Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
370 Progress of Theoretic al Physics, Vol. 54, No. 2, August 1975 On Exten sive Prope rties of Proba bility Distri bution Funct ions in Non-E quilib rium States · Hiroshi FURUKAWA Depart ment of Physics , Kyushu Univer sity, Fukuok a 812 (Received Novembe r 28, 1974) The extensive property of the logarithm of probabili ty distributi on function on macrovariables in the states far from thermal equilibriu m is investiga ted with the aid of a cu~u lant-corre lation function method. If the correlatio n lengths among physical quantities in spacetime in an initial state are of short range, then it is shown that the correlatio n lengths in space-time remain to be of short range· under a finite order perturbat ion of an external field. The extensive propertie s of the logarithm s of the probabili ty distributi on funCtions of general types of macrovar iable are found if the state observed can be reached by a finite order perturbat ion. At a critical point ( , line or domain) our analysis cannot be applied. The asymptot ic evaluatio n of the probabili ty distributi on functions of macrovar iables in nonequilibriu m state is thus shown. The applicatio ns are discussed. The logarithm of the probability distributi on function of the time rate of change in the quantity conjugate -to the external field in a steady state leads· to the generaliz ation of Onsage:r' s po~ential. Some remarks on the variation al principle s are presented . § 1. Introd uction We have shown in previou s papersn that a function al techniqu e is availabl e to the study of states far from thermal equilibr ium. Howeve r, no discussi on of the probabi lity distribu tion function for macrova riables in non-equ ilibrium states was made. Althoug h there has been no general method of constru cting· microsc opically the probabi lity distribu tion function of macrova riables in states far from thermal equilibr ium, the recent develop ment of the statistic al mechan ics on laser indicate s a close analogy of the probabi lity distribu tion function in a state far from thermal equilibr ium with that in a thermal equilibr ium state. 2> In fact several authors have shown that at least in a steady state the probabi lity distribu tion function is characterized by its logarith m function on the basis of a Fokker- Planck type ,approac h. 3>, 4> Recentl y, general conside rations on properti es of the probabi lity distribu tion function of macrova riables in non-equ ilibrium states have been made by: servera l authors ,5>-n who have shown that master equation s take certain asympto tic forms. Kubo et al. 6> have s,hown in more general situatio ns an extensiv e propert y of the logarith m of time depende nt probabi lity distribu tion function on the assumpt ion of Markov ian nature of the process and an extensiv e charact er of the transitio n probabi lity. In this paper a microsc opic theory of the probabi lity distribu tion function of macrova riables in non-equ ilibrium state will be develop ed. Althoug h there is little ambigui ty in Kubo's analysis6> of the extensiv e propert y of the probabi lity distribu - On Extensive Properties of Probability Distribution Functions 371 tion function in non-equilibrium state, the same problem should be discussed from more microscopic point of view. As is well-known the statistical nature of a thermal equilibrium state comes from the fact that a system consists of many independent elements. Such elements seldom mean the constituent ato~s or motecules, but mean their clusters. The main question is whether --we can consider a system in a state far from thermal equilibrium