A new representation of subdynamics by an N-projectors method

Physica 63 (1973) 3347

0 North-Holland Publishing Co.

A NEW REPRESENTATION

OF SUBDYNAMICS

BY AN N-PROJECTORS

METHOD

Z. DEMENDY Physical Department of Universityof Heavy Industry, Miskolc, Hungary Received 18 March 1972

Synopsis

By means of N projectors a realization of systems is given. It is shown that the method indistinguishable particles is equivalent with a correlation-patterns representation. The basic the purpose of future applications.

the subdynamics for a system of N arbitrary subin the case of a system of an inlkite number of class of solutions of the equations of the dynamical formulas of the subdynamics are reformulated for

1. Introduction. Since the appearance of the pioneering work of Prigogine, George and Heninl) several studies were devoted to the extension of the concept of subdynamics for inhomogeneous systems. As a result of these papers2-5), the subdynamics was formulated for inhomogeneous systems with a stationary external field. Moreover, it was shown6) that the subdynamics is an inherent property also for systems with arbitrary time-dependent liouvillian. All these approaches were kept on a rather abstract level using only the possibility of a decomposition of the global distribution function (d.f.) by means of some time-independent projectors P and Q. The components fv and fc generated by P and Q are usually called the vacuum and correlation parts of the global d.f., respectively. In this way the precise meaning and the applicability of the subdynamics depends on the possibility of the realizations of the projectors P and Q. For homogeneous systems, the vacuum part is defined by a projector which integrates out the entire spatial dependence of the global d.f.l); this, in Fourier representation is equivalent to retaining only the Fourier component for which all wave vectors vanish. Among the various proposals7-10) of projection operators for inhomogeneous systems we mention here only two realizations originating from Balescu. A few years ago Balescug) proposed to identify the vacuum part as a product of one-particle distributions. Although this realization is rather plausible, because of its nonlinearity it had to be abandoned. Very recently, it has been shown by Balescul”) that a linear realization of the projector P can be achieved through the dynamical correlation patterns (d.c.p.) which is a new representation of the Liouville equation. 33

Z. DEMENDY

34

A formulation of the subdynamics through d.c.p. has several advantages; for example, it avoids the difficulties coming from the thermodynamic limit, and the various terms appearing in the equations can be represented by a diagrammatic technique. In this paper an alternative approach is proposed. It is shown that the abovementioned nonlinear realization of the projector P can be achieved by an N-projector technique for a system of N arbitrary subsystems. This realization leads to an explicit formulation of the starting equations of the subdynamics. At this stage of the theory there is no need to use a thermodynamic limit, of course. At a next stage we assume in particular that the system under consideration consists of an arbitrary number of indistinguishable subsystems, which allows us to transform the basic equations into two infinite hierarchies of equations describing the evolution of the vacuum and correlation component of the “distribution vector” of the system. These two hierarchies correspond to a decomposition of the usual BBGKY hierarchy if the subsystems are structureless particles. At this stage of the formalism a thermodynamic limit should be taken, of course. It is shown that the hierarchies obey equations identical with those derived by using the d.c.p. As for the comparison of the d.c.p. representation with the present treatment it can be said that although there is an equivalence between them the characters of the two approaches are entirely different. The N-projectors method leads to explicit expressions for the basic formulas of the subdynamics and in the applications these formulas will have to be concretised and simplified only. Within the framework of d.c.p. the equations for applications are achieved by a rather elaborate summation procedure of certain classes of diagrams. In this paper all the important formulas of the subdynamics are given for practical applications but actual calculations are postponed to forthcoming papers. 2. The N-projectors method. We consider here a system of arbitrary subsystems Al, 4, a**, A,. A subsystem A, will not be further decomposed into smaller systems. The whole system is under the action of an external field. In this case the Liouville equation for the d.f. of the whole system f(x, , x2, . . , , x,; t) has the following form :

(2.1) LO (i; t) = L (i; t) + L’ (i; t),

(2.4

where L (i; t) denotes the Liouville operator of subsystem AI, while L” (i; t) is the liouvillian of an external field acting on the subsystem Ai. All the variables of a subsystem Al are abbreviated by xi or simply i. The interaction between the subsystems AI and Aj is characterized by an operator L’ (iI j; t) which acts only on the

REPRESENTATION

OF SUBDYNAMICS

35

variable x, and with respect to x1 it is a function. The connection between the present form and the usual one of an interaction liouvillian is the following L:,(t) = L’ (i/j; t) + L’ uli; t) L’(ili)

ViE{1,2 ,...,

= 0

i #j,

if N}.

