Title Statistical Mechanics of Non-Equilibrium Systems : ExtensiveProperty, Fluctuation and Nonlinear Response

Author(s) SUZUKI, Masuo

Citation 物性研究 (1975), 23(5): C44-C54

Issue Date 1975-02-20

URL http://hdl.handle.net/2433/88897

Right

Type Departmental Bulletin Paper

Textversion publisher

Kyoto University

r = time devel0p. op.M ,

~ ;f!5 X mk

1) L.Onsager: Phys. Rev. 37· (1931) 405; ibid. 38 (193J) 2265.

2) N.Flashitzume: Pr0g. ,Theor. Phys. 8 (J952) 46:1; ibid 1~ (1956) 369

3) L.Onsager and S.Mach-Iup: Phys. Rev. 91 (1953) 1505;'ibid 15J2

4) M.S.Green:Journ. Chern. Phys. 20 (1952) 1281

5) J.L.Lebowitz and P.G.Bergmann: Ann. Phys. 1 (1957) 1

6) 'P.Glansdo'rff and I.Prigogine:Physica 30 (1964) 351

7) F.Schl~gr: Ann. Phys., 45 (1967) 155

8) RcGraham and H.Haken: ZcPhys. 243 (J971') 289, ibid 245 (J971)

141

9) RcGraham: Springer Tr'acts in Modern Physics 66 (1973)

Springer-Verlag

10) H.Nakano: Prog. 'Theo-r. Phys. 51 (1974) 1279

11 ) E . Ne I s on: Ph y s. Rev. 150 (] 966) 1079

Statistical ~~chanics of

'Non-Equi I ibrium Systems

- Extensive Property, Fluctuation and NonIinear Response -

Ma suo SUZUK I (Dep t. 0 f Phys i c~, Un iv. 0 f T0kyo)

( to be subm it ted to Prog. Theor ~ Phys. )

1. Kubo's Ansatz:'Extensi-ve Property

P(X, t) = Cexp CO¢(x, t)) ,

for large n with x = YO; X = macrovariable.

2 . 'Whe n Xis a Mar k0 f f ian macr 0 va riab Ie,

8p (x, t )8 -t = FM P ( x , t )

-C44-

•

Stat,istical Mechanics of Non":-Equil ibrium Systems

eVQI. eq. fo,ry (0 = deterministic path

Cn == n - th mornen t ,

variance a (t) :

I .

2 C 1 (y ( t) a (t) + Cz ( y (t) )

30 When X IS a non-Markoffian macrovariable,

conf igP (X, t ) = L ·0 (x-X) p ( {a. }; t ) ,

. . J

4. PorthemacrovariableX in quantum systems,

p(X, d -'T o(X-X)p(t); ih 8p(t) = eX, p(t))rat .

Extensive Property:

p ex, t) = C exp (n ¢ ( x, t }:J

5. C;enerating PunctionFormalism

=\

. '1' AX ( ) .e Pt·········'l'(A,t) r

L. eAXe tFpcon f ig. . 0

(quantal)

(stocahstic or classical)

(canonical) (normal ized)

1 .c+ i oo -AX' .p (X, t) =: '--. .f e 'l' ( A, t) dA

27l" 1 c - i 00

qeneralized thermodynamic potential

:7 ( A, t) = I og 'l' (A , t) 0 r 'l' ( A , t) = e:7( A ,t )

6. Assumption for the initial distribution:

J/ i ) . J/(i).P (0)= e . or Po = e

-C45-

7. ( i)Properties of W(A, t) and yO, t) when X J! and Jf are

Hermi tian:

1) 'PeA,t)'> 0 and J'7(A,t) = real for A real.

2) When A is real, '1' and J-'" are convex functions of A•.

