数理解析研究所講究録 1227 巻 2001 年 199-208

199
On Steady-State Entropy Production of A One-Dimensional
Lattice Conductor
早大応物

田崎秀一

(Shuichi Tasaki)

Advanced Institute for Complex Systems andDepartment of Applied Physics, Waseda Univ.

I. INTRODUCTION

The understanding of irreversible phenomena including nonequilibrium steady statesis alongstanding problem of statistical mechanics. Since general features of irreversible
phenomena are not well understood, rigorous approaches are important.
In their purely dynamical study on nonequilibrium steady states for aclassical infinite harmonic chain, Spohn and Lebowitz [1] used semiinfinite left and right segments as
reservoirs. They showed that any initial state, where the left and right reservoirs are in

equilibrium with different temperatures, evolves towards asteady state with nonvanishingenergy current. Recently, following the same line of thoughts as Spohn and Lebowitz, andapplying the method of ’-algebra, Ho and Araki [2] proved the approach to nonequilibrium$\mathrm{C}$steady states for an isotropic XY-chain,

As the works by Spohn-Lebowitz [1] and HO-Araki [2], we studied nonequilibrium steadystates for aone-dimensional conductor with the aid of the C’-algebra [3]. Left and rightsemiinfinite segments of the lattice are assigned for electron reservoirs. Initially the tworeservoirs are set to be in equilibrium at different temperatures$\mathrm{a}\mathrm{n}\mathrm{d}/\mathrm{o}\mathrm{r}$different chemical

potentials. The evolution of the initial states for $tarrow\pm\infty$ was investigated and two differ-

ent quasi-free steady states$\omega_{\pm\infty}$

were obtained. Transports and current fluctuations were

investigated.
The steady state$\omega_{+\infty}$

carries nonvanishing electric and energy currents, which agree with

the nonlinear generalization of the Landauer conductivity and which are consistent with the200
second law of thermodynamics [3]. The state$\omega_{+\infty}$

is equivalent to the nonequilibrium steadystate proposed by MacLennan [4] and Zubarev [5]. The other steady state
thermodynamical currents and is the time-reversed state ofspace of states” , the state$\omega_{+\infty}$$\omega_{+\infty}$

behaves as an “attractor” and$\omega_{-\infty}$

carries anti-. Roughly speaking, in.(a$\omega_{-\infty}$

as a“repeller” Andinitial states evolve unidirectionally from the “repeller” to the “attractor” in away consistent
with dynamical reversibility.

Now it is desirable to introduce and study entropy production as its positivity is the very

definition of irreversible processes. However, definition of nonequilibrium entropy and its
production is still controversial. And the related works are classified into two. On the one
hand, an appropriate entropy is introduced and its derivative is calculated. For example,

Ojima, Hasegawa and Ichiyanagi [6] defined entropy production for driven systems as thetime-derivative of relative entropy with respect to the initial state (see also Ichiyanagi [7] andOjima [8] . For other examples, see e.g., Ref. [9]. On the other hand, an entropy production$)$

is directly introduced based on thermodynamic considerations. Along this line of thought,

Spohn and Lebowitz [10] investigated an entropy production of systems weakly coupled withreservoirs in the scaling limit and found that it can be characterized as atime-derivative of

arelative entropy. Recently, Ruelle [11] investigated entropy production of nonequilibrium
steady states of spin systems within the framework of C’-algebra and showed its positivity.
In this article, as in the work of Ruelle [11], we study the entropy production of thesteady state$\omega_{+\infty}$

and show that it has properties fully consistent with nonequilibrium ther-modynamics. Sec. II is devoted to the summary of the previous results [3]. In Sec. Ill,we generally discuss the possible expressions of entropy productions. In Sec.
late the entropy production of the steady state$\omega_{+\infty}$$\mathrm{I}\mathrm{V}$, we calcu-

and show that it is non-negaive andvanishes only when two reservoirs are in equilibrium with each other, and that it has aknown quadratic form in the linear response regime. All those features are fully consistent

with nonequilibrium thermodynamics. Sec.

