Equivalence between the two generalized master equations

Fujita,

Physica

S.

28

28 l-297

1962

EQUIVALENCE

BETWEEN THE TWO GENERALIZED MASTER EQUATIONS *) by S. FUJITA

Facultb des Sciences de l’tJniversit8

Libre de Bruxelles,

Uelgique

Synopsis The

two

forms

Prigogine phase initial behaviour Both

of generalized

and RBsibois condition

of

of the equations

derived

gives valuable

such non-Markoffian A few remarks

they

a diagonal

master

are examined

derived

are shown to equivalently

element

exhibit

equation

by

Van

with the aid of diagrams.

of density

matrix

non-Markoffian

hints to the problem

and

by

describe the exact asymptotic .Their

evolution. in what

Hove

Given the random

interrelation

is clarified.

The way in which

conditions

and in what

equations will be reduced to the usual Markoffian

they

are

manner

master equations.

are given on this point.

1. Introduction. In 1956, Van Hovel) derived what he calls generalized master equation valid to arbitrary order in the perturbation by making the so-called random phase assumption relative to the wave function only at the initial time. This equation is essentially equivalent to the asymptotic Von Neumann equation for the density matrix in the limit of large system with a special initial condition, and is regarded as a form especially useful for the study of irreversible phenomena. Recently Prigogine and Rksibois2)3) obtained what they call generalized master equation, which is again a transformed form of the asymptotic Von Neumann equation with certain initial condition different from that proposed by Van Hove. Therefore these two versions of the generalized master equation are expected to be equivalent with each other if their solutions are made to satisfy an identical initial condition e.g. one on which the random phase assumption is imposed. However these two look at first sight very much different. It is the purpose of the present note to point out the linkage between them and elucidate the underlying physical picture. A summary is given in sec. 2 for the derivation of the two generalized master equations. These equations are based on the two series (2.58) and (2.62), representing certain mutually different forms of the expansion for a diagonal element of the density matrix. In sec. 3 these series are examined in detail with the aid of diagrams and their interrelation is clarified. A few *)

This work is supporterd

its European

by the Air Research and Development

Office.

-

281

-

Command,

U.S.A.F.

throuqh

S. FUJITA

282

brief remarks are given in sec. 4 relative two theories and the present work.

to the physical

implication

of the

2. The two generalized master equations. Let us suppose that the Hamiltonian H of a many-body system is a sum of two parts: H = Ho + IV,

(2.1)

where A is a number (coupling constant). The eigenkets of Ho are supposed to be known and will be denoted by /a> with a representing a set of quantum numbers. Some of the quantum numbers become continuous in the limit of large system. In this limit the states are normalized such that (ala’)

= S(c( -

a’),

(2.2)

the right-hand side symbolizing a product of delta-functions and Kronecker’s symbols for continuous and discrete quantum numbers respectively. Let us denote a ket of the system at time t = 0 by /qo>,which is normalized: <~o/yo> = The ket may be expanded

1.

(2.3)

in terms of {IX>}: Leo> = ./ 1~) da+).

The ket I~lt> representing the state the Schrodinger equation *)

the ket I@

(2.5)

an operator

A, diagonal

value of A referring

(2.6)

in the a-representation la).

This will be a linear function da/P(tjaa’)

so that (2.7)

to yt is defined by

t = =
(A>t =/A(a)

itself obeying

= e-dHt 1~0).

A Ia) = A(a) The expectation

evolves

at time t is related to ITO) by I@

Let us consider

of the system

lw> = H lqt).

it Therefore

(2.4)

/TO).

of A(a) and may be written

(2.8) as

da’/c(a’)12 +/A(a)

da/I@ aa’a”) da’ da” c*(a’) ~(a”).

This equation uniquely determmes Hamiltonian and a-representation. *) The units are chosen such that 5 =

(2.9)

P(t I aa’) and I(t 1aa’a”) in terms of the

1, fi being Planck’s constant divided

by 271.

EQUIVALENCE

BETWEEN

The density matrix

THE TWO

GENERALIZED

MASTER

EQUATIONS

283

p(t) of the system may be defined by (2.10)

p(L) = I(@ (vtl. In terms of p(t) the expectation

value of A is given by (2.11)

t = Tr(p(t) A} as easily checked with (2.8). This p(t) is related to the density matrix

(2.12)

p(0) = ITO) (plol at time t = 0 by

(2.13)

p(t) = eeiHt p(O) eiHt. From related

the last four equations,

one sees that

a diagonal

element

of p(t) is

to P and I in the following manner:


da’Ic(a’)

aa’a”) da’ da” c*(a’) c(a”) (2.14).

