qt (+) and ~(+), of the von Neumann equations, Eqs. (4.8) and (4.9), respectively. The value of the functional W[~k (+), ~(-); ~(+), ~(- ) ] is equal to

w[¢~+), q/-); ~(+), t/,(-)] = ~-~((i[ln p, Ai], ~k(-)))f

+ Z ( ( i [ k T In p + H(0), i/~-I [ln p, Ai]], ~ ( - ) ) ) f . (4.13)

It is readily seen that from Eq. (4.8)

~(-)t = _qt(+)(~ - O ) , (4.14)

and from Eq. (4.9) the similar formula for ~(+) and ~'(-), It will be convenient to redefine the inner product as

1

((A;B>) = - T r / d x p l - X - B + p X A . (4.15) J

0

M. Ichiyanagi/Physica A 246 (1997) 78 96 91

This definition has been given by Nakano [16]. If we replace all the inner products involved in functional, W[cp(+),q~(-);8~+),8(-)], with new inner products defined by Eq. (4.15), we get the expression

W[~o; 8] = -((q~; i/,sq~)) - ((8; iL~(0)8))

+2 ~((i[ln p, Ai]; q~))f

-2 i ~ ( ( i [ k T l n p + H(O),iL-][lnp, Ai]], 8 ) ) f . (4.16)

Let us decompose an operator A into A' and A" which are, respectively, the odd and the even parts of A under the time reversal operation;

A = A / + A", A-- = - A / + A" . (4.17)

Then, the variational functional (Eq. (4.16)) are rewritten in the following form:

W[~o', q~"; 8'8"]

= -2((~o"; ii.sq~')) - 2 ({8"; iLs(0)~)'})

+~{((,p'; ,p')) + ((3'; 8")) - ((u,"; ,p")) - ((8"; 8"))}

+2 Z ( ( i [ l n p, Ai]; q~' - (s + iL)-~(s - if,)-~iL[i(kT In p + H(0)) , 9 ' ] ) ) f

- 2 s Z ((i[ln p, All; iL,~-' i/,5~ [i(kT In p + H(0)), ~)"]))f. (4.18)

As emphasized by Nakano, in Eq. (4.18) only the odd component of q~' directly couples with the external fields f , and the even component q¢' does not.

In order to see the thermodynamic implication of the above variational principle, we shall make an approximation in which we set LH(0) ~ 0. At the end of Section 2, we have seen a possible reason for this approximation to define the quasi-stationary density matrix. Then, within this approximation, we have from Eq. (4.11),

W[U,; 8] = -((q,;iLsU,)) - ((8;iL,.(0)8)) + 2 ~ ((i[lnp, A i ] ;~) ) f , (4.19)

where

: q~ + £~ l [kT lnp + H(0) ,8 ] . (4.20)

This suggests the expressions for the dissipation functions [17]

2,/~[q~; U'] = ((~o; i/,sq~)), (4.21)

2~[8; q~] = ((8;iL~(0)~))} , (4.22)

and the entropy production

2S[{,f] = Z ((i[ln p, Ai]; ~ } ) f . (4.23)

Then, the variational principle can be reformulated in the following form.

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Make the functional

W[q~; 8] = -2{~[q~; q~] + ~[S; 8] - S[~,f]} (4.24)

stationary with respect to the variations of ~p(= ~p' + ~p") and 8(= 91 + 911). Then stationary condition for Eq. (4.19), with respect to the variation of ¢p", soon

leads to the solution

~p" = - i£sqY /s . (4.25)

While, in the approximation considered, the variation of 9" leads to the relation

81' = -iL(O)O'/s + A0 (4.26)

where

1 ^ - - 1 A0 = + i ~ L s ( 0 ) - L_s[kT lnp + H(O),i[lnp, A i ]] f . (4.27)

By making use of Eqs. (4.25) and (4.26) in Eq. (4.19), we obtain

W[~pl; 81] = ((iLq~l/s; i£~p/)) + s ((~pt; ~p/)) + 2 ~ ((i[ln p, Ai]; qol))

+ ((iL(O)8'/s; iL(0)81)) + s ((81; 8/))

^ - - I ^ - -1 ^ t +2i /(i[ln p,.4/]; Ls L_, [kr In p + H(O), ]))S

+2s ~ <<[Ag; AS]>> . (4.28)

By defining the operators/} and D as

b = (iL)Z/s - s, D = (iL(O))Z/s - s , (4.29)

the variational functional, Eq. (4.28), can be rewritten in the following form:

W[~o'; S'] = -<<q~'; bq~')) - 2 Z <<i[ln p, Ai]; qJ')) f - - <<S'; DS'))

- 2 i Z ((i[ln p, Ai]; L s l L - ~ [kT In p + H(0), iLS']))f

+2s ~ ((AS; AS>> . (4.30)

The last term on the right-hand side of Eq. (4.30) goes to zero as s J. 0. Note also that the third term of it can be rewritten as

/(i[ln p, A;]; (s + i/~)-l(s - iL) -1 [kT In p + H(0), i£3']))

= ((i[lnp, i£Ai]; (s + iL) - l ( s - i £ ) - l [ k T l n p + H(O), S'])) , (4.31)

in which the operators (i/~)(i/~Ai) appear, when we use Eq. (2.8), together with Eq. (2.7), for p.

