Statistical mechanics of irreversible processes

B r o u t , R.

Physica XXlI

Prigogine, I.

35-47

1956

STATISTICAL MECHANICS O F I R R E V E R S I B L E PROCESSES PART V : ANHARMONIC

FORCES

b y R. B R O U T and I. P R I G O G I N E Facult~ des Sciences de l'Universit~ Libre de Bruxelles, Belgique

Synopsis Ce t r a v a i l est c o n s a c r 6 tt l ' 6 t u d e d u m ~ c a n i s m e p a r lequel les a m p l i t u d e s des m o d e s n o r m a u x t e n d e n t a s y m p t o t i q u e m e n t v e r s la d i s t r i b u t i o n c a n o n i q u e (gaussienne) d a n s u n r6seau s o u m i s t~ d e faibles forces a n h a r m o n i q u e s . O n .consid~re u n e n s e m b l e s t a t i s t i q u e d e s y s t ~ m e s de m a n i ~ r e tt t r a i t e r les a m p l i t u d e s c o m m e des v a r i a b l e s al6atoires suppos6es i n d 6 p e n d a n t e s . C h a q u e s y s t ~ m e c o n t i e n t u n h o m b r e s u f f i s a n t d e degr6s d e l i b e r t 6 d e m a n i ~ r e ~t r e j e t e r e n fait, ~t l ' i n f i n i ; le t e m p s d e r ~ c u r r e n c e d e P o i n c a r 6. Si m i t i a l e m e n t les a m p l i t u d e s s o n t des v a r i a b l e s a i 6 a t o i r e s i n d 6 p e n d a n t e s o n m o n t r e q u ' e l l e s le r e s t e n t ~ t o u t i n s t a n t u l t 6 r i e u r e t q u e la f o n c t i o n de d i s t r i b u t i o n de c h a c u n e d'elles t e n d v e r s u n e g a u s s i e n n e . O n o b t i e n t a i n s i u n e g 6 n 6 r a l i s a t i o n d u t h ~ o r ~ m e c e n t r a l au cas d ' u n p r o b l ~ m e non-lin6aire. L a v a r i a t i o n d a n s le t e m p s d u s e c o n d m o m e n t de la d i s t r i b u t i o n o b 6 i t ~ t o u t i n s t a n t ~t u n e ~ q u a t i o n d u e ~ P e i e r 1 s. Des 6 q u a t i o n s p o u r les m o m e n t s d ' o r d r e s u p 6 r i e u r s o n t ~tablies de la m ~ m e m a n i ~ r e .

1. I n t r o d u c t i o n . Let us consider a crystal and describe its mechanical state in terms of the complex amplitudes of normal modes bk (with b* = b_k) where k is the index for the wave-vector. The corresponding energies of the normal modes are

Ek = o~lbkl 2

(1. I)

We introduce small anharmonic forces in order to assure the ergodicity of the system. The equations of motion for the amplitudes b~ are easily written down 1) (cf. § 2) d b k / d t = i2 Za,a V-kaa/o~k.exp[i(oJk

- - Wa - - ooa)t]b~b a.

(1.2)

Besides the wave vector k the amplitudes depend also on a second index corresponding to a sel~aration of positive and negative frequencies (see 1)). In order to avoid a too heavy notation it is convenient to use a single index k. It is also useful to introduce here a coupling parameter 2. Let us consider an ensemble of such systems; the bk are then considered as random variables submitted to a probability distribution. The central problem of interest for us will be the evolution in time of this distribution function ancl the investigation of the mechanism b y which the asymptotic --

35

--

36

R. B R O U T A N D I. P R I G O G I N E

equilibrium distribution is reached. W e shall be interested here in the case of sufficiently weak coupling only (4 -+ 0) so that the contribution of the anharmonic forces to the equilibrium distribution m a y be neglected. Their role is then similar to that of the collisions in the usual kinetic t h e o r y of gases. This problem is analogous to that studied in the earlier contribution of this series 2) 3) but is much more difficult because of the non-linear character of the equations of motion (1.2). We suppose that we have initial randomness of phase b,b~ = b2i~ii (t = 0).

