Quantum mechanics of nonconservative systems

ANNALS

OF PHYSICS

114, 479-496 (1978)

Quantum

Mechanics

of Nonconservative

Systems

KUNIO YASUE Department of Physics, Nagoya University, Nagoya 464, Japan

ReceivedJuly 21, 1977 The quantumtheoreticformulationof opendynamicalsystems interactingwith chaotic thermalenvironments isinvestigated. Nelson’sstochasticquantizationprocedureisadopted to derivethe Schrijdinger-Langevin equationwhich describes the irreversiblebehaviorof an open system.Then, taking a classicallimit of the Schrijdinger-Langevin equation, weobtainthe Hamilton-Jacobiequationfor classical irreversibledynamics.Theasymptotic form of t’heHamilton-Jacobiequationand Schriidinger’s quantizationproceduregiveusa nonlinearSchrGdinger equationwhich describes quantumtheoreticasymptoteof the open system.F’or the purposeof analyzingquantum-mechanical irreversiblebehaviorsof open many-bodysystemsa field theoretictreatmentof the Schrodinger-Langevin equationis presented. As an applicationof the methodto a physicalsystemthe thermaldecayof the ac Josephson currentnearthe critical temperatureisinvestigated.

1. INTRODUCTION

Quantum mechanics has been developed to deal with closed or isolated dynamical systems. There a time evolution of such an isolated dynamical system is assumed to be represented by a one-parameter unitary group on a Hilbert space. The infinitesimal generator of the group is related with the notion of Hamiltonian of the system. How can one treat open dynamical systems interacting with the external world in the framework of quantum mechanics ? Let us try to formulate the problem in the framework as faithfully as possible. If we intend to realize a quantum-mechanical time evolunon of such an open dynamical system, we have to start with the time evolution of the total system, i.e., the system plus the external world, generated by the total Hamiltonian HT =H,@Z+Z@H,+H,,

U-1)

where @ denotes a tensor product, H, the Hamiltonian of the system without the interaction with the external world, HE that of the external world without the interaction with the system, and HI represents the interaction between the system and the external world [I]. Then we have to encounter two difficulties: First, even when the system itself is a simple dynamical systemwith finite number of degreesof freedom we have to deal with a complicated dynamical system with infinitely many degrees of freedom including the external world. Second, it is difficult to specify rigorously the interaction between the system and the external world in each case. To avoid such difficulties inherent in the fundamental approach to the open system 479

0003-4916/78/l 142~479$05.00/0 All

Copyright 0 1978 by Academic Press, Inc. rights of reproduction in any form reserved.

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YASUE

one may adopt a rather phenomenological approach, incorporating phenomenological irreversible properties such as dissipations and fluctuations into quantum mechanics. In the present paper we shall derive the phenomenological formulation from the stochastic foundation of quantum mechanics. Then we shall show that a dissipative potential term and a random potential term can be incorporated into the Schrddinger equation, which might include thermal and statistic influence of the external world, without a detailed description of the interaction between the system and the external world. In Section 2 a quantum-mechanical version of the classical Langevin equation, i.e., the Schriidinger-Langevin equation, which well describes irreversible behaviors of open dynamical systems, is derived by means of Nelson’s stochastic quantization procedure. In Section 3 we investigate a classical limit of the Schrodinger-Langevin equation and give the Hamilton-Jacobi equation for classical dissipative dynamics. In Section 4 we derive a time-independent nonlinear Schrodinger equation which describes quantum-mechanical stationary dissipative states. Section 5 is devoted to investigate the field theoretic treatment of the Schriidinger-Langevin equation. ln Section 6 we analyze the thermal decay of ac Josephson current near the critical temperature as an application of the present formulation of quantum mechanics of open dynamical systems. Concluding remarks are given in the Section 7.