(2.3)

Obviously eq. (2.1) is a generalization of a Liouville equation describing N interacting point-like particles. Now, we introduce the integral operator pi(t) =f(xl;t)f(xz;t)

‘..f(xi-l;t)f(xi+l;t)

*.*f(xi~;t)Jd{Nli),

iE(1,2 )..., N),

(2.4)

where d {N/i} = dx, dxz ... dxj_l dx,,, f(xi;t)

... dx,,

(2.5)

= Sd{N/i}f(xl,xz,...,x~N;t),

(2.6)

is the d.f. of the Ai subsystem. By the normalization jdx,f(x,;t)

= 1,

iE{1,2

,...,

N},

(2.7)

the operator P,(t) becomes a projector. By means of Pi(t) we identify the vacuum and correlation components as pi(t)

= fi ftxj; 0 =_W), j=l

iE {1,2, . . . . N),

(2.8)

Qdt>fCt>

=fCt>

iE

2, . . . . N},

(2.9)

-.fXt>

sLf,(t>~

*

{I,

.

where

Q&> = I-

Pi@>,

(2.10)

and Qi(t) is also a projector. It is obvious that Pi(t) and Qi(t) satisfy the following identities :

J’,(t) Qttt) = Q,(t) pi(t) = 0,

(2.11)

P*(t), p,(t)l-

= 0,

(2.12)

[Qdt>, QAt>l- = 0,

(2.13)

[Pi(t), Lo (i; t)]-

(2.14)

Pi(t) Lo (j; t) = 0

= 0,

if

i #j,

(2.15)

36

Z. DEMENDY

where the last property is a consequence of the differential character of Lo (j; t). Hereafter we omit the explicit writing of the time arguments at all places where it does not lead to ambiguity. To get a coupled system of equations for components fV and fc we shall use the identities x2, *-*, xi4 = &_L

(2.16)

VjE(1,2

(2.17)

PiL’ (jli) = 0,

)...) N}.

The latter equality clarifies the usefulness of writing an interaction liouvillian in the form (2.3). Premultiplying both sides of eq. (2.1) with P1 and using (2.17), (2.14), (2.15), (2.8) and (2.9) we obtain pi

at.f

=

1Lu(i)

ViE(1,2

+

jil

piL’

Cili)]

_A

f

pi,il

L’

(jli)_L

3

)...) N).

(2.18)

With (2.16) the sum of eqs. (2.18) gives the following equation for-f,:

Subtracting eq. (2.19) from eq. (2.1), with (2.9) one obtains

(iilLo(i) +i :

&L =

i=1

j=l

Q&'(jli)

Qi)L

+t ; i=l

j=l

QJ'(jli)PJi.

(2.20)

Eqs. (2.19) and (2.20) strongly resemble the usual starting equations of the subdynamics .

a,.fi

=

(Lo

+

PLP)f”

+

J’LQ.fi,

&.A = (Lo + Q-&Q>_& + QLPJ;.

(2.21) (2.22)

Comparing (2.21)-(2.22) with (2.19)-(2.20) leads to the following correspondence LO

PL,P

p&Q

(2.23)

=i~lUiI,

= 5 i=l

=

$

i=l

2 P,L’(jli)

Pi,

(2.24)

Qi,

(2.25)

j=1

f

j=l

PJ

W

REPRESENTATION

QL,P = t i=l

QLQ =

37 (2.26)

c QiL’ (jli) Pi, j=l

5 5QJ

I=1

OF SUBDYNAMICS

(2.27)

tili) Qi *

j=l

In the formal theories of subdynamics 2*4*5)the operators Lo and L, as well as the projectors P and Q are assumed to be time independent. As was shown in ref. 6, the subdynamics can be formulated even if all the quantities in the starting equations (2.21) and (2.22) depend on time. It is important to note that eqs. (2.21) and (2.22) involve a basic requirementlO), namely

[w>,Lowl-

(2.28)

= 0,

which means that the operator Lo cannot change the state of correlation of a set of particles. The correspondences (2.23)-(2.27) show that the basic assumption (2.28) is satisfied by eqs. (2.19) and (2.20) which fact strengthens the validity of the realizations (2.8) and (2.9). In eq. (2.21) the term PL,Pfv is usually called Vlasov term. In the present representation this term retains its original meaning PL,Pfy = ? i=l

i

s dx,f(x,;

t) L’ (jli; t)_Mt) = :

j=l


t=l

i

;_&(t), (2.29)

j=l

(2.30)