(1.3)

This assumption applies directly to spatially uniform systems. Also if we have initial spatial gradients t h e y will generally die out, due to the effect of the harmonic forces in acoustical times (to-I) 2) and so m a y be" neglected. If however, we are interested in systems which present a systematic gradient over macroscopic distances, this assumption m a y be not sufficient. Such cases are closely related to the problem of thermal conductivity and will be treated in a separate publication. We shall go even further and suppose that we have at t = 0 a factorized distribution /(b,, b 2 . . . ; O) = H , 9,(b,; 0). (1.3') Thus we take the amplitudes to be initially independent variables. This assumption is strongly suggested by (1.3). A deeper justification of (1.3') will not be undertaken here. • If we consider times of the order R2t (with 2 ~ 0, t --> oo) and a system with an infinite number of degrees of freedom we prove that a) the factorization condition (1.3') is maintained in time /(b~, b 2 . . . .

;t) = H ~0~(b~;t)

(1.4)

The maintenance of the factorization condition implies the maintenance of "randomness of phases" with time bi(t)b*(t) = b~(t) ~q

(1.5)

b) each of the single distribution functions approaches a gaussian distribution to~[bil2 ~o,(b, ; t) --->exp (1.6) t-+oo

Er

which corresponds to the usual canonical distribution. c) the second moments change with time according to an equation proposed first by P e i e r 1 s 1). Similar equations m a y be derived for higher order moments. Due to :the maintenance of the factorization (1.4) and of the "molecular chaos" (1.5) we may as well express at any moment the statistical distri-

I R R E V E R S I B L E P R O C E S S E S P A R T V" A N H A R M O N I C FORCES

37

bution in terms only of the energies of the normal modes (1.1) in.spite of the fact that the mechanical description introduces the amplitudes. The situation is the same as in quantum mechanics under conditions which have been recently discussed b y V a n H o v e 5). The necessity of considering an infinite number of degrees of freedom is closely related, as in our previous work, to Poincar~'s recurrence theorem. Only for a number of degrees of freedom tending to infinity may we consider times as long as we wish without conflict with the quasiperiodic behaviour of closed systems. For finite systems we must always consider times much shorter then Poincar~'s recurrence time. The role of anharmonic forces in the establishment of the equilibrium distribution has been studied originally b y P e i e r i s 1) using a second order perturbation technique assuming randomness of phase at the initial time. This method is valid only for times much shorter then the relaxation time needed to reach equilibrium. To apply his method for longer times it is necessary to split the time into sufficiently short intervals and to postulate randomness of phase at the beginning of each time interval. V a n K a mp e n 4) and V a n H o v e 5) have recently emphasized the unsatisfactory character of the repeated use of the hypothesis of randomness of phase. The mathematical procedure which we use has a strong similarity to that introduced recently b y V a n H o v e 5) in his interesting paper on " Q u a n t u m mechanical perturbations giving rise to a statistical transport equation". It consists m a i n l y in construction of the formal solution of the equations of motion and the systematical study of terms which m a y contribute b y powers of ~2t. There are two points of difference with V a n H o v e's work which are interrelated: the mathematical complexity which arises from the nonlinear character of our problem and the fact that we s t u d y directly the distribution function for the individual amplitudes of the normal modes ("# space"). On the contrary V a n H o v e's linear statistical transport equation applies to the system as a whole ("F space"). We m a y also mention that the proof of property (1.6) is equivalent to a generalisation of the central limit theorem for independent variates to a non-linear situation and therefore presents some interest from the general point of view of mathematical statistics. Other problems and especially the problem of stationary states (i.e. thermal conductivity) will be treated in a subsequent publication. Also we m a y mention t h a t the relation of this problem to that of the Boltzmann equation for gases is so close that we m a y hope to obtain a similar derivation for the Boltzmann equation for large classes of initial states.