. 3) Por complex A (= not real) , we have

I '.I' (A , t) I < '1'( Re A, t)

8. If 'P(A, 1) = (C1 e01f (A,t) (Bxtensive proper-ty) ,

C1

c+joo , .

p(X,t) = -'-. J exp [-O{lx-1f(A,x)}]d A27l'1 c-joo '

for large 0,

o1f (A, t)

OA

AO being given by the solution

= x

9. "Local operator Ii Q(d is defined by a functional of field

operators {1f(r / )}; r l cD (r) - circle domain of radius b.

10.. ( i )

Gen.eral condi tions f0r X ,J! and Jf

JJf(r)dr.

11. Theorem for- "extensive property" In quantal systems.

Theorem I (quantal) If the averages of local operators,

(1) C) (2) . , (3),<X(r» ,,<Jf I (r» ,and <Jf(r» are bounded, then

lim 1JtO O,t) = exist (uniformly convprgent)O~oo '

for IAI <A (fixed) and t finite. ThereforeW(A,t) has the

extensive property: W(A,t) = C1

exp C01JtCA,t)j

and' consequen t ly. so d0es {J (X, t) .,

-C46-

Statistical Mechanics of, Non-Equilibrium Systems

12. Defini t ion of the averages

First separate the system ,n into [11 and n'2

X ="Xl -L.-X2

,· 1/ (i) = 1/ (1 i) + 7/ 2( n, and 7/ +"'"I . thJ. ....kJ _ thJ. ~ - J1 1- J1 2 ,

whe r e Q j = j~, Q ( r) d ~ ., J

- s ( PI + ,uP2) s ( P1+ ,up2 )

Operational; p(s,,u).Q == e Q e

<Q>(j) - T Qp,/T p.r J r J

where Pj

= ,oj (s,,u) ,.' 0 <s $1, 0 $; ,u'5; 1,

_ .',' (i),01 (s,,u) - exp [ACX1 +,uX2 )) exp CAX(S',u) J1 (s,,u) J1 ),

. ) - [.,(i) (j)) c (j) 'Cj,0 2 ( S , ,u , exp J1 1 +,u J1 2 'exp AJ1 ( s ,,u) J1 (_ it, 1) Xl ...J

( .) ,

,03 (s,,u) exp [J1(ist,,u) J1 1l

) exp [A..ff(i(S-1)t,,u)X1)

13. Remarks on Theorem I

a) W is a continuous function of A o

( indep. of 0) .

c) If'{c j } are indep. of A for IAI < A, the limit YO(A,O =9-

Y(Ad) is uniformly convergent in wider sense in the

~om'plex circle domain IAI < A. Hence,

dPYn(A, t) dPy ('A,t )

1 i m - ( vVe i erst r ass )0- ~OO dip dA P

_ -1/d9) II bounded II for .02/.01 - 0 (n / I ) =9 O.K.

e) ,II bounded II for s = j.J. = 1=9 "If" for 0 < s < 1, 0 < ,u < 1

( conj ecture)

f) V\hen X (r) commutes wi th X (r / ), <X (r»(l) 1 s bounded.

g) If (J1(j)(r), J/(i)(r / ))= O~<J1(i(r»(I) is bounded.

It is not easy to prove that <..ff(r»(3) IS bounded when

[J1(r), J1(r / )) = 0 0

-C47--'-

, I

I

h) C lea r Iy, <cff (r)>(3) 1 S bound ed i f cff2 ' O.

j) Only the boundness of <X(r»(l) etc. are assumed.

j) ~,J1mn~oo thermodynamic limit in van Ilove's sense.

k) Theorem I is extended to classical systems. (Classical

Liouvi lIe op.).