2. SCHR~DINGER-LANGEVIN

EQUATION

We consider the external world, with which an open dynamical system interacts, as a chaotic thermal environment (a heat reservoir). It is to avoid the difficulty in realizing the complicated interaction between the system and the external world; namely, the influence of the external world to the system is assumed to be purely statistical in nature. In classical mechanics dynamical behavior of such an open system as irreversible dynamics is described by the so-called Langevin equation W?(t)

= ---B&f) - grad UQ@>,0 + -40,

(2.1)

where Q(t) = (Ql(t),..., Q%(t)) denote reduced coordinate variables of the open system, V(Q, t) a usual potential function, A4 a mass parameter, and A(t) = (Al(t),..., A”(t)) a Gaussian white noise with mean 0 and variance E[k(f)

/P(u)] = 2Dsqt

- u),

l
j
(2.2)

Note that the complicated interaction between the system and the thermal environment is characterized by the friction coefficient p and the variance parameter D = fik,T,

where T denotes a temperature constant [2].

of the thermal

(2.3)

environment

and kB Boltzmann’s

NONCONSERVATIVE

SYSTEMS

481

As far as we are concerned with classical properties, many of the nonequilibrium phenomena characteristic to open dynamical systems interacting with chaotic thermal environments have been clarified in terms of the Langevin equation (2.1). Now we: shall derive a quantum-mechanical version of the Langevin equation which might well describe a quantum-mechanical behavior of the open system. Here we have to note that neither the conventional canonical quantization procedure nor path-integral quantization procedure is not applicable to the open system because of the absence of the Hamiltonian or the Lagrangian for the open system described Iby the Langevin equation (2.1). Therefore we are obliged to make use of a different method in which neither Hamiltonian nor Lagrangian is needed. Such a quantization procedure has been proposed by Nelson [3,4] which demands no Lagrangian or Hamiltonian but Newton’s equation of motion in a generalized sense. Nelson’s quantization procedure consists in utilizing fundamental notions of stochastic processes and so it has been called “stochastic quantization” by Guerra l5,

a

Let us consider the stochastic quantization of the classica open dynamical system described by Eq. (2.1). The first basic assumption of the stochastic quantization is that quantum-mechanical behavior of the coordinate variables Q(t) of the open dynamical system is represented by an n-dknensional diffusion process described by the stochastic dtrerential equation

dQW = b(Q(t>, t>dt + dW(t),

(2.4)

where b(Q, t) = (b’(Q, t),..., b”(Q, t)) denotes a drift vector to be determined and W(t) = (W(t),..., Wn(t)) an n-dimensional Wiener process. Note that the diffusion coejkient of the Wiener process W(t) is assumed to be h/2M, where fi denotes Plan&s constant divided by 277. Thus an infinitesimal increment dW(t) of the Wiener process is a Gaussian process with mean 0 and variance E[dW(t)

dW(t)]

= tdt.

(2.5)

The probability density function p(Q, t) defined as p(Q, t) d”Q = Prob{Q(t) satisfies the Fokker-Planck equation $ f(Q, t) = --dW(Q,

t) p(Q, t)} + 2G div grad p(Q, t).

E d”Q}

(2-6)

Prior tat proceeding to the second assumption of the stochastic quantization it may be useful to summarize the definitions of mean velocity u and mean acceleration a of the process Q(t):

v(Q(t>, t> = W + oz+c>Q(t), 4Q(t), t) = WD,

+ Dz+J>

(2.7)

Q(t),

(2.8)

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KUNIO

YASUE

where D and D, denote the mean forward derivative and the mean backward derivative generated by the process Q(t). The two derivatives are defined as

Df = t&y$ Hf(Q(t + h), t + 4 - f(QW, 01 QW =

(

-& + b * grad + &

div grad) f(Q(1), t)

(2.9)

&f, t> = ‘hJyi Hf(QW, f> - fl = -$+b,grad (

- &

div grad)

f(Q(O, t>

(2.10)

for any smooth function f(Q, t), where b*(Q, t) = b(Q, t) - (fi/M) grad log p(Q, t) and I?[* 1 Q(t)] denotes the conditional expectation with respect to Q(t). The second basic assumption is that the Langevin equation (2.1) is expressed in terms of the mean velocity and the mean acceleration given in Eqs. (2.7) and (2.8)

Ma(Q(t), t> = -Bv(Q(t>, t> - grad v,t) + 40. A straightforward

calculation Mgv+

(2.11)

of Eq. (2.11) gives us M(v.grad)v

- &L grad

div grad pliz P 112

(2.12)

=-/3v-gradV+A.