$\mathrm{V}$is devoted to the summary and concluding

remarks, where the relation between entropy production and relative entropy is discussed

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II. MODEL AND NONEQUILIBRIUM STEADY STATESThe system in question consists of electrons on an infinitely extended chain interactingwith alocalized potential and is defined on aC’-algebra as follows.
The basic dynamical variables are creation and annihilation operators,

spectively, of an electron at site$j(\in \mathrm{Z})$$c_{j,\sigma}^{*}$

and$c_{j,\sigma}$re-with spin $\sigma(=\pm)$ . They satisfy the canonicalanticommutation realtions (CAR):$[c_{j,\sigma}, c_{k,\tau}]_{+}=[c_{j,\sigma}^{*}, c_{k,\tau}^{*}]_{+}=0$,$[c_{j,\sigma}, c_{k_{7^{-}}}^{*},]_{+}=\delta_{jk}\delta_{\sigma\tau}1$,(1)where $[A, B]_{+}=AB+BA$ is the anticommutator, 0the null element and 1the unit. The

C’-algebra $A$ of dynamical variables is the CAR algebra [12] i.e., a$\mathrm{B}\mathrm{a}\mathrm{n}\mathrm{a}\mathrm{c}\mathrm{h}*\mathrm{a}\mathrm{l}\mathrm{g}\mathrm{e}\mathrm{b}\mathrm{r}\mathrm{a}$

$)$

with

C’ norm generated by$B(f., g) \equiv\sum_{\sigma=\pm}\sum_{j=-\infty}^{+\infty}\{f_{j,\sigma}.c_{j,\sigma}+g_{j,\sigma}c_{j,\sigma}^{*}\}$

where the sequences$\{f_{j,\sigma}.\}$

and$\{g_{j,\sigma}\}$(2),

are square summable.

The physical states are defined as positive and normalized linear functionals

algebra $A$ , i.e., linear functionals satisfying (i)$\vee u(1)=1$$\omega(B^{*}B)\geq 0$$\omega$over thefor any $B\in A$ and (ii)with 1the unit of $A$ .

The Hamiltonian $H$ of the system is given by$H=- \hslash\gamma\sum_{\sigma=\pm_{j}}\sum_{=-\infty}^{+\infty}\{c_{j,\sigma}^{*}c_{j+1,\sigma}+c_{j+1,\sigma}^{*}c_{j,\sigma}\}+\sum_{\sigma=\pm}\sum_{j=1}^{L}\hslash\epsilon_{j}c_{j,\sigma}^{*}c_{j,\sigma}$

where
and

$\mathrm{a}_{j}$$\hslash$is the Planck constant divided by$2\pi$,(3), $\gamma(>0)$ is the strength of the electron transfer

stands for the localized potential. The corresponding “first quantized” Schrodinger

operator is assumed to admit acomplete set of outgoing scattering states and have no boundstate. The outgoing state $\psi_{q}(j)(-\pi\leq q\leq\pi)$ is the solution of the eigenvalue equation
corresponding to an eigenvalue $E_{q}=-2\hslash\gamma\cos q$ :$-\hslash\gamma\{\psi_{q}(j+1)+\psi_{q}(j-1)\}+\hslash\epsilon_{j}\psi_{q}(j)=E_{q}\psi_{q}(j)$with the outgoing boundary condition

,

(4)

202
$\{e^{\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}}+R_{q}e\ovalbox{\tt\small REJECT} \mathrm{j}\}\rangle$$\mathrm{m}_{q}(7)\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$

when j$-\mathrm{o}\mathrm{o}(+\mathrm{o}\mathrm{o})$$\ovalbox{\tt\small REJECT}$

for q $>0(<0)\rangle$$(_{\iota}\ulcorner \mathrm{J})$$\ovalbox{\tt\small REJECT}$

where$R_{q}$

is the reflection amplitude. The time-evolution automorphism$\alpha_{t}$:$Aarrow$$A$isgenerated via atruncated Hamiltonian in astandard way [12].

Initial states are prepared in the following way: Firstly, the chain is divided into three:$(-\infty, -M-1]$ , $[-M, N]$

and $[N+1, +\infty)$ with $M>0$ and $N>L$ . The two semiinfinite

segments serve as reservoirs and the finite one as an embedded system. Corresponding 1$0$

this division, the algebra $A$ is decomposed into atensor product of the three subalgebras$A_{L}$, As and $A_{R}:A=A_{L}$ (&As (&A . Now the Hamiltonian $H$ is represented as a$R$of aleft-reservoir part