(t)f(x,; 0,

where in the last equality it has been taken into account that J dxiL’ (jli)f(xl,

~2, . . . . xN; t)

=

0,

(2.3Oa)

because of the differentiating character of L’ (j/i). In the present approach the vacuum part of the global d.f. is represented as a product of the distribution function of each subsystem. This realization in the case of indistinguishable subsystems reduces to the proposal of Severnell) and Balestug), which was named “I?-body construct”; moreover, this vacuum component for a homogeneous system becomes identical to the realization used in ref. 1. In this way we can conclude that eqs. (2.19) and (2.20) may be regarded as starting equations of the subdynamics for a classical mechanical system. On the other hand it is obvious that for the purpose of practical applications eqs. (2.19) and (2.20) are not too convenient for a system consisting of indistinguishable subsystems. In this case it is useful to transform eqs. (2.19) and (2.20) into matrix equations which describe the evolution of vacuum and correlation components of a “distribution vector” introduced by Ginibrel’) and BalesculO). We define the “distribution vector” f, as : f=

{_fi(Xl),_ml, 4, **.>,

Z. DEMENDY

38

where the sth component of the “vector” f is the reduced distribution f,(x,, x2, . . . , x,; t). It should be emphasized that the “distribution-vector” method has only meaning for a system of infinite number of indistinguishable subsystems where all the reduced distributions fs remain finite in a thermodynamic limit. The reduced d.f. f, differs from the distribution f(xl, x2, . . . , x,; t) in a normalization factor, which fact will have to be taken into consideration in the forthcoming manipulations. In the following we transform eqs. (2.19) and (2.20) into hierarchies of equations for the vacuum and correlation components of the reduced distribution f, which are denoted by fsvand f:, respectively. Integrating eqs. (2.19) and (2.20) over all subsystems but that which is numbered s, one immediately gets

•~~$,&s+J’(s +lli)f’z(s+ 1, ~)fs’-l({4~}>,

(2.3 1)

j&’ wf:

+i&id~s+,L’ 0 + 119{f,‘+,Cb + I>>-f’z@ + 1, i)f,‘-db/Q), (2.32) where

{s+ l> = (Xl, x2,

.. ..

x,+1),

(2.33)

The notation

(L’(l’)) = J dxfl(x, t) L’ (xlxt; t),

(2.35)

expresses the irrelevance of the variable of the integration. Adding eq. (2.32) to eq. (2.31) the sth equation of a BBGKY hierarchy (2.36) is obtained as should be the case. This last result means that the projectors Pi(t) lead to a direct decomposition of the BBGKY hierarchy. In Balescu’sl”) paper this decomposition was carried out via the d.c.p. representation. Before a comparison of the present approach with

REPRESENTATION

39

OF SUBDYNAMICS

the Balescu one the physical meaning of the terms in eqs. (2.31) and (2.32) is briefly discussed. Eq. (2.3 1) describes how the state of an “uncorrelated” group of s particles evolves in time. Here the word “uncorrelated” means that there is no interaction between the particles of the group under consideration, therefore any variation of the function fs’ may arise only from the interaction with the particles of the environment. The correlations of the environment act on the group of particles via a pair interaction, therefore only the pair correlation function f: appears in eq. (2.31). The interaction between a particle of the set of s particles and a particle of the environment modifies the state of the particle under consideration, of course, but this effect does not create any correlation in the s-particle group. This fact explains why the function fJ_, does appear in eq. (2.31). In eq. (2.32) only the meaning of the last term of the r.h.s. requires an extra explanation. It is obvious that any variation of the correlations in a system of s particles is a consequence of the internal interactions or of the interactions with the environment. The function fSc+i contains all the possible correlations in the s-particle system and in the environment. Among these correlations particularly those occur which correspond to the terms of type fi”fsv-1. But as has been shown, this type of terms causes variation only in the function fsv and not in fsc, therefore those terms have to be subtracted from fsc+1. For the purpose of a comparison of Balescu’s approach with the present one we write down eqs. (2.31) and (2.32) for the case s = 2. &f;

= {LO(l) + L0(2) + (L’(I1)) + )f’, (2.37)

+ S d+ [Lhf H(k3) fi(2) + L;s f 4(3,2)f,(l)l, &ff

= (LO(l) + L0(2) + Li,)ffi

+ L;,f;