Finally we assume the mean momentum Mv(Q(t), t) of the process be a gradient of a smoothfunction S as (2.13)

Mv(Q(t), t> = fi grad S(QO>,t>

and introduce a wavefunction or a probability amplitude of the reduced coordinate variable of the opensystem

!J

= [p(Q,

f>Y2 ew iS(Q, t),

Then we obtain a quantum-mechanical system as follows: ih $ #(Q,

t) = [ - & + &fi

wave equation

i2 = -1 .

(2.14)

for the open dynamical

div grad + V(Q, t) - Q * A(t)

S(Q, O] #0

= [-&divgrad+

V(Q,t)-

Q-A(t)

(2.15)

483

NONCONSERVATIVE SYSTEMS

The random potential -Q * A(t) in Eq. (2.15) may be replaced by a more general random potential R(Q, t). ih 4 #(Q, t) = [ - &

iP

div grad + V(Q, t) + R(Q, t)

+ 2M k log

$. I

#(Q,

It is worthwhile to note that the probabilistic holds autom,atically from Eq. (2.14),

(2.16)

t)

interpretation

of the wavefunction

I #
(2.17)

even though Eq. (2.16) is no longer linear. Equation (2.16) was first derived heuristically by Kostin [7, 81 and has been called the “Schrizidinger-Langevin equation.” The derivation explained above is based on the work. by the present author [9, IO] and Skagerstam [Ill. Dissipative and irreversible characters of the Schriidinger-Langevin equation (2.16) have been investigated by many authors [7-l 51. At the end. of the present section we shall point out that the quantum-mechanical irreversible dynamics of a charged particle interacting with an electromagnetic field can also be described by a single-particle Schrodinger-Langevin equation ifi g #(r, t) = [&

C--ifiV - eA(r, t)}” + #~(r, t)

+ R(r, 0 + $73

log

where A(r, t) and $(r, t) denote a vector potential and a scalar potential of the electromagnetic field, respectively, m the mass of the particle, and e the charge.

3. HAMILTON-JACOBI

FORMULATION OF CLASSICAL IRREVERSIBLE DYNAMICS

Prior to proceeding to the quantum-mechanical analysis of such an open dynamica system in terms of the Schrodinger-Langevin equation (2.16), we had better investigate the classical limit of the Schriidinger-Langevin equation because it may give us a new light for our stochastic approach. Following the well-established procedure with respect to the classical limit [16] we perform the substitution

RQ, 0 = A exp [f WQ, t)]

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KUNIO

YASUE

in Eq. (2.16) obtaining a wky’ ‘) + F& j grad W(Q, t)i2 i- V(Q, t) + R(Q, t) grad W(Q, t) + MB W(Q, t) = 0.

- &div

(3.2)

Passing to the limit ti 4 0, we find the following first-order partial differential equation

VQ, t)+R(Q, t)+a

aW(Q,t) at + & I grad W(Q,W +

W(Q, t) = o,

(3.3)

where W(Q, t) should be understood as the classical action. It will be shown that Eq. (3.3) is an extension of the Hamilton-Jacobi formulation of classical conservative dynamics to classical irreversible ones with dissipations. Rewriting Eq. (3.3) in terms of new variables X = (X0, Xl,..., Xn) E R”+l with X0 = t, X1 = Ql,..., X” = Qn as

f-fp+&$[qgq and introducing

+ w7 + a0 + & W(X)= 0

(3.4)

conjugate variables p.

3

=

aww axj’

.j =

0,

1 ,...)

(3.5)

n

we find the standard form [17] of the first-order partial differential equation (3.3) w,

P, WI = PO+ &,j i

Pi2 + V(X) + R(X) + & w

3=1 zz

0.