to consist of such independent elements. It is convenient for us to investigate this question with the aid of the correlation function. We shall find that the short range nature of the correlation length between physical quantities in space-time is maintained in all steps of a perturbation scheme. . Thus, we shall find in more general cases an extensive property of the logarithm of the probability distribution function of macrovariables in non-equilibrium states. Recently the same problem has been treated by Suzuki as a relaxation process in a finite time interval of a large system. 8> In the next section the definition of several types of probability distribution function for macrovariables will be given. In § 3 the asymptotic evaluation for probability distribution functions in a uniform .system will be discussed. In § 4 the explicit computationof probability distribution functions will be discussed in certain cases. In § 5 the extrema of the probability distribution. functions are investigated in connection with variational principles. Section 6 is devoted to the concluding remarks. § 2. Probability distribution functions Quantum mechanical techniques are used throughout this paper. We assume, however, that the variables of the probability distribution function represent macroscopic quantities so that we can neglect the effect of non-commutati vity of them (we mean that the WKB approximation may apply). The probability distribution functions treated in this paper are the following three types. A. Type I The first type of probability distribution function is the most familiar one defined by P(a, t) =Tr o(a-A)p(t) =Tr o(a-An(t) )Po, (2·1) where Tr stands for the trace, p(t) and Po denote the density matrices at time t and at the initial time, respectively, A and a denote an operator and a random variable, respectively and An(t) denotes the operator in the Heisenberg representation. We shall call (2 ·1) the single time probability distribution function. B. Type II The second type of probability distribution function is defined by P(a, b, c, ···, t1. t2, t3, ···) =(o(a- An(tl)) o(b- Bn(t2)) o(c -Cn(t3)) ··· )o, (2·2) where (···)0 =Tr ··· p0 • This type gives the probability of finding a, b, c, ··· at times tr. t 2, t 3, ···, respectively. If An(tl), Bn(t2), Cn(t3), ··· are statistically independent, then (2 · 2) becomes 372 M. Furuk awa (2·3) We shall call (2 · 2) the multi time (or join:t) probab ility distrib ution functio n. C. Type III The third type of probab ility distrib ution functio n is given by P~a, T, i) = Tr o(a~ i~TAH(t')dt')Po. (2·4) In the limit r~o (2·4) reduce s to (2·1). Using th~ multit ime probab ility distribution functio n, we can rewrit e (2 · 4) as =lim <-->0 where ti=t- rj. 1s rewrit ten as So(a-Ln;a )P (a1 a 1 ,2 , 7: J=l . 7: a n da 1 ···,"'---" -,t1, ... ,tn) JI~,(2·5) 7: . J=l 7: By makin g use of the expres sion (8/at)i JH for AH(t) , P(a, T, t) = J P(a+ a', a', t, t-T)d a'. (2·6) Herea fter, we shall call (2 · 4) the contin uous time probab ility distrib ution functio n. § 3. Asym ptotic evalu ation of proba bility distri butio n funct ion A. Corre lation functi ons the equati ons of motion : p (t) The densit y matrix and a Heisen berg operat or obey =. AH(t) = :n [Ho+ H (t), p (t) ], (3 ·1) i~ [AH( t),H0 <HJ+H1<Hl(t)], (3·2) 1 where H 0 and H 1 (t) are the unpert urbed and pertur bed Hamil tonian s in the Schrodinger repres entatio n, respec tively; the subscr ipt Hand the supers cript (H) denote the Heisen berg repres entatio n and the dot denote s the time deriva tive. We assum e that the system is in a therm al equilib rium state at t = t 0 = - oo. The pertur bed Hamil tonian is given by H1 = SiJ(r)Z(t, r)dr, (Z(- oo, r) =0) where Z is the extern al field and iJ(r) denote s a physic al quanti ty conjug ate to Z. On Extensive Properties of Probability Distribution Functions 373 The perturbed Hamiltonian H 1 drives the system to a non-eciuilibrium state. Equation (3 · 2) is solved to gi~e Ae(t) =A(t) + E(~)" f n=l th Jt>t ···>t,. [[··· [A(t), <i(t1)], ···], <i(t,.)] 