+ J dx, ILL13 + LulfC3 - L;sfN,

3)f,(2)

- LhsfX2,3)f,(l)), (2.38)

where the notation (2.3) may be applicable because of (2.30a). It is easy to see that with the relations fi(l) f X2)

(2.39)

= Plh), = P2 (x1 ; 4

f su, 2) =

= Plh)

(2.40)

P1h),

(2.41)

Pz (Xl-d,

f c3(1,2,3) = P3 (Xl ; x2x3)

P3 (Xl ; x2x3)

=

Pled

+

P3 (x2;

Pz (XIX?) 9

x1x3)

+

P3 (x3;

x1x2)

+ p3 (%X2X3),

(2.42) (2.43)

Z. DEMENDY

40

eqs. (2.37hand (2.38) become identical to eqs. (4.11) and (4.12) of ref. 10. Although the present representation of the state of an uncorrelated set of s particles in the form s fs’=Ps(xl;xz;...;xJ = flIpl(Xl) i=l

corresponds only to a special solution of the basic equation (4.6) of ref. 10, no other type of solution seems to be acceptable from a physical point of view. In this way we conclude that solutions to eqs. (2.31) and (2.32) provide an appropriate description for the evolution of a system of indistinguishable particles. In order to show that eqs. (2.31) and (2.32) can be regarded as the sth component of a single matrix equation for the distribution vector components fv and fc, respectively, we introduce the integral operator

ps,i(G= fLW>;

t) j d Cb/ilL

(2.44)

fLl({~/~l) = flWfl(2) --f1(i - l)flG + 1) --f1(s).

(2.45)

where

This operator is analogous to P,(t) in a reduced description. Beside Pslt it is useful to use the symbol j dx,+J’

(s + lli) =
(2.46)

With (2.44) and (2.46) the third term of the r.h.s. of eq. (2.31) becomes

Eq. (2.47) can be abbreviated by the “scalar product” (2.48) Similarly the last term of the r.h.s. of eq. (2.32) can be rewritten as

Q,(t) G(t) =

,ilQ,iW

,

Q,t = I - Ps,i.

(2.49)

(2.50)

With the symbols (2.48) and (2.49) eqs. (2.31) and (2.32) become

&f I = @,” + Cl f: + PJ3f w:

=

{L_fZ +

W3f

,“+1,

s”+ I> + Cf :,

(2.51) (2.52)

REPRESENTATION

41

OF SUBDYNAMICS

where (2.53)

L, = L,O+ L,‘,

(2.54)

(2.55)

c =jl

(2.56)


Using the distribution

vectors

fv = {“MXAfXX1~

4,

fc = (0, fXx19

fC3b1, x2, xd, . ..I,

and the operators (W&r)

4

(2.57)

* * .I 3

(2.58)

L?,, _?ZP, ?Z’c and _CPC defined by the matrix

= d,,, (C + C)

elements

3

(2.59)

W~Pl4 = &+ 1,”PSG) 3

(2.60)

GPQl4 = &,“C,

(2.61) (2.62)

we obtain

the basic equations

of the subdynamics

in a matrix representation:

Jt.6 = ~Y(O fy + T,(t) f,,

(2.63)

JA

(2.64)

= -G(t)

It shall be found form

fv + Z&)

A*

useful to write eqs. (2.63) and (2.64) into a more concise matrix

&f = WI f(t), L(t) =

f=

Y,(t) ~,@I

(

; 9 ( c>

(2.65)

~I@)

Z,(t)

>’

(2.66)

(2.67)

42

Z. DEMENDY

where f may be called “super distribution vector”. The formal solution of eq. (2.65) is

f(t) = lJ cc to>f(h),

(2.68)

U (t, to) = Y exp i dzL (z), to

(2.69)

where U (t, to) is the Liouville propagator, F is the time-ordering operator and to is some initial instant. It will be shown that U (t, to) can be expressed in terms of some operators containing the asymptotic limits of the distribution vector f. The next section is devoted to a brief derivation of the basic formulas of subdynamics in the present representation. 3. Basic formulas of subdynamics in distribution-vector formalism. As it has been mentioned the time dependence of the Liouville operators does not cause difficulties in the formulation of subdynamics. Here we adapt a method, with little modifications, developed in ref. 6 to achieve the formulas relevant for practical applications. Let us write the formal solution of eqs. (2.63) and (2.64) in the form (3.0

(3.2)

(3.3)

9 (t, to) = LT exp j dz_?ZV (r), to

(3.4) If one chooses to = 0 as an initial time instant, the asymptotic time limits can only be defined as t -+ + co. From both a mathematical and a physical point of view, it shall be more convenient to use the time limits t,, + i 00 rather than the limits t+

&co.