(3.6)

Characteristic curves of Eq. (3.6) in the (X, P, W)-space following first-order ordinary differential equations dXj aF ---=-3 ds aPj

dpj ds --

($+

Pj$)p

R2n+3 are given

by the

j=O,l

,..., n

(3.7)

j=O,

l,m**,n

(3.8)

(3.9)

NONCON~ERV.~TIVE

485

SysT~xs

These equations can be rewritten explicitly as dX” = dt = ds,

(3.10) (3.11)

for j = 0 and (3.12) dPj aV dt = - k axj +

(3.13)

forj = 1,2 ,..., n. Obviously these characteristic curves are solutions of the second-order ordinary differential equations

W?(t) = -grad VQ(t>, t) - B&t> + A(t),

(3.14)

with A(t) = -grad R(Q(t), t), which is nothing but the classicalLangevin equation (2.1). By those results it may be ensured that Eq. (3.4) is the Hamilton-Jacobi equation for classical irreversible (i.e., nonconservative) dynamics described by the Langevin equation (2.1), and Eqs. (3.12) and (3.13) are Hamilton’s equations of motion. Equation (3.11) provides the energy loss of the open system because PO= 3 W/at is the total energy; namely, the Schriidinger-Langevin equation (2.16) provides the correct classicallimit with I? -+ 0. It seems surprising that we have derived the Hamilton-Jacobi equation (3.4) and Hamilton’s equations of motion [Eqs. (3.12) and (3.13)] with no usage of the Hamiltonian which might not exist in such a case of irreversible dynamics with dissipations. This may be explained by the fact that the existence of the action is intrinsically assumedin our framework.

4. NONLINEAR

SCHR~DINGER

EQUATION

FOR STATIONARY

DISSIPATIVE

STATES

Tn order to make a further analysis of Eq. (2.1) we shall neglect the last term A(t) which appeared in the right-hand side of the equation. It is simply to avoid the complexity Iof calculation. Although A(t) may play a crucial role in the irreversible dynamics, we can expect to obtain a useful insight to someextent even if we neglect it becauseof the presenceof the friction term. Thus let us start with

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KUNIO

YASUE

According to the Hamilton-Jacobi formulation of classical irreversible dynamics developed in the preceding section, the equation of motion (4.1) is equivalent to the Hamilton-Jacobi equation

aw(Q, 0 + & at

I grad

B WQ, W + V(Q) + ?i? W(Q, t) = 0.

(4.2)

We can separate the time dependence of the action W(Q, t) as

WQ, t> = -&

[ 1 - exp (- {

t)] +

U(Q),

where K,, is a constant and U = U(Q) does not contain explicitly

&j I gradWQY

t, obtaining

+ V(Q) + -& U(Q) = $ K,, ,

For sufficiently small t (t Q M//3) Eq. (4.3) can be approximated

as

P Kc++ U(Q). W(Q, t) ‘v - M

(4.5)

This approximated relation suggests to us that the constant K, is relevant to the initial energy of the system E,, ,

On the other hand Eq. (4.3) shows that as the time tends to infinity the action W(Q, t} approaches to the time-independent value -K, + U(Q). Physically speaking, the open dynamical system described by Eq. (4.1) or Eq. (4.2) with initial energy E,, loses its energy and decays into an asymptote (an equilibrium state) described by the reduced Hamilton-Jacobi equation (4.4). We shall examine a quantum-mechanical significance of such an asymptote in the light of Schrodinger’s original quantization procedure. According to the basic assumption of Schriidinger’s quantization procedure (i.e., “quantization as a variational problem”) [ 181, a quantum-mechanical description of the asymptote described by the reduced Hamilton-Jacobi equation (4.4) may be given by the substitution

WQ) = fi log u(Q)

(4.7)

in Eq. (4.4). Then we obtain fi2 ’ gr;;;;F)‘2 2M and set up a variational

+ V(Q) + 4 fi log u(Q) = E,, ,

(4.8)

problem SJ[u] = 0,

(4.9)

NONCONSERVATIVE

487

SYSTEMS

where + UP) + $ fi log u(Q) - E,] u(Q)” d”Q. The variational [

(4.10)

problem [Eq. (4.9)] is replaced by the nonlinear eigenvalue problem

- &: div grad + V(Q) + &

log Us]

u(Q) = (I$ - &)

(4.11)

u(Q).