2. _Formal solution o/ the equation o/ motion. Abreviating V_ia~a2(t) = (1/coi)V_ia~a~ exp[i(°~i -- °~.1 -- °J~2)t]

(2.1)

38

R. B R O U T A N D I. P R I G O G I N E

we h a v e the equations of motion (cf. 1.2) (repeated indices are summed)

dbddt = i ~ V _ ,al~2(t)bolbo2

(2.2)

with

b* = b_i; V*,olo2(t) = V,-o,-o,(0; co* = -- o _ , Let us develop a formal solution of (2.2) in a power series of 4. Thus

b,(0 =

7=o

(2.3)

with

b~1~ = i~ f~ dt t V:o,l.l,(t.~b(°)b(°),, .1 al

(2.4)

(o) b~2' -----2(i~)2f~ dt2fd2 dr, V_,,#2(t2)b,2(o) V_a~lal(~l)bfll(o)bal '

b~3)

(2.5)

4(i;t)3f~ dt3f~)* dt2f~2dt t V --iflsa3k/t 31~b(°)V (°)v /, ~(0)~(0) B2 --aafl2t~/t 2/~bf12 r--a~fllat~Vl/V[Jl t"al

+ (i;t) 3 [~ dt3f~s dt2f~3 dt I V --iPaa3~ . /t 31~V --aap2at~ o q 21~b(°)b(°)V (°}~(°} fl~ a 2 --~3fllal~/t I1~bfll Val (2.6) These expressions have a simple intuitive meaning. Let us analyze the two terms in b!3). Inspite of the use of the classical mechanics we shall describe the interactions in terms of phonons. If we talk in terms of " p h o n o n proccsses" as successive events in time we m a y s a y t h a t each factor of V represent a single event. In t h e first t e r m of b!31, two phonons, one in state a 1, the other in fit combine to give a p h o n o n in state a 2. After this the phonon a 2 combines with r2 to give a plionon a 3 which t h e r e u p o n combines w i t h p h o n o n r3 to give finally a p h o n o n in state i. The limits of time integration gives the proper sequence. Such a process will be called a pure chain process. It is characterized b y the creation of a " v i r t u a l " phonon, i m m e d i a t l y followed b y the destruction of this same phonon in a successive event. In contrast, the second t e r m in b~3) is to be described as follows. First phonons a 1 and fll combine to give f13. W i t h i n the same time interval a 2 a n d r2 combine to give a 3. Then, a 3 and r3 combine to give i. Such a process is t e r m e d a branch process. Diagramatically we m a y draw the processes as follows (time travels up)

chain

branch

IRREVERSIBLE PROCESSES PART V : ANHARMONIC FORCES

39

3. Second order terms. We will now analyze in detail second ordor terms adopting the time scale introduced in the introduction. We take ~2t = 0(1) b u t ~"t ---- 0, n > 2 i.e. we pass to the limit ~t -+ 0, t -+ co, A2t finite. We have explicitly (cf. 2.1, 2.5) b~2) =- 2(i~)2f~ dr2 f~* dt,(1/°J,)

V-,~,a,(1/°J~) V-a~t.x b°~,b°al b°.1 ×

× exp [i(oJ, -- oJa, -- oJ.,)t 2 + i(oJ~, -- oal -- oJal)tl].

(3.1)

We consider the contribution b~21 b(°li to b~(t). The bars represent averages t a k e n according to the initial factorized distribution (1.3). Unless there exist certain combinations in b!21 which contribute a factor of t to the integral, (3.1) is of order ~2 and hence zero. The only w a y to introduce such a factor is to choose the frequency factors in the exponential to be the negative of one another i.e. (coi - - o J # , - oJ.,) = - - (•., - - oJal - - oJ.1).