12 +JI(t[

For an initial state 1~0) with random phase the second term on the righthand side is negligible and then one obtains
(2.15)

Ic(u’) 12.

Van Hove analyzed the behaviour of P(t Isa’) while Prigogine and Resibois worked with elements of p(t). Eq. (2.14) or (2.15) gives the relation between these quantities. In the theory of Prigogine-Resibois a different initial condition is employed, in which case the full expression (2.14) must be considered. In the present article however we make comparison between the two theories assuming the simpler initial condition in which (2.15) holds. In the remainder of this section we shall briefly summarize these two theories. a) Generalized The operator

master equation of evolution

of Van

Hovel).

e-iHt may be expressed dj

Rl SE (H -

e-ilt

as (2.16)

R1,

(2.17)

I)-1.

The quantity Rz is usually called resolvent ojxmztor. The contour y encircles sufficiently large portion of the real axis counterclockwise. The function P(tIaa’) can be expressed as P(t I aa’) = -(2z)-s/dZJdZ’ Y Y where X,p(aa’)

exp[i(Z -

I’) t] X,,*(aa’),

(2.18)

is defined by {Rp4Rl*)d

/a’> =

Ia’>JA(a)

da Xllt(aa’).

(2.19)

284

S.

FUJITA

Here the suffix d indicates the diagonal part of matrix showed that Xl~f can be expressed by a series: xzz+cC’) = D&)

D&z)

x [Ww(aa’)

S(cc -

a’) + ;1W&)

&,(a)

elements.

Van

x

+ I.‘?/ Wzz~(orcc~) DZ(CYI)Dz,(cq) dcciWzz,(ccrcr’) + . ..I x (2.20)

x Dz(a’) Dz,(a’), where Dz is a diagonal {(V -

Hove

part of the resolvent

AVDzF’ + . ..) A(T/ -

Rl and W~Z,is defined by

AVD~~b’ + . ..)jidla’>

= I&)/A(a)

da Wzz~(cm’)

(2.21 j

for an arbitrary diagonal A. The symbol id means irreducible diagonal part which is obtained when all intermediate states are taken to be different from each other and from initial state. The diagonal part DZ of RI can be written in terms of a quantity GZ as D1 = (H,, -

I -

(2.22)

A2Gz)-1

with Gz = {J’DzV -

(2.23)

ill/DzVDzV’ + ...}Bd.

It can be shown that Gl(a) -

(2.24)

Gz*(a) = -i/da’@zl,(a’a)

with Wzl*(a’c() = i[Dl(a’) Using (2.24) and another Dz one obtains (I -

Dl,(a’)]

Wz~‘(a’a).

(2.25)

Gy)] DzDz’,

(2.26)

identity

Dz, = [(I -

I’) + A2(Gz -

from (2.20)

I’) Xzl*(atL’) = [Dl(cr) -

-

il2/

Dz,(a)] S(a -

Fzz,(accl) deer Xzl,(ala’)

If one introduces

an energy-dependent

a’) -

+ i12Jdalmw(oqcc) partial

transition

Xzl*(aa’).

(2.27)

probability

PE

defined by PE(t / CXC’)= (%z-~/ for t > 0, it is related to the transition

Y

dZ exp(2iZt) XE+Z,J+Z(CC~‘)

probability

P such that

P(t 1am’) = />EPE(t -00 for t > 0.

(2.28)

! cm’)

(2.29)

EQUIVALENCE

Van

Hove

obtained

in the following aPE(t 1aa’) at

BETWEEN

=

THE

from

TWO

(2.27)

GENERALIZED

MASTER

the generalized

master

285

EQUATIONS

equation

for PE

form:

fE(t/a)

S(a -

a’)

t + 27c12/ dt’/ wE(t -

t’ j acq) dalPB(t’

1ala’)

0

-

j dalwe(t

2n12Jtdt’

0

with

t’ 1ala) PE(t’ 1aa’),

(2.30)

dl exp(2iZt) @E+l,E-l(aa’)

(2.31)

-

wE(t j aa’) = (2$)-l/

fE(t /CC)= (2&9-l_/ dl exp(2iZt) [D~+l(ar) The master

equation

(2.30) is supplemented PE(O

by the initial

condition (2.33)

/ aa’) = 0.

b) The generalized master equation of Prigogine-Resiboiss)a)*). The density matrix p(t) in (2.13) satisfies the Van Neumann aP (t)

i ___

at

Suppose that single-particle

(2.32)

D,+l(a)].