Since u(Q) is related with the asymptote through Eq. (4.7), we may attribute the quantum-mlechanical description of the asymptote of the open dynamical system described classically by Eq. (4.1) or Eq. (4.2) to the nonlinear Schrddinger equation (4.11). We may call such a quantum-mechanical asymptote “a stationary dissipative state.” Such a nonlinear eigenvalue problem as Eq. (4.11) has been profoundly investigated by Bialynicki-Birula and Mycielski 1191 with mathematical concerns in nonlinear wave mechamics: They proved the existence of the lower eigenvalue bound for Eq. (4.11) and the consistency of the probabilistic interpretation (Born interpretation) of the wavefunction with logarithmic nonlinearity. Moreover they found Gaussian solutions of Eq. (4.1 I) with V(Q) = 0. Those rigorous analysis of Eq. (4.11) suggests the existence of the stationary dissipative state, that is, a quantum-mechanical asymptote of the open dynamical system dissi,patively interacting with the environment.

5. FIELD 'THEORETIC FORMULATION

OF THE SCHR~DINGER-LANGEVIN

EQUATION

Since we are much concerned with a many-particle system, it seems desirable for us to formulate a field theoretic treatment of the Schriidinger-Langevin equation. Let us sta.rt with the single-particle Schriidinger-Langevin equation1 i;

~~(rr t> = [- &

d + vfr, f> + R(r, t) + $

where A denotes the three-dimensional Laplacian interacting %withthe chaotic thermal environment. from the variational principle with a Lagrangian;

?+J
C(r, 0,

(5.1)

and m the mass of a particle Equation (5.1) can be derived

6 IL dt = 0,

L=s[h&r, 04th 0- &-V&r, t)*V$(r, t)

(5.2)

- W, 0 G $(r, f>

(5.3) 1 I-Iereafter we adopt a natural system of units with fi = 1.

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KUNIO

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where V denotes the three-dimensional jugate variable

gradient. Introducing

the canonically

= $cr, 0,

con-

(5.4)

we construct a Hamiltonian H = /r(r, = I[& + W,

t) &r, t) d3r - L V&r,

r> - VW,

6 + V(r, f) dk

t> $(b t>

0 $ #(f, t> (5.5)

The Schrbdinger-Langevin motion

equation (5.1) corresponds to the canonical equation of 8H $(r, t) = ___ Wr, t) (5.6)

Note that the complex conjugate equation canonical equation of motion

of Eq. (5.1) also corresponds to the

7i(r, t) = i&r, t) SH Mf, 0

=--*

(5.7)

Field theoretic treatment can be carried out by replacing # and $ in Eq. (5.5) with a field operator $ = &r, t) in the Heisenberg picture defined on the Fock space 9 and q* = $*(r, t). Then we have a field theoretic Hamiltonian

A= S[&

V$* .01,8 f

+ -!!?{log 2m

and commutation

V(r, t) &+# f R(r, t) d*$

$* - log $ - l} $*$I

d3r,

(5.8)

relation of Bose statistics [$(r, t), tj*(r’, t)]-

= S3(r - r’)

(5.9)

NONCONSERVATIVE

489

SYSTEMS

or Fermi sta.tistics [$(r, t), $*(r’,

t)]+ = P(r - r’).

(5.10)

Heisenberg’s equations of motion

provide the Schriidinger-Langevin

jE$= [ -

&A

i$’ = [Ij, A],

(5.11)

i$* = r$*, fil

(5.12)

equation for the field operator

+ V(r, t) + R(r, t) + $;

{log zJ* - log $}I 4

(5.13)

and its adjoint form. The Hamiltonian i? (5.8) and the canonical commutation relations (5.9) and (5.10) may be interpreted as an extension of the conventional field theoretic treatment of the nondi;ssipative (i.e., conservative) many-particle system to that of the dissipative many-partidle system interacting with the chaotic thermal environment. Irreversible and dissipative features of such a dissipative many-particle system have been taken into account in the expression of the Hamiltonian (5.8). Indeed the Hamiltonian A is not invariant under the time reversal operation 4 H $* and z$* t+ $ and so it is not self-adjoint. Moreover the total number operator fi = J $*$ d3r is not conserved because [fv, A] # 0.