1 xi(t1) ···I(t,.)dt1···dt,. (3·3) and . tJ"Ae(t) r· =.(~)"P'e(t-t 1 )···8(t,._ 1 -t,.) tJI(tl)·. ·tJZ(t,.) X th . X ( [ · · · [ Ae (t), 0"H (tl)], · · ·, 0"H (t,.)] >o , (3·4) ' where X(t) =exp{ -Hot/ih}X exp{H0t/ih} is an operator in the interaction representation, P' denotes the sum over all permutations of arguments (tt, ···, t,.) and 8(x) is the unit step function; i.e., 8(x) =1 for x>O and 8(x) =0 for x<O. Next, let us introduc,e the quantity Xe<n> (t) defined by' Xe<"l (t) = (~)" f th Jt>t····>t,. [[ ··· [X(t), O"(t 1) ] , • • · ] , <i(t,.)] (3 · 5a) which consists of 2" terms of· n + 1 products and (3·5b) Then we have 00 Xe(t) =I:: Xe<"'l (t). n=O Let us consider a correlation function of the type: (3·6) We here. make two assumptions on the correlation function (3 · 6). i) If any number of components 0" ( t£, rt) in (3 · 6) are separated from the others by more than a characteristic; length ~0 or time r 0, then they are statistically independent of th~ others;e.g., II . (<i(tl, r1) O"(t2, r2)···<i(tm, r m) >o = ( 0" ( t1, r1) >o(<i ( t2, r2) · · · 0" (t,;., r m) >o Jtl-t1 J>r(,,j=2, ···,m. for jr1-r1 l>~o or ii) The integration over ti, ···, tm and rt, ···, rm is bounded; i.e., I t(O:(t1, r1) · · ·<i(tm, ~m)>odt1· · ·drm I<C"'(vt)"', (3·7) 374 H. Furukawa where C to justify invariant Now is a positive constant which is finite. In this paper we shall not try these assumptions under any conditions.*> We assume that fJ(t) has no part with respect to H 0• let us consider the correlation function of the type (3·8) It is easily found on the basis of assumption i) that if any quantities in (3 · 8) are separated from the others by distances greater than a certain distance in space or in time, then they .,are statistically independent of the others, e.g., (fJH<nl (t, r) X (JH<nl (t', r') )o= (fJH<n'(t, r) )o(fJH<nl (t', r') )o for lr-r'l.> (2n + 1) ~o or It-t' I> (2n + 1) r 0• This is due to the property of the integrand of (3 · 5a). Let us consider the quantity ({ [ [ · · · [ (J ( t,r) , (J ( t1, r1)], · · ·], (J ( tm, r m)'J} X {[[ · ·· [fJ(t',r'), fJ(t1', r/)], ··•], fJ(t/, r/) ]} )o. (3·9) If correlations among different groups of quantities each in a repeated commutator vanish, then the operators behave as e-n umbers for one another and (3 · 9) vanishes. By this reason only the configurations in which the points (t1, r 1) , ···, Ctm. rm) exist in the neighborhoods of (t,r) and (t/,r/), ···, (t/,r/) in the neighborhoods of (t',r') contribute to (3·9). Thus, (3·9) is factorized for sufficiently large lr-r'lor lt-t'l as ({[[···[fJ(t, r), fJ(th r1)], ···], fJ(tm, rm)J})o X( {[[···[fJ(t',r') , fJ(t/ r/)],-··], fJ(t/, r/)]} ) 0 • (3·10) The correlation length of the (3 · 9) is at most of order (k+ m + 1) ~0 in space and (k + m + 1) ro in time, since (3 · 9) contains only (k + m + 2) -body equilibrium correlation function of the type (3 · 6). This situation is the same even for the m-body correlation function of the type (3 · 8). If any operators of n-th order are separated from the others by distances greater than at least (n + k +1) ~0 in space or (n +k + 1) r 0 in time, then that is statistically independent of the others, where k denotes the '·order of the neighboring operator. It is to be