= HP(t) -

p(t)H

Ez

equation: (2.34)

[H, p].

the ket Ia> should be specified by occupation states, which is usually possible :

numbers

(2.35)

/a> = ]nrns . . . ) E In) with the suffices n-(occupation

1, 2, . . . denoting single-particle quantum numbers. representation (2.34) can be written as

In the

number)

i -1
= 2 [ 7L”

One now introduces

a new notation


operator

B.

B,_,,


(2.37) is equivalent

(2.36)

such that (y)

If one defines a set of numbers

n the equation

in

n + n’ n’ = v, ____ = N, 2

(2.37) (N, V) such that (2.38)

to


v/2) = B,(N).

(2.39)

*) In reference 2 only a classical version of generalized master equation is exposed. The quantum version is trivially obtained by using the correspondence rule~)bptween classical and quantum statistical quantities relevant to the evolution equations.

286

S. FUJITA

After some manipulation

one can write the Von Neumann

equation

(2.36)

in the form: i $(N, where (v IX(N)1

t) = c (v IX(N)] Y’

(2.40)

v’) PY,(N, L),

v’) is defined by

(v /X(N)1 v’) = T/+~‘,yy+~(N)q-y with q”’ denoting

a displacement

operator

q-Y’H&N)

“17+y

(2.41)

such that (2.42)

for an arbitrary function /(NJ) of NJ. Notice that (2.40) is formally diagonal immediately written down as pY(N, t) = C (v lexp Y’ Introducing

in N. Its formal solution

i%(N)

t/ v’) ,w(N, 0).

can be

(2.43)

a resolve& operator 9i?z 3

one can express the evolution e -ixt

(ti

-

operator i _ _- 2n

s

(2.44)

2)

by

LJS$ exp( -

izt),

(2.45)

Y

where the contour y is prescribed similarly as in (2.16). Denote the diagonal part of LSz in the ,,v-representation” bygz. Prigogine and Rksibois shows that gz satisfies

an integral

equation:

L?Sz= &@&O)+ ~2SS3,WYBz

(2.46)

gz = [PO -

FSz]-l

(2.47)

z)-1

(2.48)

or z -

with QTzW E (X0 9,

= {v&(“)v

-

-

~~~z(o)~~z(o)Y

+ . . .}d.f_

(2.49)

In the last equation the symbol d.f. means diagonal fragment: part of matrix elements in which all the intermediate states are taken to be different from the initial state. It can be shown that gz may be also written as gz = {vg#-

-

~~~z~~~~

where the symbol id means irreducible

diagonal

+ . ..}W defined before.

(2.50)

EQUIVALENCE

BETWEEN

THE TWO

GENERALIZED

MASTER

EQUATIONS

Consider now a diagonal element (n jpllz> of density matrix.

287

In the new

notation this is expressed as pa(N) with N = n. By the specific assumption at the initial time (2.5 1)

py,(N, 0) = Q,t,o po(N, O), the expression

(2.43) reduces to po(N,

t) =

(0 lexp

--ix(N)

4 0) po(N,

(2.52)

0).

One sees that only the diagonal operator .GSZ in Y is important in the evolution of pa(N, t). One now has

$sW 194 0) exp(-

POP>4 =

(2.53)

izt) pa(N, 0).

Y

Using (2.47) and noting the equation (2.54)

(0 IX01 0) = 0, one can show from (2.53) that

+o(N, t) at

s

C!?(N,t -

= 0

t’) pe(N, t’) dt’

for

(2.55)

t > 0,

where 9(z) is defined by

$(N,T)

d-1

dz exp(-

&r) s,(N).

(2.56)

Y The equation

(2.55) is called by Prigogine

and Resibois

generalized

master equation. c) Remarks It is noted that the both generalized master equations (2.30) and (2.55) are obtained by using a set of identities (valid for infinitely large systems.) In other words no approximation is made in the processes of transformation. Therefore they ought to have mutually equivalent content as specialized forms of the Von Neumann equation. However they are not to be compared directly each to other because they refer to the mutually different quantities PE

and

=

po(N, t), (N =

n).