$H_{L}$, aright-reservoir part

$H_{R}$, an embedded-system part$\mathrm{s}\backslash 1111$

and a$H_{S}$

reservoir-system interaction $V_{int}:H=H_{L}+H_{R}+H_{S}+V_{int}$ . There is asimilar decompositionof the number operator: $N=N_{L}+N_{R}+N_{S}$ . Next we introduce an equilibrium stateover the algebra
potential
let$\omega_{R}$$\mu_{L}$$A_{L}$of the left reservoir variables with inverse temperaturecorresponding to the Hamiltonian$H_{L}$and the number operator

be an equilibrium right-reservoir state overchemical potential$\mu_{R}$$\omega_{S}$over$A_{S}$and chemical$N_{L}$. Similarly,

with inverse temperature ($A_{R}$$\mathrm{J}_{R}$corresponding to the HamiltonianThen, for each embedded-system state$\beta_{L}$$H_{R}$and the number operator, an initial state$\omega_{in}$$\omega_{L}$and$\bigwedge_{\mathit{1}T}’$.

is given by atensor

product$\omega_{in}=\omega_{L}\otimes\omega_{S}\otimes\omega_{R}$.

We showed [3] that, for $tarrow\pm\infty$ , the initial statequasifree states$\omega_{\pm\infty}$, i.e., for any $B\in A$ ,(6)$\omega_{n}.\cdot$

weakly evolves towards unique$\lim_{tarrow\pm\infty}\omega_{in}(\alpha_{t}(B))=\omega_{\pm\infty}(B)$the choice of the separating points $M$ , $N$ and the initial system state$\omega_{S}$, irrespective to. As the state$\omega_{\pm\infty}$

are quasifree, they are fully characterized by the tw0-point functions. For example,$\omega_{+\infty}(c_{j\sigma}^{*}c_{j’\sigma’})=\delta_{\sigma\sigma’}\int_{0}^{\pi}dq\{F_{L}(E_{q})\psi_{q}(j)^{*}\psi_{q}(j’)+F_{R}(E_{q})\psi_{-q}(j)^{*}\psi_{-q}(j’)\}$,(7)

where $F_{L}(E)=1/\{e^{\beta_{L}(E-\mu L})+1\}$ and $F_{R}(E)=1/\{e^{\beta_{R}(E-\mu R})+1\}$ are Fermi distributionfunctions for the left and right reservoirs, respectively

203
Eq. (7) gives tw0-probe Landauer-type formula for the particle flow and the energy flow:(8)$\langle J_{j-1|j}^{N}\rangle_{+\infty}\equiv\omega_{+\infty}(J_{j-1|j}^{N})=\frac{1}{\pi\hslash}\int_{-2\hslash\gamma}^{2\hslash\gamma}dE|T_{q(E)}|^{2}\{F_{L}(E)-F_{R}(E)\}$$\langle J_{j-1|j}^{E}\rangle_{+\infty}\equiv\omega_{+\infty}(J_{j-1|j}^{E})=\frac{1}{\pi\hslash}\int_{-2\hslash\gamma}^{2\hslash\gamma}EdE|T_{q(E)}|^{2}\{F_{L}(E)-F_{R}(E)\}$

where$\langle\cdots\rangle_{+\infty}$$1-|R_{q}|^{2}$stands for the average with respect tothe transmission coefficient, andflow and energy-flow operators from the$J_{j-1|j}^{N}$and$(j-1)\mathrm{t}\mathrm{h}$$J_{j-1|j}^{N}=i \gamma\sum_{\sigma=\pm}\{c_{j,\sigma}^{*}c_{j-1,\sigma}-c_{j-1,\sigma}^{*}c_{j,\sigma}\}$$\omega_{+\infty}$,, $q(E)\equiv\cos^{-1}\{-E/(2\hslash\gamma)\}$ ,$J_{j-1|j}^{E}$

to the(9)$|T_{q}|^{2}\equiv$

stand, respectively, for the particle$j\mathrm{t}\mathrm{h}$sites:,(10)$J_{j-1|j}^{E}=- \hslash[\frac{i\gamma^{2}}{2}\sum_{\sigma=\pm}\{c_{j,\sigma}^{*}c_{j-2,\sigma}+c_{j+1,\sigma}^{*}c_{j-1,\sigma}-(h.c.)\}-\frac{\epsilon_{j-1}+\epsilon_{j}}{2}J_{j-1|j}^{N}]$(11)

III. ENTROPY PRODUCTION-thermodynamic considerations-

Entropy production may be calculated as atime-derivative of an appropriate entropy

However to avoid an arbitrariness in the definition of entropy, we follow the thermodynamic
arguments to introduce an entropy production as in the works of Ruelle [11] and of Spohnand Lebowitz [10].
We consider asystem consisting of afinite conductor placed between two infinitely extended electron reservoirs and begin with simple assumptions:1) Entropy of the finite part exists and is finite.
2) Reservoirs remain to be in equilibrium.