14~

/

Proof of Extensive Porperty.:

Separation of the domain n

n = large integer

x ,= Xl +X 2

cff (i}= cff ~ i)- +cff (i). 1 2

To J?rove Cauchy IS condi t ion

'on conve rgence •

r - - - -'-'-,7 7r - -,-'- ~-,, I/' ./" I

1 I, A I

I (. .1 II 0 1 t ,I .0\ II 1 I

t: : 'I t 11-, /" /,-/--'-~ -,'- - - .,.-~.- -"-1[/./ ,/// .//.t 1. ./

~~ L ---L~v . f: _.<~-<':'-:I L I . I

I I 0,/1 II 0

1V ~·I 0

11

I 1·1 II I /1 :I ( IL ~ ---: ~ __ < ,d -.J

~ 1 Domains a} and O 2 witheach margin of wi dth bins i de in it; 0; is theshaded region.

d = d i mens ion,

:. 11ft (A ,t) -1ft eA, 1) Is 2-ri(n+m n ' , ' '" ) ,

:Formul a. :. lim 1ft (A,t) =: exist =: 1ft(A,t)n .

n~oo

L r.J J

d Aex) 'f,1 (l-s)A(x) I( ) sA(x) /,1 sA(x) I( ) (l-s)Aex)- e =, e A x e ds = e A xeds

'dx ·0' ·0'.

15. fjxtensive property in stochadtic modelsa " , . .- pUa.} t). r p({a.}t) ; r8tJ , ' J '

wh e re d. = ± 1 (s pIn s y stem) andJ

r p (fa.} t) = - \iV. ( a. ) p ( ... a. ...) + vV. (- a . ) p ( ... - a. ...)J J' J J 'J' . J J. 'J'.

-C-18-

Statistical Mechanics of Non-Equilibrium Systems

16. Theorem II (stochastic)..' s t ( r

1+ IJ.r

Z) J( ( i)

If P (s t) - e ee JV

("normal ll) for any r 1 (and r =r-r) and for O:SsSl andz 1

-o;5; IJ. "$, 1 , then

(uniformly convergent)

~ 1fr(Ad) and P(X,t) havp extensive property.

17. Lemma 1. - If f({a.}) g({aj}),then e tr f ':::;;; e trg for t ;~ O.J-

Lemma 2. - If f ( fa j } ) e.jl( then Irzf j ;s' Cn f, where Cn IS a, z z

constant dependent on n 2 ,

De f .P e·if (II no rma 111 )~ p ( ... - a. '" t) S C p (... a. ... ), J" 3 ,'J' ,

where C3

is a constant independent of- n,

18. Average motion, Fluctuat ion, and ~Nonl inear respunse

most probable path yet) for . .o·large:

1 ) x y(t)+z,g (z,t) =¢(y+z,t) = {1fr(A(y.d,t)-ykCy,d}

81fr 81fr-z{A(v.d+Cv-(-) )(-) } + .- ,aA x= y 8 A x=y'

81fr f"9 '.Y (t) = (8 J ) = <x> ; <X> = Xp ( X, t). dX = T X p ( 1) 0

/I. A= 0 t" t • r

, 2 2 Zvariance: a Ct)=( 8 1fr/ 8A )A=O = .0« x-<x:>t) >t

-----(i) (i)Nonlinear response: case (a) ; JI =J( +KY

(a) -1 a,/:- ,~Ci)(t) J(j)y (t )"'-, <x> = n (-) = T x e~ /T e

t , k 8 A . A= 0 r r

Theorem III a. - The nonl inear' response for case (a) IS

expressed by the fluctuation in non-equilibrium as

-C49-

for a quantal system. where

For stochastic systems, we have

( ) aYCt)-l (a)r'h,a. ---=0 ""---aX aY(t» •':I:'Xy; yaK"" '- t ,K '

~(i) -1 ~(i)

Y( t) - (e 1Te Jj ) e 1T ( YeJi )'

1) < ax ; aY( t) )a)1,K

2) < ax ;0 Y ( t) :Ja)*K"1 ,

(a)3) <Pxy;y real.

= < aY ('t); aX)a)1,K

< aY( t) ; aX)a)t ,K,

-(i) (iJFor a general nonlinear disturbance of the, form Ji =Ji. K ,

mea) - aY (t )= < ax . a(~Ji(i)(" ) ) >(a)':I:'x y ;yaK "~aK K t . 1 ,K

ca s e ( b ) Ji·= Ji + K Y ":

Theorem II lb.