(5.14)

In the case of many-boson system with enormously large occupation it seems plausible to perform the following canonical transformation

k

t> = Kr, W” exp[&r, t)l,

J*(r, t) = [$(r, t)]ll” exp[-i&r,

t)],

where new canonical variables ,I? and 4 are chosen to be self-adjoint. Hamiltonian R is written in terms of 6 and 0 as

number,

(5.15) (5.16)

Then the

t JWCr. t) + R(r, t)l d3r (5.17)

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KUNIO YASUE

We may be allowed to approximate

the Hamiltonian

(5.16) by a harmonic

one

(5.18) where p. denotes the constant total number of bosons. This is because the occupation number of bosons is enormously large. The above expression of the Hamiltonian with the canonical commutation relation [20] [f?(r, t), d(r’, t)] = iS3(r - r’)

(5.19)

may provide an approximative description of the dissipative many-boson system interacting with the chaotic thermal environment. The application of this method to a physical system will be given in the next section.

6. AN APPLICATION:

THERMAL DECAY OF THE AC JOSEPHSON CURRENT NEAR THE CRITICAL TEMPERATURE

Many of the nonequilibrium phenomena characteristic to open dynamical systems interacting with chaotic thermal environments can be described classically by the Langevin equation (2.1). So it seems adequate to make use of the SchrbdingerLangevin equation (2.16) or the second quantized Hamiltonian (5.8) for a quantummechanical analysis of such nonequilibrium phenomena. As one of the interesting examples let us investigate the thermal decay of the ac Josephson current near the critical temperature T, . It is well known that in the ground state two electrons at the Fermi surface of a superconducting media form a bound pair (i.e., Cooper pair) and it behaves as a Bose particle. At a temperature lower than the critical one T < T, almost all the electrons in the superconducting media degenerate into the ground state and the collective motion of Cooper pairs causes the superconducting current. At a temperature near the critical one T N T, , a considerable number of pairs are broken into normal electrons. Those normal electrons give rise to complications in the collective motion of the remained Cooper pairs and then these pairs suffer dissipations and fluctuations due to the thermal motion of normal electrons. As we can no longer use equilibrium quantum statistical mechanics to analyze such a nonequilibrium phenomena, we shall apply the second quantized Hamiltonian (5.18) to the collective motion of Cooper pairs. Let us consider the collective motion of Cooper pairs near the critical temperature T N T, across a Josephson junction which consists of two superconductors con-

NONCONSERVATIVE

491

SYSTEMS

netted by a thin layer of insulator. Volume of each superconductor is assumed to be unity. As a model of such a Josephson junction we consider two dissipative many-boson systems (called system 1 and system 2 respectively) weakly coupled with each other. The Hamiltonian of the total system is given by

where

(6.2)

denotes the Hamiltonian of the dissipative many-boson system j (j = 1,2) defined on the Fock space e (j = 1,2) and It,, the interaction Hamiltonian defined on FI @ Sz which can be approximated in the weak coupling limit g < 1 as ox2 = gp, Jcos[&

- &] d3r, d3r, .

(6.3)

As both of the dissipative many-boson systems are assumed to be in superconducting states, e^, and 0, are spatially homogeneous. Thus in order to describe the collective mlotion of Cooper pairs in the junction near the critical temperature, we may adopt .the following Hamiltonian:

+ jkWI , t) + + j W2(r2 ,

R,(rl , t)] $I d3r,

t) + R2(r2,01 b2d3r2

+ $ /PI d3rI4,+ $ sB2d3r2 8, iP ,4 d3r, - 2 sp2 d3r, + gp, cos[8, - z s

d,].

Because we impose a macroscopic ac voltage V(t) across the junction and the junction is small enough compared with the thermal environment, the third and the fourth terms of the right-hand side of Eq. (6.4) can be written as (6.5) 595/114/I/2-3:2

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YASUE

with VI(i) - Vz(l) = qV(t), where 4 denotes the effective charge of each Cooper pair and R(t)‘s two independent Gaussian white noises such that R(t) = R,(t) - R,(t) be a Gaussian white noise with mean 0 and variance E[R(t) R(u)] = 2GS(t - u).

(6.7)

Note that the diffusion constant G of the white noise R(t) is characteristic of the special choice of the junction. The total number of Cooper pairs in each superconductor

n2 =

5

satisfies the following equations of motion,

p”zd3r,

(6.9)

respectively:

ii, = [7il ) R],

(6.10)

ii, = [fi, , A],

(6.11)

i& = m18 ff, - gp, sin& - 8,],

(6.12)

i, = i

(6.13)

that is,

Combining to be

A, + gp, sin[& - d2]

these equations, we find the superconducting

current across the junction

f = j, - j$ =- L (+4 - fi,) + 2gpo sin[4 - 8,] N 2gp, sin[8, - 0,] = Jo sin t?