In the present treatment, we assume that for dissipative systems

9 (4 to)fXh) + 0

for

to + +a,

(3.5)

Gfz 0, to).m) --f 0

for

to--f -co.

(3.6)

REPRESENTATION

43

OF SUBDYNAMICS

[In ref. 6 a somewhat different but equivalent form of (3.5) and (3.6) was given.] In this study, not the validity but the consequences of (3.5) and (3.6) are subjects to further investigation. As we shall see the assumptions (3.5) and (3.6) provide the necessary and the sufficient conditions for the existence of the subdynamics; in this way, the conditions (3.5) and (3.6) may be regarded as a natural extension of the usual regularity conditions1*2) for systems with time-dependent liouvillians. We denote the asymptotic time limits of the global distribution vectorf(t) as

f(t) *f-(t)

for

to --) +oo,

(3.7)

f(t) +f’W

for

to + -co.

(3.8)

Using (3.5) and (3.6) eqs. (3.1) and (3.2) become

Since the operators ZP, _Y, and .49, contain explicitly the various components of the distribution vector f,, the notations

would be correct in the asymptotic time limits. In order to very cumbersome notation in print we retain hereafter the above-mentioned operators. This fact should be born in spective equations in explicit forms. Substituting eqs. (3.9) and (3.10) into the corresponding and (2.60), respectively, we obtain

avoid the application of original symbols for the mind by writing the retime limit of eqs. (2.61)

We look for the solution of eqs. (3.11) and (3.12) in the form

fY+W = U” tt, to)fY+(to)

v t,

(3.13)

fi-0) = UC(t, to)fl(to)

v t,

(3.14)

Z. DEMENDY

44

where U, (t, to) = F exp i dtL, (r), to

(3.15)

U, (t, to) = F exp jdzL, to

(3.16)

(z).

The eqs. (3.11) and (3.12) with the “ansatz” (3.13) and (3.14) give the following expressions, respectively

L,(t) = %O~+ s,(t) C(t),

(3.17)

Lo(t)= p,(f) + 9,(t) DO),

(3.18)

(3.19)

(3.20)

D(t) = ; dzg (t, Z) =Yp(z) U, (z, t). Co

Using the operators D and C eqs. (3.9) and (3.10) can be rewritten as

f;(t) = W)fLW,

(3.21)

fc’(t) = Cwfm.

(3.22)

Differentiating the formulas (3.19) and (3.20) with respect to time, one obtains

ate = PC(t) C(t) + ~Q@>-

C(t) LdO,

(3.23)

&D = 9,(t)

D(t) L,(t).

(3.24)

D(f) + Yp(t)

-

By means of formulas (3.17), (3.18), (3.23) and (3.24) it is a straightforward algebra to show [for a detailed calculation cjI ref. 61, that

&A (f> M(t) - DW.Lt~N = L,(t) At0 W)

- D(t>L(t>l,

(3.25)

&A 0) Wit)

-

C(t)fv(t)l,

(3.26)

-

C(t).W)l

= L,(f) A(t) Vi(t)

where A(t) = [I - D(t) C(t)]-l,

(3.27)

A(t) = [I -

(3.28)

C(t) D(t)]-‘.

REPRESENTATION

OF SUBDYNAMICS

45

The formal solutions of eqs. (3.25) and (3.26) are, respectively A(t) [f,(t) -

W)_&(t)1 = U, 0, to>A(h) [.U~o)- WhJL(~oN,

(3.29)

A(t) M(t) -

W)M)l

(3.30)

= Uc(t, to>A(h) V3o) - C(M&)l.

In eqs. (3.29) and (3.30) the time instants t and to play an entirely symmetrical role. Now, let us choose to as a given but arbitrary instant and let us take the time limits t + - co and t -F + 00 in eqs. (3.29) and (3.30), respectively. Using eqs. (3.21) and (3.22) as well as A(-co)

= A(+co)

= I,

(3.31)

one obtains the results f:(-

co) = U, (- 00, to) A(&) M(to) -

Wd_tXto>l,

(3.32)

f:(+

~0) = U, (+ ~0, to) A@,) V&)

WcdfYWl.