Tr- iUf j(i) i 1Ji

x e e e

i r1 < eX, y ( s) ) )b) d s ,n'o" 1,K

lor a quantal system where yes) = exp (-is:J()Yexp (i~J().

(b) tTK

.1 -sTK

d s TK Ji (i)<l>Xy;y=~xe .f

odse (dKTK)e e

for a stochastic system, where rK <---~ Ji =J/+KY.

For a genera I nonl inear di sturbaJlce of the form Ji =Ji K,

19. Relation between if; (x,i!) and cumulants

Theorem IV. -If 'J!(A,O (~nd p(x,t)) has extensive property,

-C50-

Statistical Mechanics of Non-Equilibrium Systems

<x n > ==-- extensive (i.e., 0(0)). Conversely, if <Xn >. tc, 1 c,

are extensive, and, 1/

--,- n /n-lif lim «X> 1/n!) = A <00, then W(A,d and p(x,t)

n -.00 c, 0

exi~t and h~ve extensive property.

20 Theo rem V. - The func t ion ¢ (x, t) is expres sed by the cumul an t s,

00 n 'Z 2 -1 n -1

as ¢(Y+zd) = - L -. lim f '«X> to , ... , <X >c,tfi ),' n= 2 n! ~oo n, c, ,

where f is defhned byn -

and' 1/J~n-3 fn

is a homogeneous polynomial of order (n-2).

21. Expi ici t forms of {f n} :

(2-51/J 1/J - 31/J ) 1/J .,2 4 2 2,

22. Thorem VI.oc

When ¢(y+z,t) = L a zryn!n=2 n

I S known, alI

cumulants <Xn> are obtained,exP,licitIy with use of thec, 1

recurSIon formula;

:~-n 2. -1 n-l -1+ ( a (dj g «X>, 0 , .. ; <X > n )}n c,t ,c,t,

/

for n " 3, and for large fi, where 'gn is gIven by

,I, ) =- _ 1/J2n~3 f _,I, n-3 ,I"'Y n -1 2 'Y 2 'Yn n •

23.n

fiJxpl ici t forms of <X > t ,arec,

~C51-

24. Composed macrovariable fCx) ; x= x/o:

< f (x) > = f (y (t) + 0 (0 -1 )

and th~ variance arC I) isgivPD by

8 f 2 -1afcO = (-8) a (t) + 0(0. ),

x x-:::y(t) x

. 25. Exactly soluble models:

Ex.l Non-inte'racting system:

!l ..J( = - J L a~

NI . J'J

Sol uti 0 n : qr ( A, t) =. ex p C01f/ ( A, t ) J ( ex act f or any 0) ,

Yi'(A,t) =: log {tosh A + sinh A tanh h cos C 2tJ/h)}. ,

J Cx ,t) - t an h -1 C{X - a ( t )} / { 1- a ( 0 x} ; a (0 =__ tanh h cos (2 tJ/h ),

if> ( x , t ) 10g J a ( t) sinh A (x-, t) + cos h A (x. t ) } - x A (x , t)

2 .Y Ct) - a (1) and var i apce a( t) = 1 - y (1)

Ex. 2 Kine tic \iVe iss 1sing mod e I ( Kub 0 eta I, .....• )

Ex.3 (Anisotropic) Weiss Heisenberg model

.J( = .J((J J J ;H) = _0-1 L (J a~a~ + J ara~ +J,a~a~' )~,lllILa~x, y. z· i , j XI J yl J Z I J' j J'

X = L a~' and' X(i) = - Ii J/ (J 0 J 0 J 0; II ). J, x, y, z 0J '.

Ex.4 Nonlinear relaxation In the linear kinetic, Ising model.

Ex.5 Nonlinear relaxa'tion In the linear anisotropic XY-model:

26. App I i c a ti 0 n s: rei axa t i on and. flu c t ua t i () n Ins upe r cond 1I C tor s .

~C52-

Statistical.Mechanics of Non-Equilibrium Systems

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~C53-

~~;fd~tt

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