(3.2)

T h i s is s u p p l e m e n t e d b y the c o n d i t i o n for the n o n v a n i s h i n g of the V's 1) -- i +/32 at + / 3 1

+ a2 = - - a2 =

K K'

(3.3)

where K, K ' are vectors in the reciprocal lattice so chosen that t h e y bring the third phonon into the basic cell when two are given. (3.2) and (3.3) constitute seven equations in the nine unknowns of a 1, /31, /32 and hence cannot be solved unambiguously. However, the initial condition of chaos (1.3) for which the analysis is carried through, implies two of the three b's in (3.1) must be complex conjugates of one another. Thus we m a y have a t = i, /31 = --/32 or a I = --/32, /31 = i either of which satisfies (3.2) and (3.3) and gives a non vanishing contribution to b~(t). We have thus constructed a non-vanishing term in b~2) which on utilizing (3.2) is bl2)= 4(i~)2 f~ dt2f ~' dt 1(1/oJ, oJ,,)V2_ia~zIb~°)[2b~°)exp[i(oJ~-oJa-oJ~,) (t2--tl) ] (3.4) Changing variables from t2, t I, to 3' = t 2 -- t 1 and z = t 2 we have b~2~=4(i2) 2 f~ dv fd d~'(1/oJioj~ ) V~,a~. * [b~°)12bl.°l exp[i(oJ,--~oa

--o~.2)~' ] (3.5)

Let us introduce the time interval (Ow)-t where c~oJ is the range of variation of all variables in the integrand. For t >~ (Oo) -I we have also for the great m a j o r i t y of 3, z ~ (OoJ). This allows us to use the a s y m p t o t i c formula

f~f /(oJ)do~ exp(icot)dt = Itl/t.n/(O) + i fp /(o~)/co.d~o

(3.6)

f~ means the principal value of the integral. We shall work only with the first p a r t of (3.6). The second part merely contributes a sr0all change in

40

R. B R O U T A N D I. P R I G O G I N E

the frequencies due to the transitions. Thus for t >~ ((5w)-1 the average value b~21 b(°li becomes

b~2~ b(°__~ i = -- 4n~21(1/w,wo,) V~,a,~,[bCa°)121b~°)[26(w, -- w~, -- w,,)

(3.7)

The two conditions on the indices i, ~2, ~ for the non vanishing of (3.7) are -- i +/32 - - w i + w~, +

+

a2 = K

(3.8)

w . , -~ 0

(3.9)

The existence of solutions for these equations with K ~: 0 (Umklapprocesses) have been discussed in detail b y P e i e r 1 s 1). We shall show in § 6 the i m p o r t a n t role of these solutions in the establishment of the a s y m p t o t i c gaussian distribution. In terms of the diagrams introduced in § 2, the only contribution comes here from one of the two diagrams,

We shall now generalize this procedure .to higher order pairing schemes.

• 4. Totally internally paired schemes. We have seen in the previous section how to construct in b~21 a pairing of the two V factors to give a non-vanishing contribution to b~. In general, when we form the product bi(t)b_i(t ), it is necessary to find non-vanishing factors of ~t as well as to pair all b~)'s to give non vanishing averages. One m a y do this in one of two ways. E i t h e r one pairs factors of V within a b~kl t e r m as in § 3 or one "crosspairs" a V in bi(t ) with a V in b_i(t ). This pairing m u s t be systematically investigated, for some pairing schemes lead to finite results and others to zero (in the proper limiting sense). We discuss, in this section, pairing within a bi t e r m which we t e r m internal pairing. We consider first a chain t y p e t e r m in b~ which is written

( -,~)2 k £t0 dt sk..f

t~

dt , V

t

b(o) V

_

_ t

b(o)

V

(o) t b(o) bo1(.1) 4

We m a y pair the V's sequentially i.e. V_ia,,a, * with V_,2,a2,_t,2,_ l . . . etc so t h a t risk = --fl2k-l, ~k-1 = i or a2k_ 1 = --fl2k, fl2k-I = i . . . . . There is also the possibility of pairing out of order (non-sequentially). However, it is easy to show t h a t in such a case the time integrations overlap in such a w a y t h a t one loses at least one necessary factor of t, t h e r e b y giving a vanishing result (For the proof, we refer to V a n H 0 v e 5) eq. 5.8).

'IRREVERSIBLE PROCESSES PART V: ANHARMONIC FORCES

41

Thus (4.1) leads us after integration over times in the same manner as section 3 to (-

+

...