The equation (2.30) originates in the equation written in the n-representation as X&zn’)

= Dz(?z) Dz+z) S(%, n’) + nz,D(n) Dr+z)

(2.20),

which

may be

x

x Dz(d) Dz+z’).

(2.57)

288

S. FUJITA

Multiplying n’, one has

this by lc(n’)iQ = and subsequently

xzz,(fi) = &(n)

&(fi)

+ i2&(n)

summing

on

(n Ip(O)l n>

&,(nj

c

WZZ+W)

Dz~)

Dz&)

:m

Ip(

w

+

...

(2.58)

n1

with

Xll,(?z) = c XLr’(nn’) lc(n’)1Q. ?L’

Comparing

the last equation (n Ip(

The equation

(2.59)

with (2.15) and (2.18), one has

n> = -(2~)-~_/dZ/dZ’ I’ i’

exp[i(Z -

(2.60)

I’) t] xlr,(n).

(2.46) has a formal solution (2.61)

LSn,= .L?Jnz(Q) + AQ~~(QM@~(Q) + 14 . . . . Multiplying

this from the right by pQ(N, 0) = (n Ip(

B&&V,

n), one has (2.62)

0) = LSz’Q’po(N, 0) + )3QG@z(Q)~zLSz(Q)pQ(N, 0) + . . . .

In view of (2.53) the left-hand side is a resolvent of p&V, t) = in the different complex planes. Therefore comparison between these two may be made conveniently. We shall do this in the following section. In the theory of Van Hove two complex variables Z and 1’ are necessary for dealing with two time-dependent operators e-iHt and eiHt in (2.8) while in that of Prigogine-Resibois only one variable z is used. This simplicity of the latter theory is somewhat compensated by the fact that the resolvent operator &Yzand therefore operators.

its diagonal

part gz are no longer numbers

3. Diagram representation. Interrelation

between the two theories.

but

In order

to clarify the relation between the two theories it is convenient to introduce diagrams representing the perturbation expansion of the element . There exist several equivalent diagram representationsa-9). We shall first pick up the one introduced by the present authorQ) for the later convenience. For definiteness let us consider a system of interacting particles obeying the Bose or the Fermi statistics. Other interesting cases such as electronphonon, electron-impurity systems may be treated in a similar manner. In our example we have the Hamiltonian in second quantization”) H = C p2aJap + + G P

C u(p + q -

r-s)

afaiapaq6(3)(p + q, r-s),

P,q,r,s

*) The units are chosen such that 1M =

!, where A4 denotes

the mass of a particle.

(3.1)

EQUIVALENCE

BETWEEN

THE

where v is the Fourier transform

TWO

GENERALIZED

of pair potential,

MASTER

EQUATIONS

289

52 the volume and S(a)@, q)

a three-dimensional Kronecker’s delta symbol; LZ~and a: are annihilation and creation operators satisfying the usual commutation or anticommutation

relations.

The kinetic

energy part is chosen as unperturbed

Hamil-

tonian : (3.2)

Ho = c pu;L$. The eigenstates of Ho can be specified by numbers n of particles single-particle momentum states. Consider now the element

occupying

(92 /p(t)1n) =
U(t)

=

1 +

e-6*&U(t),

0

(-

=

. ..T&., 0

J,-(~) _

Substituting such series for e-t*t with quantities of the form

and ei*t may be expanded

&ft

(-iI)~;dt$lts

; 1

with

E

e-i*t

u+(t)

&*t

V(tr) V(t,) . . . V(t,).

(3.3) in per-

(3.4)

(3.5)

0

eiHv

v

e--iH~~.

(34

and ed*t into (3.3), one will have to deal

iii)k (G)“’
(3.7)

The operator p(O) being diagonal in n by assumption (2.15), such a quantity as (3.7) can be decomposed into a sum of complete contractions by means of a theorem*) similar to Wick’s theorem and the resulting contractions will be represented by diagrams, which may be constructed as follows: Draw a horizontal boundary line. Corresponding to each V(t,‘) coming from eiHt write in a point at t = t3’ above the boundary, where time is measured from the right to the left. Similarly for each V(tj) coming from e-iHt write in a point at t = tj below the boundary. Connect these points by directed solid lines such that (a) each point may have two lines leaving and two lines entering and (b) the completed diagram may consist of a certain number of closed subdiagrams. A diagram thus constructed corresponds to a non-trivial construction of (3.7). Each directed line connecting two points denotes a contracted pair of annihilation and creation operators referring to the same momentum. A closed diagram consists of only these lines and therefore represents a non-trivial complete contraction. *) The contraction theorem for the (n, n) matrix element is rigorously valid for a Fermi gas 8) and approximately so (- 0( l/N)) for a Bose gas above the temperature of Bose-Einstein condensation. The argument is subject to this limitation,

