Title
Author(s)
Citation
Issue Date
URL
Statistical Mechanics of Non-Equilibrium Systems : Extensive
Property, Fluctuation and Nonlinear Response
SUZUKI, Masuo
物性研究 (1975), 23(5): C44-C54
1975-02-20
http://hdl.handle.net/2433/88897
Right
Type
Textversion
Departmental Bulletin Paper
publisher
Kyoto University
~
X
;f!5
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 . N e 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 i v.
( to be subm it ted to Prog. Theor ~
1.
0
f T0kyo)
Phys. )
Kubo's Ansatz:'Extensi-ve Property
P(X, t) = Cexp CO¢(x, t)) ,
for large
2.
n
with x = YO;
X = macrovariable.
'Whe n Xis a Mar k 0 f f ian m a cr 0 va ria b Ie,
8p (x,
8 -t
t )
=
FM P ( x , t )
rM =
-C44-
time, devel0p. op.
•
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)
30
When X
IS
P (X,
4.
a
(t)
+ Cz ( y
(t) )
a non-Markoffian macrovariable,
=
t )
conf ig
L
.
.
·0 (x- X) p ( {a. };
t ) ,
J
PorthemacrovariableX in quantum systems,
p(X,
d -'T o(X-X)p(t); ih 8p(t)
r a t
=
eX,.
p(t))
Extensive Property:
p
5.
ex,
t)
(n ¢ ( x, t }:J
C exp
=
C;enerating PunctionFormalism
. '1'
'l'(A,t)
r
=
L.
\
p
( )
. (quantal)
e AX Pt·········
1
(X, t)
eAXe tFp
con f ig.
=: ' - - .
27l" 1
.f
.
.c+ i oo
e
(stocahstic or classical)
0
-AX' .
'l' ( A,
t)
dA
c - i 00
qeneralized thermodynamic potential
:7 ( A, t)
6.
= I og 'l' (A ,
'l' ( A, t) = e:7( A ,t )
t) 0 r
Assumption for the initial distribution:
J/ i )
P (0)= e
.
.
or
Po
=
J/(i).
e
-C45-
(canonical) (normal ized)
7.
Properties of W(A, t) and yO, t) when X J!
( i)
and Jf are
Hermi tian:
1)
'PeA,t)'> 0 and J'7(A,t) = real for
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
8.
'1'( Re A, t)
If 'P(A, 1) = (C1 e01f (A,t)
C
27l'1
t)
"Local operator
Ii
exp [-O{lx-1f(A,x)}]d A
'
Q(d is defined by a functional of field
operators {1f(r / )}; r
10.
.
x
=
OA
Jc-joo
,
AO being given by the solution
for large 0,
o1f (A,
(Bxtensive proper-ty) ,
c+joo
1
p(X,t) = - '-.
9.
A real.
l
cD (r) .
Gen.eral condi tions f0r X ,J!
circle domain of radius b.
(i)
and Jf
JJf(r)dr.
11.
Theorem for- "extensive property"
I (quantal) If the averages of local operators,
Theorem
(1)
<X(r»
,,<Jf
lim 1Jt O,t)
O~oo
for
O
IAI
In quantal systems.
=
C)
I
(r»
(2)
. ,
,and <Jf(r»
(3),
are bounded, then
exist (uniformly convprgent)
'
<A
(fixed) and t finite. ThereforeW(A,t) has the
extensive property:
W(A,t) = C 1 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
,n
First separate the system
X ="Xl -L.-X 2 ,·
I
.
1/
thJ.
=
(i)
into [11
(1 i)
1/
....kJ
+
n,
2(
7/
_ thJ.
and
n'2
and
~7/
J1
-
+"'" J1
1-
2 ,
whe r e Q j = j~, Q ( r) d ~ .
,
J
T Qp,/T
<Q>(j) where
,01
P
_
(s,,u) .
,0 2 ( S ,
,03
13.