3) Any change in the reservoir state can be regarded as aquasi-static process.

Let

$S$

,

$S_{L}$

and$S_{R}$

be entropies of the finite part, right reservoir and left reservoir, re-spectively, then the total entropy change per time ais obviously given by$\sigma=\dot{S}+\dot{S}_{L}+\dot{S}_{R}$In asteady state, all terms in the right-hand side are constant in time. Thus

(12)

204
$S\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}5^{\ovalbox{\tt\small REJECT}}(0)+St$

(13)

,

which should be finite because of the assumption 2) for all $t>0$ . And one has $\dot{S}=0$ at
steady states.

The entropy changes of the reservoirs are calculated via assumptions 2) and 3). Let

and$J^{N}$

be energy and particle flows, respectively, from the left to the right reservoirs, thenthe heat flows

where$J^{E}$$\mu_{R}$and$J_{R}^{q}$$\mu_{L}$and$J_{L}^{q}$

to the right and left reservoirs are given by$J_{R}^{q}=J^{E}-\mu_{R}J^{N}$

,

(14)

$J_{L}^{q}=-J^{E}+\mu_{L}J^{N}$

,

(15)

are chemical potentials of the right and left reservoirs, respectively. And,assumptions 2) and 3) lead to$\dot{S}_{R}=\frac{J_{R}^{q}}{T_{R}}=\frac{J^{E}-\mu_{R}J^{N}}{T_{R}}$,$\dot{S}_{L}=\frac{J_{L}^{q}}{T_{L}}=-\frac{J^{E}-\mu_{L}J^{N}}{T_{L}}$

(16)

,

(17)

where $T_{R}=1/(k_{B}\beta_{R})$ and $T_{L}=1/(k_{B}\beta_{L})$ are temperatures of the right and left reservoirs
with$k_{B}$

the Boltzmann constant. Eqs.(12), (16), (17) and$\dot{S}=0$

give

,

$\sigma=(\frac{1}{T_{R}}-\frac{1}{T_{L}})J^{E}-(\frac{\mu_{R}}{T_{R}}-\frac{\mu_{L}}{T_{L}})J^{N}$(18)

which is the entropy production at asteady state.

IV. POSITIVITY OF THE ENTROPY PRODUCTION

Now we return to the one-dimensional conductor discussed in Sec. . Prom$\mathrm{I}\mathrm{I}$and (13) as well as$J^{E}=\langle J_{j-1|j}^{E}\rangle_{+\infty}$

and$J^{N}=\langle J_{j-1|j}^{N}\rangle_{+\infty}$, (9), we find$\sigma=-\frac{k_{B}}{\pi\hslash}\int_{-2\hslash\gamma}^{2\hslash\gamma}dE|T_{q(E)}|^{2}\{\beta_{L}(E-\mu_{L})-\beta_{R}(E-\mu_{R})\}\{F_{L}(E)-F_{R}(E)\}$As aresult of an inequalit$\mathrm{e}\mathrm{q}\mathrm{s}.(8)$$\mathrm{y}$$-(x-y) \{\frac{1}{e^{x}+1}-\frac{1}{e^{y}+1}\}\geq 0$

,

(13)

205
where the equality holds only when $x=y$ , the entropy production is nan-negative:$\sigma\geq 0$

and vanishes only if

and$\beta_{L}=\beta_{R}$$\mu_{L}=\mu_{R}$

(20)

,

, or both reservoirs are in equilibrium.

Note that the definitions of heat flows (14) and (15) lead to(21)$J_{R}^{q}+J_{L}^{q}=V\langle J_{j-1|j}\rangle_{+\infty}$

where $V=(\mu_{R}-\mu_{L})/e$ is the voltage difference between the two reservoirs and $J_{j-1|j}=$$-eJ_{j-1|j}^{N}$is the electric current operator. This implies that the total heat flow from thefinite system is the Joule heat.