As has been demonstrated by Nakano and Hattori [10], the new form of the variational principle is an extremal principle, a s /9 is positive and Hermitian. However, the

M. Ichiyanaoi/ Physica A 246 (1997) 78-96 93

variational functional involves iL -1, which may be calculated by using a variational principle.

By utilizing these results in Eq. (4.24), after a simple manipulation we arrive at

2~[~p'; ~p'] = ((~p'; b~p')), (4.21a)

2~[11'; 0'] : ((~1'; bl~')) (4.22a)

and

2S[~';f] = Z ((i[ln p, Ai]; q~'))f

+ E ( (i [ln p, i£Ai ]; (s + i/~ ) - 1 (s - i L )- 1 [kT In p + H(0), I)']) ) f .

(4.23a)

For instance, from Eq. (4.19), we get at once

(' : q~' + i(s + iL ) - l ( s - iL) -~

× { - s[kT ln p + H(O),~)"] + iL[kT ln p + H(O),~)']}. (4.32)

The main conclusion following from the consideration presented in this section is that the variational principle is of Machlup-Onsager's form if one accepts the interpretation that the "masses" mij in the latter theory be related to Mij defined by Eq. (3.27). The M/j occur because the stationary density matrix p is never of a canonical form expressed in terms of H(0). Machlup and Onsager made no comment on the origin of their masses.

5. Concluding remarks

The lack of generally accepted theory of dynamical responses to external fields for systems out of equilibrium has long been a blemish of statistical mechanics of irreversible processes. The viewpoints of irreversible thermodynamics and statistical mechanics should be drawn together in the framework of a simple formalism like the Kubo-Nakano theory. In this paper we have tried to focus our attention to the role of the fluctuation-dissipation theorem (or the Onsager reciprocity relation) on the linear response theory and have shown how the linear response theory, the Kubo--Nakano theory, for near equilibrium systems is extended with respect to stationary states of an open system under the action of finite external forces. In the introduction we noted that the Kubo-Nakano theory of linear responses bases upon the ordinary fluctuation- dissipation theorem. The author agrees, however, that the statistical mechanics is more fundamental than the irreversible thermodynamics and holds out more hope for an eventual physical interpretation of the responses to external perturbations.

It has been pointed out by Lavenda [18] that, without the Markovian assumption, time-dependent external fields invalidate the fluctuation-dissipation theorem. However,

94 M. lchiyanaoi/Physica A 246 (1997) 78-96

there remains much to be worked out along this line of thought. Although the Kubo- Nakano theory may be extended with respect to some reference states, it is difficult to state theoretical criteria for the validity of a linear approximation. One suspects that essential features would include requirements that the time scale of external fields be much larger than that of the system dynamics and that, in a thermodynamical sense, the reference state is stable against the external forcing. The most troublesome task in developing a formal response theory is the treatment of Heisenberg operators which depend on all the times in the history between an initial and a final times. Recently, a time-dependent projection technique has been used by Heiner [19,20] to evaluate response functions which contains the projection onto an observation subspace and the orthogonal projection onto a complementary subspace. He suggests that if the time scales of observation and of elementary excitations are well separated the notions of the Kubo--Nakano theory could be extended to systems far from equilibrium. Of course, it is an old and established idea that a separation of time scales is necessary in describing dissipative processes on the basis of reversible microscopic dynamics [21].

The second theme invited in the present paper is to develop a variational principle for irreversible processes and to exploit the link with irreversible thermodynamics. An attempt to formulate such a theory for nonequilibrium stationary states, however, has been only partially successful. In the process, we have to introduce generalized forces, in terms of which the transport coefficients are expressed. We are led to discover that the transport coefficients satisfy Onsager's reciprocity relations. This may be an indication that the present extension of the Kubo-Nakano theory with respect to the reference state can be based on Machlup-Onsager's theory of irreversible processes.

Appendix. The canonical ensemble

In this paper, we have been interested in stationary states of a macroscopic system characterized by the extensive quantities, ,4 i (i = 1,2 . . . . . n). When this system is treated by Gibbs' method of the canonical ensemble, the Hamiltonian H =/-/[,4i] of a certain configuration will depend on the variables, Ai. Then, the quantities, -~3H[{Ak}]/SAi,

are the generalized forces in Gibbs' terminology and are denoted by Xi. Next, let us consider the case in which the time-independent external fields Fi are applied to the system, the Hamiltonian of which is

H'[Fi] = H[A,] - ~ A iF , . (A. 1)

Then, the basic thermodynamical relation is

T dS = d H ' + T ~-'~'Xid~i . (A.2)

Here, bars denote averages in the canonical ensemble determined by the Hamiltonian Hr[Fi], S denotes the entropy and T is the temperature of the system in question.