,u
r
) ,
J
0 <s $1,
0
.','
exp
(s,,u)
[.,(i)
J1 1
cA
(j))
+,u J1 2
'e xp
exp [J1(ist,,u) J1
( l.)
1
$; ,u'5; 1,
CAX(S',u)
exp [ACX 1 +,uX 2 ) ) exp
Remarks on Theorem
a)
Qe
p.
r
J
= ,oj (s,,u) ,.'
j
s ( P1 + ,up2 )
- s ( PI + ,u P 2)
== e
Operational; p(s,,u).Q
(i)
J1 (s,,u) J1
(j)
),
'Cj
J1 ( s ,,u) J1 (_ it, 1) Xl ...J
,
)
exp [A..ff(i(S-1)t,,u)X 1 )
I
W is a continuous function of A
o
( indep. of 0) .
c)
If'{c j
}
<
are indep. of A for IAI
=9-
A, the limit YO(A,O
Y(Ad) is uniformly convergent in wider sense in the
~om'plex circle domain IAI
dPYn (A,
II
e)
,II
bounded
II
bounded II
A. Hence,
dPy ('A,t
t)
1im
)
-
0- ~OO
9)
<
dip
( vVe i erst r ass )
dA P
_
for .0 2 /.0 1 - 0
for s
=
(n -1/d
/
I
)
=9 O.K.
= 1=9 "If" for 0 < s < 1,
j.J.
0 <
,u
<
1
( conj ecture)
commutes wi th X (r / ),
f)
V\hen X
g)
If (J1(j)(r), J/(i)(r /
(r)
<X (r»(l)
= 0
bounded.
))= O~<J1(i(r»(I) is bounded.
It is not easy to prove that <..ff(r»(3)
[J1(r), J1(r / ))
1s
IS
bounded
when
0
-C47--'-
,
I
I
> (3)
h)
C lea r I y,
j)
Only the boundness of <X(r»(l) etc. are assumed.
j)
~,J1mn~oo
k)
Theorem I
<cff (r)
1S
bound ed i f cff 2
O.
'
thermodynamic limit in van Ilove's sense.
is extended to classical systems.
(Classical
Liouvi lIe op.).
/
14~
Proof of Extensive Porperty.:
-'-'-,7 7r - -,-'-
r - - -
n
Separation of the domain
,
I/'
1
I
I
I,
(.
0
I
./"
.1
I
I
,I
1
t
1
.0\
I
n = large integer
x ,=
Xl
t
-
-
1.
I
L
I
I 0,/1
0
I
I
I
L
2
~
---:
1·1
I /1
(
__
~
<
1
.,.-~.- -"-1
./
_.<~-<':'-:
I
V ~·I
1
I
cff (i}= cff ~ i)- +cff (i)
1
:'
~~ L ---L~v . f:
+X 2
.
/,-/--'-~
-,', / / / .//.t
[/./
,d
I
I
t :
1-, /"
~-,
I
A
.I
I
0
1
1
I
:
-.JI
Domains a} and O 2 with
each margin of wi dth b
ins i de in it;
is the
shaded region.
~ 1
To J?rove Cauchy IS condi t ion
0;
'on conve rgence •
d = d i mens ion,
:. 11ftn+m (A ,t) -1ftn eA, 1)
:Formul a.
Is '2-,ri
),
'"
(
'
:. lim 1ft (A,t) =: exist =: 1ft(A,t)
n~oo
n
.
d Aex)
'f,1 (l-s)A(x) I( ) sA(x)
/,1 e sA(x) AI( x
) (l-s)Aex)
- e
=, e
A x e
ds =
eds
'dx
·0'
· 0 ' .
15.
fjxtensive property in stochadtic models
a
"
- pUa.} t).
8tJ , '
wh e re
r
J
d. =
J
p (fa.} t)
J'
±1
,
r
. .
p({a.}t) ;
J
(s pIn
'
r
L
J
s y stem)
r.