The relation with thermodynamics is more transparent in the linear transport regime.
Let$T_{0}$

be the mean temperature of the reservoirs,mean chemical potential and$T_{R}=T_{0}- \frac{\triangle T}{2}$Then, when,$|\triangle T|<<T_{0}$$V$$e|V|<<\mathrm{M}\mathrm{o}$$\langle J_{j-1|j}\rangle_{+\infty}=GV+L_{1}\frac{\triangle T}{T_{0}}$where the heat flowdE.
$L_{1}=- \frac{e}{\pi\hslash}J_{-2\hslash\gamma}^{2\hslash\gamma}$$L_{2}= \frac{1}{\pi\hslash}\int_{-2\hslash\gamma}^{2\hslash\gamma}$

In the above,,the$\mu_{0}$,,$\mu_{L}=\mu_{0}-\frac{eV}{2}$$\langle J_{j-1|j}^{q}\rangle_{+\infty}=L_{1}V+L_{2}\frac{\triangle T}{T_{0}}$

,

(23)

,

$(E- \mu_{0})|T_{q(E)}|^{2}(-\frac{\partial F_{0}(E)}{\partial E})$,$(E- \mu_{0})^{2}|T_{q(E)}|^{2}(-\frac{\partial F_{0}(E)}{\partial E})$

$F_{0}(E)=1/\{e^{\beta_{0}(E-\mu_{\mathrm{O}})}+1\}$(22)

was introduced and the coefficients are [3]$|T_{q(E)}|^{2}(- \frac{\partial F_{0}(E)}{\partial E})$dE

dE$\mu_{R}=\mu_{0}+\frac{eV}{2}$we have,$J_{j-1|j}^{q}=J_{j-1|j}^{E}-\mu_{0}J_{j-1|j}^{N}$$G= \frac{e^{2}}{\pi\hslash}\int_{-2\hslash\gamma}^{2\hslash\gamma}$

the temperature difference,the potential difference:$T_{L}=T_{0}+ \frac{\triangle T}{2}$and$\triangle T$

(24)
(25)

with $\beta_{0}=1/(k_{B}T_{0})$ .

In this case, the entropy production is given by$\sigma=\frac{\triangle T}{T\frac{)}{0}}\langle J_{j-1|j}^{q}\rangle_{+\infty}+\frac{V}{T_{0}}\langle J_{j-1|j}\rangle_{+\infty}=\frac{1}{T_{0}}[GV^{2}+2L_{1}V\frac{\triangle T}{T_{0}}+L_{2}(\frac{\triangle T}{T_{0}})^{2}]$(26)This agrees with the expression of the entropy production known in the linear nonequilibrium thermodynamics [13].
All those features are fully consistent with nonequilibrium thermodynamics

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V. CONCLUSIONS

We have shown that anonequilibrium entropy production previously introduced for spin
systems by Ruelle [11] can be extended to one-dimensional conductors and that it is fully

consistent with nonequilibrium thermodynamics.

Now we explore physical implications of the results. For this purpose, we assume all
the states are described by density matrices. First we observe, because of the conservationof energy and particle number, the average energy flow

flow
$N_{R}$$\langle J_{j-1|j}^{E}\rangle_{+\infty}$

are given in terms of reservoir energies $Hl$ ,$\langle J_{j-1|j}^{N}\rangle_{+\infty}$$H_{R}$and the average particle

and particle numbers$N_{L}$

,

:

where

$\langle J_{j-1|j}^{E}\rangle_{+\infty}=-\langle\dot{H}_{L}\rangle_{+\infty}=\langle\dot{H}_{R}\rangle_{+\infty}$,$\langle J_{j-1|j}^{N}\rangle_{+\infty}=-\langle\dot{N}_{L}\rangle_{+\infty}=\langle\dot{N}_{R}\rangle_{+\infty}$,$\dot{H}_{L}=\frac{d}{dt}\alpha_{t}(H_{L})|_{t=0}$$\langle_{r}\dot{4}\rangle_{+\infty}=0$. Furthermore, if an observablebecause of the invariance of the state$\omega_{+\infty}$

$A$

(27)
$(\underline{9}8)$admits afinite average$\langle A\rangle_{+x}$.

.