290

S. FUJITA

Mathematical expressions corresponding to a diagram can written down. Such a prescription may be found in reference 9.

be easily

Consider now diagrams of the second order in il. Assuming that the two interactions V are labelled by tl and tz, we have the four different diagrams drawn in fig. 1. P f ,

$zj

5 j rt

t, --c------

t2 --@-b

4

a

c Fig. 1. c-number

diagrams

of the second order.

From the general rule established in reference 9 it is clear that these four diagrams correspond to one operator diagram *) in fig. 2. The correspondence is established such that (u) the c-number diagrams are identical in structure with the operator diagram if the boundary lines in the former are suppressed, and (b) in general the number 2” of c-number diagrams of order n corresponds to one operator diagram.

Fig. 2. An operator

diagram

of the second order.

Such operator diagrams are evidently to represent those contractions arising from the perturbation expansion of (0 lexp ---i%(N) t IO) in (2.52), the expansion being made through a formula similar to (3.4). We can study the behaviour of (0 lexp --itit 0) alternatively by resolvent method. Then the perturbation expansion for the resolvent W, or its diagonal part Liz becomes somewhat simpler, for example: g2 = gz(a) + ;/2~#J)V&(a)V$(a) *)

Operator

diagrams

were first introduced

by Prigogine

+

j13 . . . .

and RBsibois

3)7).

(3.8)

EQUIVALENCE

BETWEEN

It is well-known7) identical

THE TWO

GENERALIZED

3) that one can represent

in structure

MASTER

29 1

EQUATIONS

this series by operator

diagrams

with those such as shown in Fig. 2. The prescription

of obtaining mathematical expressions for these diagrams are given in reference 7 or 3. If we now rearrange this series by the number of diagonal fragments, we simply obtain Prigogine-RGsibois’ series (2.61). Let us introduce two resolvents RI and RI, associated with eiHt and e-iHt, respectively.

In terms of these, the element


The resolvents

s Y

dl

s Y

(n If(t)/ n> can be expressed

dl’ exp[i(Z -

RI and RI* can be expanded

I’) t] .

as

(3.9)

in series, for example, (3.10)

RL = R,@) + AR&“)VR$“) + 12 . . . . with R&O) = (Ho -

(3.11)

Z)-1.

Upon substitution of these series the element those V’s coming from RL~ are denoted by points, below the boundary, arranged in the same order as they appear in the expansion of Rll while those V’s from RI by points, above the boundary, arranged in the reversed order as they appear in the expansion of Ri. Except for this rule the process of completing diagrams are the same as for time-dependent diagrams. The prescription for obtaining mathematical expressions from diagrams may be seen in reference 5*). It should be noticed that in a diagram the two sets of V’s coming from RI and V’s from Rl’ are ordered right-to-left in each set, but not ordered mutually. Consequently diagrams in I- and It-planes are not in one-to-one correspondence with time-ordered c-number diagrams. However a correspondence between these can be easily made by introducing mutzlal ordering between all V’s in the former and subsequently making correspondence between each of such ordered diagrams and a time-ordered diagram. It will *) In this reference characterized

a slightly

different

H = r, E&Q

+ E EkAiAk

k

Our about

type

of diagrams

representation

+ M-1

k

may be reduced

p(O), which is supposed

the boundary.

diagram

is used for a different

system

by

to that

to be situated

C ~~a&z~Af_~A~. %k,K

appearing

at the right

here by making

reflection

end of the boundary

of the I-plane

line, and suppressing

292

S. FUJITA

be then seen that of time-ordered

a diagram

diagrams.

in (I, I’)-planes

This is illustrated e

Fig. 4. Schematic arrow

a corresponds diagrams

sketch of the interrelations

indicates

by an example

sum

in fig. 3.

C

(&I’)-diagram c-number

two-way

to a certain

e

3 Fig. 3. X

corresponds

existence

arrow

of simple

that of one-to-one

to three

time-ordered

b, c and d.

between

four types of diagrams.

correspondence

and a thick

X thin

two-way

correspondence.