J
and
= - \iV. ( a. ) p ( ... a. ...) + vV. (- a . ) p ( ... - a. ...)
J
J
'J'
-C-18-
.
J
J.
'J'.
Statistical Mechanics of Non-Equilibrium Systems
16.
Theorem
II
(stochastic).
st (r +
.'
If P
("normal
-
e
(s t) ll
IJ.rZ )
1
o;5; IJ. "$, 1 ,
JV
r 1 (and r z =r-r)
1
for any
)
J( ( i)
ee
and for O:SsSl and
then
(uniformly convergent)
~ 1fr(Ad)
17.
and P(X,t) havp extensive property.
Lemma 1. -
If f({a.})
Lemma 2. -
If f ( fa j } )
g({aj}),then e tr f ':::;;; e tr g for t ;~ O.
J
-
constant dependent on
De f .P
e·if (II
e.jl(
Irzf j ;s' Cn
then
, z
f, where C
nz
IS
a
n2 ,
no rma 111 ) ~ p ( ... - a.
,
t) S C p (...
'"
J"
3
a. ... )
,'J'
,
where C 3 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-(-)
)(-)
} +
, a A x= y 8 A x= y
9
'.Y (t) =
81frJ )
(8
/I.
A= 0
=
,
<x> t;"<X> t
2
=
f"
•
.
'
Xp ( X, t). dX
2
=T Xp
r
( 1)
Z
variance: a Ct)=( 8 1fr/ 8A )A=O = .0« x-<x:>t) >t
(i)
-----(i)
Nonlinear response:
y (t )"'-,
<x>
Theorem III a. -
(a)
t ,k
case (a) ;
=n
-1
a,/:-
(-)
8 A . A= 0
JI
=
=J(
+KY
,~Ci)(t)
T x
r
e~
/T
The nonl inear' response for case (a)
expressed by the fluctuation in non-equilibrium as
-C49-
J(j)
r
IS
e
0
for a quantal system. where
For stochastic systems,
()
r'h,a.
we have
a YCt)-l
---=0
(a)
""---aX aY(t»
'-
':I:'Xy; y a K " "
~(i) -1
(e 1Te
Y( t) -
1)
< ax ; aY ( t) )a)
2)
< ax ;0 Y ( t) :Ja)*K"
1,
3)
<Pxy;y
1,K
Jj
)
•
t ,K '
~(i)
e 1T ( YeJi
)'
= < aY ('t); aX)a)
1,K
< aY ( t) ; aX)a)
t ,K,
(a)
real.
-(i)
(iJ
=Ji
.K ,
For a general nonlinear disturbance of the, form Ji
mea)
-
a Y (t ) = < ax . a (~Ji(i)("
"~aK
x y ; yaK
':I:'
ca s e ( b )
Theorem
Ji·= Ji
.
Tr
x e
1 ,K
+ K Y ":
- iUf j(i)
e
i 1Ji
e
II lb.
i
r
1
< eX, y
n'o"
<l>Xy;y=~xe
tTK
.1
.fo
dse
-sTK
for a stochastic system, where
( s) ) )b)
ds
,
1,K
yes)
lor a quantal system where
(b)
) ) >(a)
K t
=
exp (-is:J()Yexp (i~J().
d
(dKTK)e
rK
<---~ Ji
sT
K
Ji (i)
e
=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. - I f 'J!(A,O (~nd
p(x,t))
-C50-
has extensive property,
Statistical Mechanics of Non-Equilibrium Systems
<x n > c, 1
==--
0(0)). Conversely, if <X
extensive (i.e.,
n
>.c, t
are extensive, and
- - ,-
if
lim
n -.00
exi~t
20
and
,
n
«X>
h~ve
Theo rem V. -
c,
1/
/n-l
1/n!)
=
A
<00,
0
then W(A,d and p(x,t)
extensive property.