(3.33)

-

A comparison of eq. (3.32) with eq. (3.13) and eq. (3.33) with eq. (3.14) leads to the equations

f:(to) = No) [fv(to)- Wo)S,(to)l,

(3.34)

fc’(to) = Wo)_moh

(3.35)

K(to) = Wo) _t’L(to)

(3.36)

A-(to) = AGo)lX(to>- Wo>f,(to)l.

(3.37)

9

It is more convenient to write eqs. (3.34)-(3.37) in matrix form, as f +(to)=

Wo) f(to)

v to

(3.38)

f -(to> =

Wo) fool

v to

(3.39)

2

9

where the matrices n(t)

l-I(t)

=

=

CA@)

-

D(t)

C(t) A(t)

-C(t)

A(t)

-

4)

D(t)

>’

D(t) A(t) C(t) D(t) A(t) - A(t) C(t) A(t) >’

(3.40)

(3.41)

46

Z. DEMENDY

are projectors, as they should. The components f+ and f- are invariant under the motion since

no>lJ 0, to) = u (4 GJWCJ, w> u (t, GJ = u 0, h3>WJ,

(3.42) (3.43)

where the matrix evolution propagator U (t, to) has been defined by (2.68). A more explicit form of U (t, to) expressed by the operators U,, U,, D, and C can be achieved by solving eqs. (3.25) and (3.26) for fv and f,, respectively. This formula is not written down here for reasons of space, it can be found in ref. 6. 4. Discussion. In this paper the starting equations of the subdynamics were derived in a reduced description by means of some projectors Pi. It is obvious that the appearance of the operators Pt is a consequence of the actual representation of the N-particle vacuum. Now, if one defines the “vacuum of correlations” fl as a distribution function for a set of s particles which interact only with their environment whereas there are no interactions in the group of s particles, there is no other possibility for the mathematical representation of this kind of independence as a factorization property. But, if one assumes that an uncorrelation means a factorization, then the evolution equation for fs’ can be found without the operators P,(t) in the following way: any variation in time of the function f,' is caused by the free streaming of the particles of the group under consideration and by their interaction with the environment. Let us decompose the reduced distribution of the group of s + 1 particles as f

s+1

(4.1)

=fsy+1 +fs"+1.

Here the crucial question is what kind of correlations from fl+1may give contributions to the evolution off:. Let us denote these types of correlations by fsyl. Since the term fs"+ 1is free from correlations a BBGKY hierarchy for fsvcan be written as

4fl = Lff: + i jdx,+&‘(s + lli>fLl +iilj dx,+J (s f lli)fs*fl. 1=1

(4.2)

It is clear that f,J1 may contain no terms arising from the correlations of the system of s particles. In this way 8, fsvinvolves the pair correlations only occurring between the s + 1st particle and particles of the uncorrelated group of s particles, and eq. (4.2) can be rewritten as

&fI = Lffi + f Jdx,+J’(s + lli)f,V+l i=l

+

i&s d-Q+L’(s + lli>f”2(s + lli>fl-1 ((s/i)), 1

(4.3)

REPRESENTATION

which is identical

OF SUBDYNAMICS

to eq. (2.31) if fs+ I is written

47

as fs"fl . The fact that in a reduced

description the starting equations of subdynamics can be “derived” on purely physical ground without using projections, is really not too suprising. It has been emphasized by Severne’) that one must be able to deduce from the hamiltonian an intrinsic realization of the projection operators which is justified a fortiori by the fact that one recovers known kinetic equations. It will be shown in a forthcoming paper that the present formalism provides valid kinetic equations in the case of various models. This last fact means that the projectors P and Q really play an auxiliary role and not a fundamental one in the formulation of the subdynamics.

REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9) IO) 11) 12) 13)

Prigogine, I., George, C. and Henin, F., Physica 45 (1969) 418. Balescu, R. and Wallen~om, J., Physica 54 (1971) 477. Prigogine, I., Rosenfeld, L. and George, G., to be published. Demendy, Z., Physica 59 (1972) 463. Geszti, T., Physica 62 (1972) 527. Demendy, Z., Physica 62 (1972) 545. Seveme, G., Transp. th. and stat. Phys. 1 (1971) 145. Baus, M., Bull. Acad. Roy. Belgique Cl. Sci. 53 (1967) 1291. Balescu, R., Physica 38 (1968) 98. Balescu, R., Physica 56 (1971) 1. Seveme, G., Physica 31 (1965) 877. Ginibre, J., La Thtorie Cinetique Sans Peine (preprint, 1967, unpublished). George, C., Physica 37 (1967) 182.