In contrast to the case of time-ordered diagrams, time-dependent diagrams i.e. 1, I’-diagrams and z-diagrams, do not have a simple correspondence. Unfortunately this makes complicated the relation between Van Hoven’s and Prigogine-Resibois’ theory. However, they may be compared indirectly by the three correspondence rules existing between four types of diagrams so far discussed, which is schematically shown in fig. 4, In order to obtain Van Hove’s series (2.58) we shall take the following steps : Consider a class C of diagrams which have no lines crossing over the boundary. For a diagram of this class C we can discuss its configuration in the l-plane independently of that in the I’-plane. In the 1 plane we may analyze various configurations by means of diagonal fragment and obtain Dl for the total configurations. Similarly we obtain DI, for the total configurations in the I’-plane. Therefore the total contribution of the class C to is given by D@)

Dz+)


a>.

(3.12)

EQUIVALENCE

BETWEEN

Consider an arbitrary

THE TWO

GENERALIZED

MASTER

EQUATIONS

293

diagram. The Z-plane or the I’-plane can be thought

as being composed of a number of regions, whose boundaries are marked by vertical lines hitting through points of interaction. Each region represents a state. In either half-plane there may be several regions within which no particle lines are present connecting between points in the same plane. Such regions are called free regions. If two successive free regions are separated only by a diagonal fragment, embedded entirely in its own half plane and extending over several non-free regions, these two represent one and the same state. For example, every free region in a diagram of C and therefore the first and the last free region represent the same state. If a configuration in the half plane contains two or more free regions representing an identical state, it is called improper. Otherwise it is called proper. Improper diagrams can be uniquely reduced to proper ones by suppressing diagonal fragments occurring between successive free regions. The reduction is obstructed only by the presence of a set of subdiagrams crossing over the boundary and their associates. The process of reduction is illustrated in fig. 5.

b

a

Fig. 5. Reduction

of an improper

diagram

a into a proper

In the class C there exists only one proper diagram, contribution is given by D&O)(n) D@)(n) The similarity

between


n>.

(3.12) and (3.13) is remarkable.

diagram

b.

whose corresponding (3.13) From this example

we can guess (and prove) that the total contribution corresponding to a sum of a proper diagram and those improper diagrams which upon reduction give rise to the proper one, can be obtained from the mathematical expression corresponding to the proper one by replacing D$a) and DJ,(@ respectively by Dl and Dir. Let us now consider a proper diagram not belonging to C. It can be seen that in this diagram every free region above the boundary is paired with one below such that the both represent one and the same state. It can also be seen that a set of subdiagrams, at least one of which crosses over the boundary and which separate two successive free regions, represents a component of I4’~r, introduced in (2.21). In fact this set may have a certain number of diagonal fragments entirely embedded in either I- or If-plane.

294

S. FUJITA

Such diagonal fragments may be eliminated by a reduction procedure just discussed and consequently Wllq will be expressed in terms of Dl and Dt, as in (2.21). We call such a set w-set. If we reorder all proper diagrams by the number of z&s, the corresponding contribution to
n>

The contribution of total diagrams to (?-t j&p(O) &I IZ.) will be obtained by adding to this the one corresponding to improper diagrams, which is done by L)g and DIP. The result is expressed by Van by replacing DE(0) and I)@ Hove’s series (2.58). We may summarize the analysis done in this section as follows: i) Corresponding to four types of perturbation expansion for (n lpi n>, four types of diagrams i.e. time-dependent c-number and operator diagrams and their time-independent associates, are introduced. Their interrelations are schematically shown in Fig. 4. (2.62), ii) The both series, Van Hove’s (2.58) and Prigogine-Resibois’ are obtained through systematic reorderings of the perturbation expansion for . 4. Remarks. We shall make a few remarks about the two theories and the present work. i) We have seen that the two basic series (2.58) and (2.62) equivalently describe the exact behaviour of (n Ip( n>. It is important to note that their corresponding generalized master equations (2.30) and (2.55) have both non-Markoffian. terms. This may be regarded as fundamental characteristic of the exact evolution of (n ]p(t)l n). In Van Hove’s equation for PE(~) there is an inhomogeneous term while no such term is present in Prigogine-Resibois’ equation for . Therefore, this may seem to disturb the equivalence between the two theories. However, the apparent discrepancy is transitory. It is shown in the Appendix that no inhomogeneous terms appear in the evolution equations for P and consequently for 0. ii) It is well-known that in certain conditions the ~Iarkoffian-tie evolution equation for P or approaches to a simpler Markoffian equation, is a central problem of the statistical mechanics of irreversible processes. In fact more value may be assessed to the study of this problem than to the derivation of an exact generalized master equation. The way in which this equation is derived, gives valuable hints to the problem.