noted, however, that this upper limit of the correlation length is overestimated, since the contributions from the integrations over space-time are not taken into consideration. It is, therefore, preferable for us to consider that a quantity (JH<n'(t,r), which consists of (n + 1) operators in the interac'tion representation, extends in spacetime at most with the volume (n +1) ~03 r0 • On the assumptions i) .and ii) the integration of (3 · 8) over t1o · · ·, tm, r1, · · ·, r m is also hounded. *' In Kubo's article in 1957 it is noted'' that if A, Band A(t)B are ergodic, then we shall have limt .... ~<A(t)B(O))o=<A)o<B)o. Our assumption i) plays the same role as Kubo's postulate. If the correlation in time among physical quantities decay faster than r' for large t, our analysis given below is correct. On Extensive Properties of Probability Distributio n Functions 375 It looks as if we can justify the assumption i) by repeating the same procedure as the one made for (3 · 8). However, there arises the difficulty that the operators appearing in this case are denoted by the Schrodinger representatio n. No operators in the Schrodinger representati on in any time points are statistically independent and the expansion parameter .is time t which we cannot fix. Therefore, it Is difficult for us to justify the assumption i) in the same way as in the case of (3 ·8),. Although we have considered only the upper limit of the correlation length, the same discussion can follow for the lower limit of the correlation length among physical quantities. The discussion of the lower limit can be done in a parallel way. B. Asymptotic evaluation of probability distribution function Here, we shall consider only the . single time probability distribution function. The sam~ discussion can be made for the others. Let the correlation length of the correlation function consisting of A (r) be also of order ~0 in space. No restriction on the temporal correlation among the A's is needed here. Now, we use the following special perturbation al scheme for the calculation of the probability distribution function. If we use another type of perturbation al scheme, then we find the same result. Let us introduce P<"l (a, t) = " <IJ (a- .I; AH<kJ (t)) ) k=o 0 = s -1d s exp (isa) Q<"l (s, t), 2n (3 ·11) where (3 ·12) IS the characteristi c fm;.ction of p<n> (a, t) and AH<kJ (t) = 1 AH<kJ (t, r)dr. (3 ·13) p<n> (a, t) reduces to P(a, t) in the limit n-HxJ. The correlation length of the correlation function consisting of L~=oAH<k> (t, r) is at most order (2n + 1) times the correlation length of the correlation function consisting of A ( t, r) in space. This almost proves that in a spatially uniform system the probability distribution function is scaled as (3 ·14) since Q<">(s, t) should be scaled as exp{Va<">(i s)}.*> Here, the volume V may be larger than ~,3 V0 , where V 0 is a volume for which pw> (a) can be asymptotica lly evaluated as *> This can be shown in the same way as in the equilibrium statistical mechanics by Ursell. 376 H. Furukawa (3 ·15) ' and e:n( < (2n + 1H'o) denotes the correlation length between Lj=lAHCJ> (t, r) in different space coordinates. We have assumed that V 0 is so large that the saddle points method for .the integration over s in (3 ·11) can be used with practically sufficient accuracy. A non-interacting system has a a-function type correlation .space in an equilibrium state. Therefore, the correlation length among atoms does not ~hange by any order perturbation of the external -field acting on single body operators. In such a case we may have V = V 0 • Rigorously speaking,· so as to evaluate the probability distribution