M. Ichiyanagil Physica A 246 (1997) 78-96 95

Hence ai = A i and doti = dA i. In general, Xi are the functionals of ~i. The free energy F = -H' - TS of the system is given by

exp(-fiE[a, T]) --- / exp( - f lH ' [F , ]) d F . (A.3 )

Here the integral is over the F space. Hence, at constant energy, Eq. (A.2) is equivalent to

( ~ S ~ , (A.4) x i [ { a } ] = , , ? - ~ , / ~ ,

The infinitesimal work performed on the system is given by

dH ' = Z Fidai. (A.5)

From Eq. (A.1) and (A.4) we have the simple relation [21]

dS = ~ ( - ~ + ' i ) d~q-. (A.6)

Accordingly, we are interested in the stationary states which are characterized by the entropy maximum, dS = 0, for which

Fe - - T + Xi[{~ti}] = 0 (for all i ) . (A.7)

Hence, by differentiation with respect to the averages ak, we have

oei Tc3-Xi[{~j}] 3ak O~

or equivalently

eai ( c~2S ~ - ' rc~Fk -- \ ~ f . (A.8)

This is the form of the fluctuation-dissipation theorem which relates the curvature of entropy with the responses, O=k/c~F~. It is readily seen that the left-had side of Eq. (A.8) is given by

T ~ 7 = AiAk - aiak , (A.9)

which are the canonical anomalies in the Gibbs terminology. By taking advantage of the deterministic character of the systems, Takahasi [7]

showed that Gibbs's fluctuation formula, Eq. (A.9), can be extended to a formula for the after-effect functions. Indeed, we can rewrite Eq. (A.9) as

t?ai( t + x) T -- Ai( t + *)Ak(t) - ai(t + ~)ak(t) (A.10)

gFk(t)

tbr the time-dependent quantities, a i ( t ) ( = Ai(t)) . He used this equality to prove Onsager's reciprocity theorem.

96 M. Ichiyanagi/Physica A 246 (1997) 78-96

It is easy to see that Eq. (A.10), indeed, is a generalization of Eq. (A.9). To see this, for brevity, let us consider the monochromatic forces

Fi(t) = Fi(og)exp(-kot) . (A.11 )

Then, we postulate that Eq. (A.6) can be written as

in the frequency domain, where we have set ~i(t) = ati(~0)exp(-iot). Accordingly, for stationary systems, we have

E ( o ) + ~ (o~) (for all i) (A.13) 0 - T

which is a generalization of Eq. (A.7). For the near equilibrium case, we assume that

-Xi((D) --~ -- ~ ( g - l((D))ikOtk((D), (A. 14) k

where g(o)) denotes the correlation matrix whose elements are given by

gik(e)) = Ai(e))Ak(-co) - ati(co)~tk(-o)). (A.15)

Consequently, we conclude that Takahasi's postulate (Eq. (A.10)) bases upon the fluctuation-dissipation theorem in the sense of de Groot and Mazur [21].

References

[1] H. Nakano, Prog. Theor. Phys. 15 (1956) 77; 17 (1957) 145. [2] R. Kubo, J. Phys. Soc. Japan 12 (1957) 570. [3] H. Nakano, Int. J. Mod. Phys. B 7 (1993) 2397. [4] M. Ichiyanagi, Phys. Rep. 262 (1995) 227. [5] M. Ichiyanagi, J. Phys. Soc. Japan 51 (1986) 2963. [6] N.D. Zubarev, Nonequilibrium Statistical Thermodynamics, Plenum, New York, 1974. [7] H. Takahasi, J. Phys. Soc. Japan 7 (1952) 439. [8] H. Nakano, Proc. Phys. Soc. 82 (1963) 757. [9] B.A. Lipmann, J. Schwinger, Phys. Rev. 79 (1950) 469.

[10] H. Nakano, M. Hattori, Prog. Theor. Phys. 83 (1990) 1115. [11] E. Fick and G. Sauermann, The Quantum Statistics of Dynamic Processes, Springer, Berlin, 1990. [12] M. Ichiyanagi, J. Phys. Soc. Japan 61 (1992) 37. [13] I. Ojima, H. Hasegawa, M. Ichiyanagi, J. Stat. Phys. 50 (1988) 633. [14] S. Machlup, L. Onsager, Phys. Rev. 91 (1953) 1512. [15] See, e.g., M. Ichiyanagi, Phys. Rep. 2,43 (1994) 125. [16] H. Nakano, Prog. Theor. Phys. 49 (1973) 1503. [17] L. Onsager, Phys. Rev. 37 (1931) 405. [18] B.H. Lavenda, J. Math. Phys. 21 (1980) 1826. [19] L. van Hove, Physica 21 (1955) 517; 23 (1957) 441. [20] See also, I. Ojima, J. Stat. Phys. 56 (1989) 203. [21] S.R. de Groot, P. Mazur, Non-Equilibrium Thermodynamics, North-Holland, Amsterdam, 1962.

Linear Response Theory for Non Equilibrium Stationary Systems

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