(6.14)

with $ = & - 8, and Jo = 2gp, . f$ and 8, are determined by the equations of motion

ii, = [d, , R],

(6.15)

ii, = [Oz , h],

(6.16)

NONCONSERVATIVE

493

SYSTEMS

that is, B, =

-VI(t)

-

R,(t)

-

Bm 8, +

iP 2m

3

(6.17)

* e2 =

-

-

R,(t)

-

8yg 82 +

i6 G

3

(6.18)

Vz(t)

where we have neglected the contribution from the surface integral. Equations (6.17) and (6.18) yield the following stochastic differential equation for the phase difference f?: (9 = - ;

(6.19)

8 + qV(t) + R(t).

A solution of Eq. (6.19) with the initial condition Gaussian random process d = d(t) with mean

d(O) = 8, is known to be a

~(t> = 8, exp ( - a t) + q exp (- $ t) jO’ V(U) exp (L zd) &

(6.20)

and variance o(l)=~[l

(6.21)

-exp(-$‘)I.

Accordingly the total superconducting current across the junction random promcess J(t) = Jo sin b(t).

is subjected to a (6.22)

current (3((t)) is the average of Eq. (6.22) and can be

The mean superconducting calculated as

Q(t)> = E[&)] z J,E[sin 6(t)] = Jo sin[fi(t)] exp[-o(t)],

(6.23)

where we have used the characteristic function of @(t) E[exp i@(t)x]

= exp[--a(t)x2

(6.24)

+ ifi(t

for x E R. Tlhus we obtain an explicit form of the mean total superconducting across the junction, /j(t))

= J, sin [O, exp (- i x exp [- 7

1) + q exp (-

(1 - exp [- $t])]

i

t) JOAL.(u) exp (i .

current

24)du] (6.25)

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KUNIO YASUE

If we put on the ac voltage V(t) = v, + v cos wt with v Q V, across the junction, (J(t))

(6.26)

the mean total current becomes

= Jo sin [ 19o exp (-it)+TG(l

-exp(-it))

P/m Zyv(coswf+~sinwl-exp(-~t)) + Wm>2 + w x exp[-F(l

-exp(-+t))]

‘v Jo [sin (du exp (- i

t) + y

[ 1 - exp (-

i

t)])

Pim 2q~(co~~I+~sinwt-exp(-~f)) + (B/m)” + CfJ x cos(&exp(-Lt)+y[l x exp [- y(

-exp(-{t)])],

1 - exp [-2&t]‘)]

.

(6.27)

The last expression of the ac Josephson current coincides with the one at the absolutely zero temperature [21-231 in the limit /!I + 0, G -+ 0. For sufficiently large r (t > m/p) we may have (f(t))

- Jo [sin 75 x cos T]

which has nonvanishing

+ ca,m;[y

W2 qv (cos wt + y

exp (- F)

sin ,t) (6.28)

Cesaro mean

am - Josin[T]exp[--1,

r>F

(6.29)

Consequently we find that the observed ac Josephson current near the critical temperature T II T, approaches to a stationary equilibrium value independent of the initial condition and the ac frequency w .

7. CONCLUDING

REMARKS

In order to demonstrate quantum mechanics of nonconservative systems we have shown a quantum theoretic formulation of open dynamical systems interacting with chaotic thermal environments. Because there is no Hamiltonian for these systems,