function in the form (3 ·14), it is necessary for us to assume further that there occurs no divergence in the strengths of the correlation functions of .the type (3 · 8) or those integrations over their arguments. This assumption is fulfilled on ass1,1mption ii). However, such a divergence, if occurs, originates from the micro,scopic structures of matters and otherwise comes from a wrong perturbation scheme. Therefore, such a divergence is of no interest from statistical mechanical point of view. The asymptotic, evaluation (3 ·14) is· ·quite general. By taking the limit in: . In pcnJ (a, t) 11m ----'-~'- n-oo v . lim cnJ (__!!___ t) = '(__!!___ t) _In P(a, t) n-->= '1J V '. '1J V ' · V ' (3 ·16) ' we obtain the probability distribution function whose logarithm is. extensive. By assuming that the limit (3 ·16) exists we may approximate it by a term with a finite number no: lim pen> (a, t) =P(a, t) ·_pen,> (a, t). n-->= C. Critical point Since AHen> (t) or L~=oAHek> (t) is _in the 'n-th order with the external field 2:, the existence of the limit (3 ·16) needs the convergence of an infinite series of the form However, it is not necessarily true.*l Such a divergence can be avoided by the use of an initial ensemble described *> In this case such an infinite sum appears in the calculation of the cumulant. Let us write the m-th order cumulant as S<n,~·,n., if\ A 9 (ntl(t, r 1)>,dr1• .. dr.,. If (Ae<n•) (t, r 1) ) 0 ~0, then the cumulimt may be evaluated roughly as l<,,~,n., ,ft Ae<n<l (t)),J;$m"'-'Vn,~--n., {(n, + ~ +n.,) so'}.,_, exp ( n,+· .. +n..) No · Here, (n,+ .. ·+nm)so' is the upper limit of the effective region in which (IU'=,Ae<"<l(t,r,)), can have non-vanishing :value and we have put l(AHe•> (t, r))ol=i=exp(- (n/No)), where No .is some cons· tant, N 0 oclnF. We can expect that the cumulant converges only for No>O. On Extensive Properties of Probability Distribution Functions 377 by the density m,atrix of the form C0 (A.) exp (- J..B) p 0, where C 0 ().) is a normalization factor and ). is a constant pa:r:ameter. Putting A=O after calculating the probability distribution function, we. obtain P(a, t) in a non-equilibrium state beyond a singular point. Such a procedure is. similar to the calculation of the partition function of a spin system near the Curie point in the presence of a magnetic field. We can also use a microcanonical ensemble of the form (J(b-B)p 0 or of the form (J(b-BH ( t)) Po• Then, the probability distribution function for A is given by P(a, t) = J P(a, b, t, t')db ,- where P(a, b;·t, t') =<iJ(a-AH(t))(J( b-BH(t')))o 1s the. multitime probability 'distribution function of A and B. This procedure corresponds to the introduction of such a variable as the order parameter for the mode which will appear after an instability.*) § 4. Applications Single time probability distribution/uncti on We calculate the logarithm of the propability distribution function. Equation (2 ·1) can be expanded in the form A. (4·1) where P. ( o t t a, ; 1, t ) ···, " (J"P(a, t) = ~:E(t1)···(J:E(t.,.) I .E=o ~ ( i~ )"'P'O(t-tt) ···8(t,._1~t.,.)Tr[[··· [(J(a-A(t)), O"(t1)l···J, O"(t,.)]po. . ' (4·.2) The expansion (4·1) is resumed up as where Y.(a t·t ... t ) 0 ' '. 1, ' " - (J"'InP(a,t) ·~ (J:E(t1) .. ,(J:E(t,;) .E=O' (4·4) *> The recent develope1Ilent of the statistical mechanics of the phase transition shows that the divergences in thermodynamic quantities at a critical point stem orily from the elimination of the degrees of freedbm of dynamical variables. This situation would be the same even in non-equilibrium states. That is, if we .cam1ot remove the divergence. by the above procedure, we should increase the number of variab.les for the probability distribution function until we remove the divergence. 