× ... 2

(0) (0)

× [V~,~,,(1/oJ= o2.,)lb~xl2O~,,,~ + V.~,,,~(1/oJ.=w,,,)b,,,b,,]b~,

(0)

(4.2)

where it is understood that each V2_~,~ is multiplied b y ~(o~,, -- oJ~ -- o~). The square is always in the sense of absolute value squared. The nature of this pairing is again the most easily visualized in terms of the diagrams of § 2. For k = 2 we have four diagrams. Two of t.hem are

i.

i./

L

"°p4

In the first case half of the virtual phonons are i's. The second, kind of scheme is characteristic~of the non linear equations of motion (1.2). On multiplying out (4.2) one sees in general all b's are paired save bl°1. This is eventually paired to h(o) ~ upon multiplication to form the product b~(t). Summing over all orders of chain type terms, (4.2) gives the contribution of internally paired chains to the totality of internal pairing. Let us consider the same problem for a branch scheme. A prototype containing one branch is of the form

2 4 [~ dt4f~' dt3f~ 3 dt2f~* dt, V_,a,o,(t,)V_aoaa~3(ta)b~°2b~° 2 . •

V - - a4 ~o2 ~2 (t2) b(2)V ~ ~ ~ (tt) b(°~ b(°) ~2 -- 2~1 1 ~1 al

(4.3)

When one pairs the first and second factors together and the third and the fourth the resulting expression will give a non vanishing expression to b~ b(_°)i and is of the foim

All branch schemes m a y be dealt with in a similar manner. The totality of all such pairing will be called Z i Uiibl .°). Uii contains two unpaired b's, bl°l and b~°). This is the principal property required of the U's in order to. prove the persistance of factorization. The precise form of U will not be necessary to calculate, b u t rather only its time derivative, which we-calculate in a later section (see § 6).

42

R. BROUT AND I. P R I G O G I N E

Consider now b[.kl with its t o t a l i t y of chain and branch schemes followed b y the sum of terms, all of which have the same branching structure; b u t between each factor of V is successively sandwiched more and more intertlally paired factors of V, in all possible chain and branching schemes. The' sum of all such terms is called b~. The formal expression for "b~ is ~k~= X (a)h fgdtk . . . fg' dt. Ui~,,(t -- t h ) V _ , ,a,a,(th)b~a°) U,,~,_, (t h -- th-a) × . . . all schemes

,, b~h)

×. U=,., (t 2. -- tl) V ~ l Vol =1 (t~)b~)U~ (t)b 00'°) . ¢1 I aO

.

(4.5)

where the factors of V are t a k e n two b y t w o different. The sum of ~hl over all k then includes that part of the pairing in b i ( t ) b i ( t ) t h a t corresponds to all possible internal pairing within bi(t). To complete the analysis, it is now necessary to consider the product explicitly. 5. P r o d u c t terms. We now form the product b i ( t ) b i ( t ) and form pairs to pick up factors of d functions and t. Certain facts present themselves immediately. 1) Only terms of the same k contribute. P r o d u c t s -i h(hl---i hlk'J with k' :/: k always contain a factor of ~tEh-vt. 2) Only identical chain and branching sequences in b~h)b(hJi m a y be paired. If not, there is always at least one chain and one bran¢h involved in a pairing. This implies a double condition on one index resulting in the loss of a s u m m a t i o n variable, hence it is negligible. • We make this point explicit b y the following example. Consider pairing the two schemes (we put U = 1 for simplicity) ....