Then, (27) and (28) give$\sigma=k_{B}\langle\beta_{L}(\dot{H}_{L}-\mu_{L}\dot{N}_{L})\rangle_{+\infty}+k_{B}\langle\beta_{R}(\dot{H}_{R}-\mu_{R}\dot{N}_{R})\rangle_{+\infty}$

Now letto$\omega_{+\infty}$

and$\overline{H}_{R}\equiv H-H_{L}$, then the difference$\overline{H}_{R}-H_{R}$(29)

admits finite average with respect. This, asimilar equation for$\langle\{\overline{H}_{R}-\dot{H}_{R}\}\rangle_{+\infty}=0$.

$N_{R}$and (29) lead to$\sigma=k_{B}\langle\beta_{L}(\dot{H}_{L}-\mu_{L}\dot{N}_{L})\rangle_{+\infty}+k_{B}\langle\beta_{R}(\overline{H}_{R}.-\mu_{R}\overline{N}_{R}.)\rangle_{+\infty}$(30)$=-k_{B} \frac{d}{dt}\mathrm{T}\mathrm{r}(\rho(t)\ln\rho_{\mathrm{L}\mathrm{o}\mathrm{c}})|_{\rho(t)arrow\rho+\infty}$

where Tr stands for thethe steady state$\omega_{+\infty}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$,$\rho(t)$and$\rho_{+\infty}$. The density matrix

are density matrices for the state at time and$t$$\rho_{\mathrm{L}\mathrm{o}\mathrm{c}}$corresponds to the local equilibrium state:$\rho_{\mathrm{L}\mathrm{o}\mathrm{c}}=\frac{1}{Z_{\mathrm{L}\mathrm{o}\mathrm{c}}}\exp\{-\beta_{L}(H_{L}-\mu_{L}N_{L})-\beta_{R}(\overline{H}_{R}-\mu_{R}\overline{N}_{R})\}$

with$Z_{\mathrm{L}\mathrm{o}\mathrm{c}}$

,

(31)

the normalization constant. The expression (30) suggests that anonequilibrium

entropy is given by$S=-k_{B}\mathrm{R}(\rho\ln\rho_{\mathrm{L}\mathrm{o}\mathrm{c}})$nonequilibrium entropy [5].

, which is nothing but Zubarev’s definition of207
Since von Neumann entropy$\mathrm{R}(\rho(t)\ln\rho(t))$

is constant in time, one also has(32)$\sigma=-k_{B}\frac{d}{dt}S(\rho(t)|\rho_{\mathrm{L}\mathrm{o}\mathrm{c}})|_{\rho(t)arrow\rho+\infty}$

where$S(\rho(t)|\rho_{\mathrm{L}\mathrm{o}\mathrm{c}})$is the relative entropy [14,15,12]$S(\rho(t)|\rho_{\mathrm{L}\mathrm{o}\mathrm{c}})=$

-Tr

(33)

$(\rho(t)\{\ln\rho(t)-\ln\rho_{\mathrm{L}\mathrm{o}\mathrm{c}}\})$

Asimilar formula to (32) was derived by Spohn and Lebowitz [10] for systems weakly
coupled with reservoirs in the scaling limit, where the local equilibrium state is replaced byan equilibrium state.
The entropy production acan be represented in adifferent way. By noting that thelogarithm of the initial density matrix of the embedded$\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}:\ln\rho_{S}(0)$admits afinite

steady-state average, one has$\sigma=-k_{B}\frac{d}{dt}S(\rho(t)|\rho(0))|_{\rho(t)arrow\rho+\infty}$

where$\rho(0)$,(34)

stands for the initial state of the whole system. For driven systems, Ojima,

Hasegawa and Ichiyanagi [6] introduced entropy production as time-derivative of the relative
entropy with respect to the initial state $S(\rho(t)|\rho(0))$ (see also Ichiyanagi [7] and Ojima [8]).
Eq.(34) suggests that the same formula holds for internally disturbed systems.

We emphasize again that the above arguments are formal and rigorous discussions will
be presented elsewhere.

ACKNOWLEDGMENTS
The author is grateful to Professors L. Accardi, T. Hida, N. Obata, M. Ohya, K. Saito, S.
Sasa, A. Shimizu and I. Volovich for fruiteful discussions and valuable comments. This work

is partially supported by Grant-in-Aid for Scientific Research (C) from the Japan Societyof the Promotion of Science and by Waseda University Grant for Special Research Projects(Individual Research, N0.2000A-852) from Waseda University

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