378 H. Furuka wa P.0 (a, t.' t 1, ... ' t n ) = tJ" exp {Y(a, t)} I· tJ2 (t1) • • ·tJ2(t,.) .E=o • (4·5) We find Po(a) =Tr tJ(a-A (t))p 0 -, Po(a, t; t1) = i~ (4·6) 0 (t- t') Tr [tJ (a- A (t)), 0" (t')] p0 = -(10( t-t')T r tJ(a-A (t'))d" (t')p 0 (4·7) • Here, we have assume d p0oce-HHo and used the formul a 1 ih Tr[A( t), B(t')] p0 = -(1 Tr A(t)B. (t') Po. (4·8) For exampl e, if we put A=tr and 2(t) = -F=co nstant , then we have P eq (a) =P(a) ocexp{ Y0 (a) + (1Fa + O(F 2)}. (4·9) Equati on (4·9) has a well-kn own form in the equilib rium statisti cal mechan ics. The probab ility distribu tion functio n of an arbitra ry A and 2 is not so simple. B. Contin uous time probab ility distrib utionfu nction Let us assume that A(t) has no invaria nt part. We deal with such quantit ies as the time rates of change in energy and particle numbe r of a subsyst em; i.e., (8/8t)H o(t), (8/8t) N(t). By treatin g the time interva l T in the same way as in the case of the volume V in the single time probab ility distribu tion functio n, we can asympt o~cally evalua te (2·4) for large T as P(a, T, t) = exp {Te:(;)} (4·10) in a steady state. Let us put 2(t') = { 0 ' -F, t'<t- T, t'>t- T. (4·11) Then, the expone nt of ( 4 ·10) may be expand ed' into 'a power series of F: (4·12) and also into a power series of a/T: (4·12') Hereaf ter, we shall conside r the case where (4·13) i.e., if 2 is the electric field, then d" denote s the electric current . Then (2 · 4) On Extensive Properties of Probability Distril'littion Functions IS 379 written as (4·14) ·We have by the use of the interaction representation rJ(t) of rJ, (4·15) We can show that (4·16) where {3 is the inverse temperature of the initial state as can be seen in the following argument. Let us consider the -quantity (4·17) It is seen that the integration of (4·17) with t1, ···, tn-1 in the interval t-T"-'T, (4·18) depends only on the time interval t'- t" except near the boundaries t and t- T if the correlation among the d''s and [d'(t'), rJ(t")] are small compared with T. This is due to the translation symmetry of ( 4 ·17) with respect to time. The translation symmetry is maintained also for remaining arguments. Therefore, neglecting the contributions from boundaries t and t - T, we write ( 4 ·18) as ={{ 1~/ (t') dt'} n- 1 [d' (t'- t"), rJ]) 0 = \{ 1~/ (t') dt'} n-1 {- 8~" [rJ (t')' (J (t")]} )0 ' This means that ( 4 ·18) is an even function for t'- t". Thus, it follows that ({ 1~/H(t')dt'})o ft d'(t')dt'}") _y;_l[{ ft d'(t')dt'}", =/{. Jt-T 2tft \ ) \ Jt-T 0 ft rJ(t")dt"]) Jt-T +O(P). 0 (4·19) Here, the .symmetry property of (4·18) and [A", B] =nA"- 1 [A, B] have been 380 used. H. Furukawa · Using the Fourie r transfo rmation of 6-function, we find· lo(a- r_t {jH(t') dt')) =fa). e-O,a 'L, _!!__/{ re {jH(t')dt'}")\ \ Je-T o n=O n! \ Jt-T . o = Ja;.e-o.a 'L, R=O _£_{/{ · f' rf(t')dt'}")· nl \ Je-T o (4·20) = Here, we have used the identit y (1/ih) Tr[A,B ]p -/3 Tr ABp0 (p0oce- 8 H•)·. 0 compar ing (4·20) with (4·14) , (4·16) is obtained~ ' Using the random variabl e J(t) for rjH(t'), we can rewrite (4·14) as · By - P(J, T) =exp{ -T[Fo (J) -t/3FJ +0(F2) ] } , (4-21) where . J=~ re T Je-T J(t')dt_ ' · . (4·22) IS the time averag e of J(t'). Equati on (4·21) r~sembles the. ~quiiibrium pro·babilit y distribu tion functio n (4·9), althou gh (4·-21) is only an approx imation ·for large F. In a steady state we have possibl y (4·23) Equati on ( 4 · 23), howev er, is not satisfie d in ·such cases as the followi ng: i) T is smalle r than a microsc opic relaxat ion time t"m, -where (((J(t) -(J(t- T)) 2 ) 0 ". <{j(t) 2 ) 0T 2 =(fi"2)0T 2 • ii) Tis larger than a macros copic relaxat ion time "C"y, which depend s on the size of the system disturb ed, but smalle r than the Poincare. recurrence time. In this case we find (((J(t) ---:(J( t-T))2 ) =2(((J (t) -((J(t ))0) 2 ) 0 if 0 we assume ((J(t)( J(t-T) )0 =((J(t ))0((J(t- T))0 =((J)2. , iii) Tis large:J;" than the ~oincare recurre nce time. In this case we cannot study (((J{i) ~(J(t-'-T) ) 2) 0 :(;rom a statisti cal mechan ical point of view. Theref ore,. it should be assume d that' "C"m<T<rx. , C. Conse rved quanti ty Let us assume that the system change s· so slowly in space-t ime that we can regard- that macros copic subsyst ems are homog eneous and indepe ndent of one anothe r. Then, we can extend ( 4 · 21) as P( {J}) =exp{~[Fo,(J,) _:_tJ,.E~+O(.E 2)]}, ' where the subscri pt i denote s the i-th subsys tem in spac'e-t ime. can be written as • (4·24) Equati on (4·24) On Extensive Properties of Probability Distribution Functions P({J(t,r)})=Po({J(t,r)})exp{-]:_!3 2 381 lt f J(t',r')Z(t',r')dt'dr'+0(Z Jv Jt-T 2)}. (4·25) Here, J(t', r') and Z(t:, r') denote some variable and external field changing slowly in space,time. When fJn(t) satisfies the continuity equation J(t, r) +Y ·l(t, r) = 0 in terms of J and L (4 · 25) is rewritten as P( {1}) =Po( {I}, T) exp {- ! ss {3 I(t', r') ·F(t', r')dt'dr' + 0 (F 2) } , (4 ·26) where F(t, r) = -YZ(t, r). Here, a partial integration was used and a surface integration was neglected so as to derive ( 4 · 26) from ( 4 · 25). § 5. Extrema of probability distribution function and variational principles If the probability distribution function is highly peaked at the most probable value, then the variational technique is available. As an example we shall consider the continuous time probability distribution function ( 4 ·14). One may find that (4·1'4) 1s related to Onsager's pbtentials. 10h 12>.*l Let us rewrite (4·14) as (5·1) Here, 1J!0 and fflo are the lowest order expansion coefficients given in ( 4 ·12) and (4·12') and we have denoted the rest terms of the exponent of the P(a, T) as Tg (F, a/T). Since g does not contain terms . only of F or a/T, we observe that g(O, a/T) =g(F, 0) =0. We have, in virtue of the normalization conditiC!n, f Pda=1, imd the boundary condition, P(±oo,T)=O, I \ I \ a·{g(F,a/T)-OJo(F)} aF )=o ' a{g(F, a/T)-lJ!o(a/T)})=o ' . a(aiT) (5·2) (5·3) <.. ·)=f···P(a, T)da. if P(a, T)has a sharp maximum at a=a*, then where (5 · 3) reduce respectively to and (5 · 2) a{g{F, a* /T) - 0J 0 (F)} aF a {g(F, a* /T) -1J!0(a* /T)} a(a*/T) 0' 0. (5·4) (5·5) *> Recently, an advanced discussion of Onsager's potentials was made by H. Hasegawa (preprint) by the use of the Fokker-Planck equation imd in virtue of quantum-mechanically explored variational principles. 382 H. Furukawa Two kinds of potential function 1J!(F, ; ) =g(F, ; ) -(/) 0 (F), (5·6) (J)(F, ; ) =g(F, ; )-1J!o(;) (5·7) can be looked as generaliza tions of Onsager's ones. 10>-r 2> In fact, for small F, we have seen in §4 that g(F,a/T) =U3F(a /T)(=t/3 FJ). In this case 1J! and (/) are equivalent to Onsager's apart from the factor t. Therefore, . a half of the potential function due to Onsager gives the logarithm of the probability distributio n function of the time average of a flux in a steady state near equilibrium.. Two potentials (5 · 6) and (5 · 7) are related to each other by the transforma tion: (5·8) Our potentials need no requireme nt of the microscopi c reversibili ty due to its definition as several authors have discussed in various ways. 13>' 14> § 6. Concludi ng remarks The main purpose of this paper is to show