V-a~a3 bCa°3 ) V - ~ 2 . , b('°lp,V - ~ l a l b(-°lplb(°)a~ b(PI b(°1

b(°1 b(°) t

chain branch

Let us pair the two last factors of V. This requires f13 = -- a2. H o w e v e r subsequent pairing of the two first factors requires fl~ = -- f13 or a, which then requires a 2 = f13 or -- a 4 which means the loss of a 2 as a s u m m a t i o n index. This argument is true w h a t e v e r the first two paired V's m a y be and whenever a chain crosses a branch. 3) Pairing m u s t I6'e sequential i.e. V factors must be paired only if their time intervals correspond. In the proof, all indices and time variables in blk) i will be unprimed and in b~. will be primed. a) Chains. The last V factor in ~!h~ involves two factors of b and all others one factor of b. In order to lose the m i n i m u m n u m b e r of s u m m a t i o n variables, we must then pair the two last factors of V (i.e. V azal~ (t2) with I)) in such a w a y t h a t a 2 = - - a 2 , fliP--fit, a l - - - - - - a l or t fit = -al, al = -- ill. This establishes a 2 as an index in c o m m o n b e t w e e n .

.

.

.

I R R E V E R S I B L E PROCESSES PART V : ANHARMONIC FORCES

43

V_a~,a,(t'2) and V_~.,a,,a,,(t~), which again necessitates these two pairing . . . etc.

b) Branches. Here m a n y members of a branch m a y occur in the same time interval (i.e. have the same limits of integration) and each such V term has two b's multiplying it. Thus any member of this branch in b~kl m a y be combined with a member of the corresponding branch in b~. For example, in ,(~!3),. ~3).~_,, branch we m a y pair V_~6a,~8(t2) with either V_~,~,~,(t'2) or V=5,a,.x,(t'l). However two branches that occur at different corresponding times m a y not be paired, for the reasoning of a) applies. -i, and hence bi(t)b_i(t), vanishes unless i = i. For, from the above considerations, we are always led to match V_~a,~,(t,) with Vi_a,,~,,(t'~) which gives a non vanishing contribution, in the appropriate limit, only if •

i = j. An exceptional case arises for "~01.~(o~., which, however is again zero except for i = i, owing to the initial condition of factorization. This last result is of fundamental importance. It shows that if one begins with an initial distribution of amplitudes having the chaos property (1.6) the equations of motion, of themselves, insure the persistence of this property. The same method permits us to obtain immediately other important results. First

b~(tl)b-i(t2) = ~i b,(t2)b-,(t2) t2 < t,

(5.1)

Then we m a y also prove that the whole factorization property (1.3) persists in time. For example, using the expansion (4.5) and the results of this paragraph, we see that

= Z**.,,.,,,, bl*'

=

b (t)

b!*"',

(5.2)

Such conclusions rest however on the assumption that the intermediate indexes which appear in ~*1. . . . . . b~i') are different, or more precisely, that the contribution of terms in which intermediate indexes are in common is negligible. This can only be true for times which are not too long for a given number of degrees of freedom. This upper time limit goes to infinity when the number of degrees of freedom tends itself to infinity.

6. Long time equations. Asymptotic Gaussian character o/ the variates b~(t). Due to the complexity of the operator U, it is convenient to develop long time differential equations from another starting point. It is directly verified

44

R. B R O U T A N D I. P R I G O G I N E

that the following expansion satisfies our fundamental differential equations

(2.2), b,(t) = b~°) + 7~'= , (i~)k f~ dtk . . f~, dt, V~ak a,.(t,)bak (tk) V.ka,_, .~_, (tk_,)ba~_x(t,_,) .... X ....

V.~ax,x(t,)ba, (tl)bax (0)

(6.1)

We m a y use this expansion in a completely analogous fashion to the original one (2.3) which only used b~°)'s, since in § 5 (equ. 5.1) we have proved that

bi(tl)b_i(t2) is zero unless i = i. Eq. (6.1) has the distinct advantage that it only contains chain type terms for which explicit evaluation of U is quite simple. Again, in seeking the product b~(t) one seeks for factors of A2t to give non vanishing results. This is achieved either b y internal pairing or cross pairing, as before. Let us use again sequential pairing and proceed exactly as in the derivation of (4.2). In performing the integrals over the frequency exponentials it is necessary to assume that bi(t ) is a slowly varying function of t in order to obtain the 0 function in the frequencies upon integration over differences in time, for example over t2h -- t2k_ 1. The assumption that changes in the amplitudes are very slow compared to crystal frequencies, is however implicit in the whole model. Further most of the contribution from such integrations arises from times of the order t 2 ~ - t2k_ 1 = 0 ((60) -1) so that after integration we shall find products of the form b_a~(tk)bak[t ~ + 0(l/~w)] However as we are always interested in time scales much larger than 0 (1/6w) we replace these products b y

Apart these small differences the procedure is exactly the same as in the derivation of (4.2). So we get the formal expression for the operator

U,~(t) = O0. + X~°=, (-- •42) * fg dt, .... f~' dt, [V2__,a,,,(1/wio&,)b~k,(tk)6,,k

....