NONCONSERVATIVE

SYSTEMS

495

we have developed quantum mechanics on the stochastic bases founded by Nelson. Then it is found that a quantum-mechanical description of such an open system is given by the Schriidinger-Langevin equation (2.16) without a detailed description of the interaction between the system and the environment. It may be of some interest to point out that the Hamilton-Jacobi formulation (3.3) of classical irreversible dynamics is derived as a classical limit of the Schriidinger-Langevin equation. In addition, making use of an asymptotic form of the Hamilton-Jacobi equation and Schrodinger’s quantization procedure, we have shown that a quantummechanical asymptote (an equilibrium state) of the open system is described by the nonlinear !Schriidinger equation (4.11). Furthermore, toward the aim of applying our scheme to physical systems we have given a field theoretic formulation of the Schrodinger-Langevin equation and examined a Josephson junction as an example. It is found that ac Josephson current near the critical temperature approaches a stationary equilibrium value independent of the initial phase difference and the ac frequency. Several attempts from different points of view have been made to incorporate irreversible properties such as dissipations and fluctuations into quantum mechanics. Stevens [24] has investigated damping effects of two-level masers in utilizing a timedependent Hamiltonian. Fronteau and Tellez-Arenas [25] have extended the Schriidinger equation to include a non-self-adjoint potential term. Although those formulations provide approximative descriptions of quantummechanical irreversible behaviors of open systems, they do not seem to be founded on sufficient theoretical bases from our viewpoint. Here it may be worthwhile to remark on the relationship between the SchriidingerLangevin elquation and the Ginzburg-Landau equation. The latter describes static behaviors of quantum-mechanical nonequilibrium phenomena and the former describes dynamical ones because the latter has a nonlinear potential term 1# I”# = p# and the former H/2i(log $/#)# = M#, with # = (p)‘l” exp i0. Finally wle would like to point out that a microscopic derivation of the SchrodingerLangevin equation from a fundamental point of view rather than the stochastic approach may be also possible as it was done in the case to derive the GinzburgLandau equation and the classical Langevin equation. We shall explain its justification in the forthcoming paper.

ACKNOWLEDGMENT The author would like to express his sincere thanks to Professor T. Toyoda for his stimulating and valuable suggestions.

REFERENCES 1. E. B. DAWES, “Quantum Theory of Open Systems,” Academic Press, London, 1976. 2. S. CHANDRASEKHAR, Rev. Modern Phys. 15 (1943), 1.

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KUNIO

YASUE

3. E. NELSON, Phys. Reu. 150 (1966), 1079. 4. E. NELSON, “Dynamical Theories of Brownian Motion,” Princeton Univ. Press, Princeton, N. J., 1967. 5. F. GUERRA, J. Phys. Suppl. Colloque 34 (1973), Cl-95. 6. F. GLJERRA, On the Connection between Euclidean-Markov Field Theory and Stochastic Quantization, in “C*-Algebras and their Applications to Statistical Mechanics and Quantum Field Theory,” (D. Kastler, Ed.), North-Holland, Amsterdam, 1976. 7. M. D. KOSTIN, J. Chem. Phys. 57 (1972), 3589. 8. M. D. KOSTIN, J. Statist. Phys. 12 (1975), 145. 9. K. YAWE, Phys. Lett. 64B (1976), 239. 10. K. YASUE, J. Statist. Phys. 16(1977), 113. 11. B. SKAGERSTAM, J. Mathematical Phys. 18 (1977), 308. 12. B. SKAGERSTAM, Phys. Lett. 58B (1975), 21. 13. K. K. KAN AND J. J. GRIFFIN, Phys. Lett. 50B (1974), 241. 14. J. D. IMMELE, K. K. KAN, AND J. J. GRIFFIN, Nucleur Phys. 241A (1975), 47. 15. R. W. HASSE, J. Mufhemuticul Phys. 16 (1975), 2005. 16. V. P. MASLOV, “Theorie des perturbations et methodes asymptotiques”, Gauthier-Villars, Paris, 1972. 17. R. C~URANT AND D. HILBERT, “Methods of Mathematical Physics II,” Interscience, New York, 1962. 18. E. SCHR~DINGER, Ann. Physik 79 (1926), 361. 19. I. BIALYNICKI-BIRULA AND J. MYCIE~KI, Ann. Phys. (N.Y.) 100 (1976), 62. 20. L. D. LANDAU, J. Phys. USSR 5 (1941), 71. 21. B. D. JOSEPHSON, Rev. Modern Phys. 36 (1964), 216. 22. B. D. JOSEPHSON, Phys. Lett. 1 (1962), 251. 23. R. P. FEYNMAN, ‘Statistical Mechanics,” Benjamin, New York, 1972. 24. K. W. H. STEVENS, Proc. Phys. Sot. London 77 (1961), 515. 25. J. FRONTEAU AND A. TELLEZ-ARENAS, Nuovo Cimento B36 (1976), 80.