the extensive property of the probability distributio n function of macrovari ables in a non-equili brium state. · This is done by showing the existence of an infinite sequence of probability distributio n functions which converge to the true probability distributio n function maintainin g the extensive properties of the logarithms . One of the conditions under which the logarithm of the probability distributio n function in a non-equili brium state is extensive is that the perturbati onal technique can be used. The range of the applicabili ty of our analysis might be so wide. For the perturbati onal method can be used even in a state beyond a critical point starting from an initial thermal equilibrium state, though no proof exists, and that a state which we cannot reach by a perturbati onal method correspond s to an unstable state which will be difficult to maintain in a laboratory . The state of a system is described by nearly zero Fourier componen ts of physical quantities in space-time . If such Fourier componen ts with small wave numbers and small frequencie s do not change so drastically under the perturbati on of an external field, then the perturbati onal series can be truncated and the correlation lengths among physical quantities in a non-equili brium state remain of short range if it is of short range in an initial thermal equilibrium state. This is the reason why the logarithm of the probabilit y distributio n function can be extensive in a non-equili brium state. It seems that the continuous time probability distributio n function is suitable for the study of a steady state. So far as we deal with fluxes in·· a steady state, On Extensive Properties of Probability Distribution Functions 383 the results of any problems obtained by means of the single time probability distribution function and the results- by means of the continuous time probability distribution function would be the same. However, it seems that the continuous time probability distribution function is more tractable than the single time distribution function, since the former reflects Boltzmann's principle more clearly than the latter does as is seen in § 4. Acknowledgements The author would like to express his sincere gratitude to professor H. Mori for valuable discussions. This study is partially financed by the Scientific Research Fund of the Ministry of Education. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) H. Furukawa, Prog. The or. Phys. 50 (1973), 424; 51 (1974), 391; ]. Stat. Phys. 10 (1973), 139. H. Risken, Z. Phys. 186 (1965), 85. M. Lax and R. D. Hempstead, Bull. Am. Phys. Soc. 11 (1966), 111. ]. A Fleck Jr., Phys. Rev. 149 (1966), 322. M. Lax and W. H. Louisell, IEEE]. Quantum Elec. QE-3 (1967), 47. R. Graham and H. Haken, Z. Phys. 243 (1971), 289; 245 (1971), 141. R. Graham, Springer Tracts in Modern Physics 66 (1971), 1. K Kawasaki, Prog. Theor. Phys. 51 (1974), 1064. N. G. van Kampen, in Fluctuation Phenomena in Solids, R. E. Burgess (Academic Press, New York-London). R. Kubo, K Matuo and K Kitahara, J. Stat. Phys. 9 (1973), 51. H. Mori, Prog. Theor. Phys. 52 (1973), 433. M. Suzuki, Prog. Theor. Phys. 53 (1975), 1657. R. Kubo, ]. Phys. Soc. Japan 12 (1957), 570. L. Onsager, Phys. Rev. 37 (1931), 405; 38 (1931), 2265. L. Onsager and S. Machlup, .Phys. Rev. 91 (1953), 1505, 1512. N. Hasitsume, Prog. Theor. Phys. 8 (1952), 461; 15 (1956), 369. H. Nakano, Prog. Theor. Phys. 49 (1973), 1503 and References quoted therein. P. Gransdorff and I. Prigogine, Th{!rmodynamic Theor;y of Structure, Stability and Fluctuations (Wiley-lnterscience, 1971). · K Tomita and H. Tomita, Prog. Theor. Phys. 51 (1974), 1731.