X [V2._a2tJ1 , ax,

(l/¢o~o~x,)b~x(ll) ~a2i -~- V~Ol'i(1/c°a2°~i)ba2(tl)bi(ll)]

(6.2)

Here again the function 8(wa + w~ -- o~i) is included in V2_ia~. From (6.2) we immediatly have the differential equation for the time change of Uo(t )

dU,~(t)/dt= -- 22g[V_,a~,(1/w, w.,)b~ (t)O,. + V2__,'aa(l /o~, wa)b,(t)b~(t) ] Uoi(t ) (6.3) If the initial time is v instead of 0 we get in a similar way

dU~s(t-~)/dt=-n~2[V2__,a,,(1/o~,o&,)b~(t)Oi,+

V2_,a.(l/o~,~oa)b~(t)b.(t)] U~i(t-,: ) (6.4)

I R R E V E R S I B L E PROCESSES PART V: ANHARMONIC FORCES

45

We now form b!k) in analogy to b~~) (cf. 4.5)

b[~'=(ia) ~ g dt~ ... fd, dr, V,o,(t--t~)V_o;a,.,(t~)b,,(t~)Uo, o,_~(t~--t~_O × ....

X

U~,~,,(t 2 --ta)V~(a,~x(t,)ba~(tOUa,~o(t,)b~o(O )

.

.

.

(6.5)

when as in (4.5) the V's are taken two by two different. Thus we have the expansion

bi(t)

=

(6.6)

-k=o~" ~i~(k)

This then accounts for all possible internal pairings and when we form b2(t) we must cross pair. E x a c t l y as in § 5 we m a y show that only terms of the same k contribute so t h a t

b~(t) =

E~= o (b!k)) 2

(6.7)

Using (6.5) we get

b2(t) = Z~'=0 (2~t2)k/g dt, . . . fg* dt U2.k,(t -- t,) V2o , o,~,(1/o,2 ,)b#k(t,) × x

U~,o, _ ,(&

--

&_,) x . . . x U 2.,.((t 2 -- tt)V2_a(iha~(l/co~,,)b~x(tl)

X U:at ~0 (t,) b]0 (0)

X (6.8)

F r o m these equations it is now easy to derive the change of the moments of the distribution with time. These time derivatives have two contributions, one from the derivative of U(t -- t~) (called (d/dt) v) and the other from the derivation of the limit of the integral (called (d/dt)t). Let us first consider the change of the second m o m e n t with time. Using (6.5) (6.6) and (6.4) we get immediately (d/dt) vb, = =

-

-

nA2[V2__iaa,(1/miw.,)b~ (t)6,~ U h~ b i

__ y ~ 2 V 2_ifla l -# L tOi (9 a

+ V~_,a.(I/o, o~a)b ~(t)b~(t)] ba

b~bi l

(6.9)

+ c o i o~a 2

whence using (5.2) r-

p. = _

\ dt / v '

b 2 52 v, +

- a, i co,

2 -~ b°e I

(6.1o)

oJ~o~

On the other hand direct differentiation of (6.8) with respect to the limit of the integral gives, using again (5.2),

(d/dt),b 2 = 2 u t = V2,a~.,(1 /oJ,2) ba.2 b2ak

(6.1 l)

Combining (6. l 0) and (6.11) we have

de dt

27z~2V2_i~.

F b b: L co~2

6: o~i o~

e oJico.