EQUIVALENCE

Consider

BETWEEN

the change

THE TWO

GENERALIZED

due to interactions

MASTER

EQUATIONS

in the probability

295

of

finding the system in a state n. This change will in general be given by a balance between the gain in the probability due to transitions from n’ # n to n and the loss due to transitions from % to ,” # n. It is well-known4) g, lo) that a closed diagram or a diagonal fragment entirely embedded in the upper or the lower plane such as shown in fig. la or Id, contributes to the loss in the probability referring to the state represented by a region immediately left to the diagram while a closed diagram extending over the two halfplanes such as shown in Fig. 1b or 1c contributes to the gain in the probability. In Van Hove’s theory this distinction is carried out in a complete fashion. Loosely speaking, all the ,,loss” processes represented by Gr and Gl, are eliminated from the diagrams by the reduction procedure (D&O), Dr,(O)) --f (Or, Or?). Then all the ,,gain” processes are ordered as in the series (2.58). Finally through the identities (2.24) and (2.26) the ,,gain” and the ,,loss” term are brought out in the second and the third terms, respectively, of the generalized master equations (2.30) or equivalently in (2.27). It is noted that the above separation is possible only when one works with c-number diagram method or equivalently with theory in which the two evolution operators e-int and etHt, or their resolvents RI, and Rl, are expanded separately. In Prigogine-Resibois theory the ,,gain” and the ,,loss” are treated simultaneously and in a systematic manner such that the balance may vanish identically*) for the uniform ensemble which is caracterized by
S.

296

FUJITA

APPENDIX

Absence of inhomogeneous term in the evolution equation for P. Let us take the equation (2.27). If one multiplies the both sides by a factor -i(2n)-2 and subsequently side is equal to

integrates

exp[i(Z -

I’) t]

on Zand l’, one sees that for t > 0 the left-hand aP(t 1ax’)

(A.1)

at in view of (2.18), and the first term J of the right-hand J := --i(27c)-2

S(a -

exp[i(Z -

a’)/dZ/dl’ 1’ 1’

side is given by

1’) t][Dl(cc) -

Ill,(a)].

(A.2)

We wish to show that J vanishes for t > 0. From (2.17) one obtains an identity Rl Taking

its diagonal

part and applying

Dl(a) Substituting one obtains

RL! = (I -

the definition

Dl,(a) = (I -

(A.3)

1’) RIRl~. (2.19) one has

I’) / da1 Xll,(ala).

this into (A.2) and carrying

out the integrations

J = d(a - a’) /dai ;

(‘4.4) on Z and Z’,

P(t/crlcc)

f > 0.

for The right-hand constant :

side vanishes

identically

j‘dalP(t which is obviously Received

(A.5) because

the last integral

/ala) = 1,

is a (A.6)

seen from (2.8) and (2.9).

31-10-61 REFERENCE

1) -4 3)

Van Hove, L., Physica 93 (1957) 441. Prigogine, I. md Rksibois, P., Physica 27 (1961) 629. Rksibois. P., Physica (1961) 541; Prigogine, I., Balescu, P., Physica 26 (1960) S 36.

R., Hellin,

F. and RBsibois,

EQUIVALENCE

4) Van

Hove,

Seattle, 5)

Van

L., Lecture

Wash.,

Hove,

Hermann 6)

Prigogine,

7)

Prigogine,

BETWEEN

U.S.A.

L., La

Paris,

8)

Nishikawa,

9)

Fujita,

notes given at the Department

MASTER

of Physics,

EQUATIONS

University

297

of Washington,

tht!ooriedes gaz nelctres et imist%. (Edited by C. De Witt and J. F. Detoeuf,

(r6dig8

par

P., Physica RBsibois,

SC. Nucl.

Belg.,

K., J. phys. Sot. (Japan)

S., Physica

Hove,

GENERALIZED

(1958).

I. and RBsibois, I.

TWO

1960) p. 151.

(Inst. interuniversitaire

10) Van

THE

27 (1961)

L., Physica

940.

21 )1955) 517.

24 (1958) 795. P.),

Superfluid&

Bruxelles

1960).

15 (1960) 78.

et iqquatiolz de transport

quantique