7 /

(6. 12)

46

R. BROUT

A N D I. P R I G O G I N E

or in terms of normal mode energies (1.1), dEi -- 2~'2 c°i'V2-i~sQ 2 2 [ e o i E # E a - - eoI~EiE,, - - a~aEiE~] dt coi a~ eoa

(6.13)

T h e s u m in (6.131 is t a k e n only for coi - a~o - co~ = 0. Relations (6.12) or (6.13) express the change of the second m o m e n t s with time. T h e y are the Peierls equations 1) now justified for long times. We see that t h e y admit the a s y m p t o t i c solution wl2 bi2 = constant

"Ei

(6.14)

corresponding to a s y m p t o t i c equipartition of energy b e t w e e n normal modes. S t a t i o n a r y solutions of (6.13) are obtained for every distribution for which we have the equations -

.

-

=

(6.15)

0

valid for all processes, such that (cf. 3.8, 3.9) -- ~oi + cog + toa = 0 i +fl +a =K It is only if we take account of (6.14) is the only stationary solution same as in Peierls paper 1). The procedure we have used for higher order moments. For example

dt

(6.16)

the Umklapp-processes (K ~ 0) t h a t of (6.13). The discussion is exactly the the second m o m e n t s is valid also for we get *)

coi

co~co~

(6.15)

coi co~

Using the a s y m p t o t i c value (6.14) for the second m o m e n t we obtain b~ = 2(b2) 2 t--~oo

or

(6.16)

b-*oo

E~ --> 2(Ei) 2

(6.17)

In general b y a similar procedure (6.18)

E~ --->n! (E,)"

This shows that the a s y m p t o t i c distribution is of the form E~ ~,

/(E,) = exp

(6.19) .

•)

=

z

__,

=

2

l~,k',k",k"" " k,k" A g a i n (d/dr) bt s p l i t s i n t o 2 p a r t s (d/dt)l + (d/dt)u. T h e t e r m (d/dt)u i n e o l v e s 4 t e r m s e a c h of t h e f o r m (6.3), t o g i v e all t h e n e g a t i v e t e r m s in (6.15), w h e r e w e u s e (1.4). T h e t e r m in (d/dOl t a k e s o n t h e s a m e f o r m as (6.11), b u t w i t h a n a d d i t i o n a l f a c t o r of 4 o w i n g t o t h e o u t s i d e f a c t o r of 2 in ( 6 . 1 5 a ) a s well as t h e t w o t e r m s in e a c h m e m b e r of t h e d o u b l e s u m .

IRREVERSIBLE PROCESSES PART V: ANHARMONIC FORCES

47

From the asymptotic equipartition of energy (6.14) we see that.(6.19) m a y also be written /(El) = exp(--

EdkT )

(6.20)

(kT

= ~i) which is the equilibrium canonical distribution. We m a y stress that the existence of Umklapp-processes is essential in this proof. If t h e y would not exist, we could not prove the simple relation (6.16) or (6.18) between successive moments. Finally let us consider odd moments, say b~(t). Using the expansion (6.6) of bi(t) in terms ~kl we get a product of these series. Each term laas either an odd number of b factors either an odd number of V factors. In the former case we get zero because of the persistence of chaos, in the latter case we have contributions of the order 0(2) and hence zero. This then completes the proof for the asymptotic gaussian character of the variates hi(t).

Acknowledgments. We wish to thank Professor L. V a n H o v e for sending us the manuscript of his article before publication. One of us (R.B.) wishes also to thank Professor V a n H o v e for his kind hospitality during a very stimulating stay at Utrecht. The research reported in this document has been made possible through the support and sponsorship of the Air Research and Development Command, United States Air Force through its European Office. Received 17-8-55.

REFERENCES I) 2) 3) 4) 5)

Peierls, R.,Annalen Physik3 (1929) 1055. K l e i n , G. and P r i g o g i n e , I., Physical9(1953) 1053. Prigogine, I. and B i n g e n , R.,Physiea21 (1955) 299. Van Kampen, N . G . , P h y s i c a 2 0 ( 1 9 5 4 ) 603. V a n H o v e , L., Physiea21 (1955) 517.