The func t ion ¢ (x, t) is expres sed by the cumul an t s,
00
n
Z
'
2
= - n=
L - . lim f '«X> to
2 n! ~oo
n,
c,
as ¢(Y+zd)
'
-1
, ... ,
<X
n
-1
>c,tfi
,
where f n is defhned by
-
and' 1/J~n-3 f n is a homogeneous polynomial of order (n-2).
21.
Expi ici t forms of {f n} :
(
1/J,2 1/J4 - 31/J22) -1/J5
2,
.
oc
22.
Thorem VI.
When ¢(y+z,t)
cumulants <X
n
>c, 1
=
L
a
n=2
zryn!
I S known, alI
n
are obtained,exP,licitIy with use of the
recurSIon formula;
+ ( a (dj
:~-n
g
/
2.
«X>,c,t 0
n
-1
n-l
for n " 3, and for large fi, where 'gn is gIven by
,I,
'Y
23.
n -1
)
=- _
fiJxpl ici t forms of <X
n
1/J2n~3
2
>c, t
f
n
_,I, n-3 ,I"
'Y 2
'Y n
,are
~C51-
-1
, ..; <X, c
> ,
n t)},
•
),
24.
Composed macrovariable fCx) ; x=
<f
(x)
>=
and th~ variance
afcO
. 25.
=
+0
f (y (t)
arC
I)
(0 -1
x/o:
)
isgivPD by
8f 2
-1
(-8)
a (t) + 0(0. ),
x x-:::y(t) x
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
Yi'(A,t)
=:
log {tosh A + sinh A tanh h cos
J Cx ,t)
-
t an h -1
10 g
if> ( x , t )
C{ X -
J a ( t)
=__
sin h A (x-, t) + cos h A (x. t ) } a(
t) = 1 - y
2
f or any 0) ,
C 2tJ/h)}.
a ( t )} / { 1- a ( 0 x} ; a (0
a (1) and var i apce
Y C t) -
x A (x , t)
.
(1)
Kin e tic \iVe iss 1 sin g mod e I ( K ub 0 eta I, .....• )
Ex.3
(Anisotropic) Weiss Heisenberg model
X
= .J((J
=
J
x, y.
J ;H) = _0- 1 L (J a~a~ + J ara~ +J,a~a~' )~,lllILa~
z·
i ,j
L a~' and' X(i) = - Ii J/ (J
.
J,
J
,
tanh h cos (2 t J/h ),
Ex. 2
.J(
26.
C01f/ ( A, t ) J ( ex act
0
x,
XI
J
J
J
0
y,
yl
0;
z
J
Z
I
J'
j
II )
0
'.
Ex.4
Nonlinear relaxation In the linear kinetic, Ising model.
Ex.5
Nonlinear relaxa'tion In the linear anisotropic XY-model:
A pp I i c a ti 0 n s:
rei ax a t i on and. flu c t ua t i () n Ins upe r con d 1I C tor s .
~C52-
J'
Statistical.Mechanics of Non-Equilibrium Systems
References
])
R.Kllbo, J.P.hys. Soc. Japan]2 (]957) 570.
2)
P.Glansdorff and
I.Prigogine, Thermodynamic Theory of
(~il e yIn t e r sci en c e ,
S t r DC t II r e, S ta b iii t Y an d Ii' I uc t ua it on s
New Y0 r k J 9 7 J ) 0
3)
R.Oraham and F.Haake, Springer Tracts in Modern Physics,.
Vol.
4)
_6~
(Springer Verlag Berlin,
New York, 1973)0
Il.Haken, Cooperative Phenomena in Systems Far from Thermal
Equilibrium and in Non-Physical
f))
Heidelbprg~
R.Kubo, in Synergetics
Schloss glmau),
Systems (to be published).
(Proc. Symp. Synergetics, 1972,
JLHaken and B.GoTeubner, eds. Stuttgart
(] 073L
6)
lLKubo,
K.l'vJatsuo and K.Kitahara,
J.Stat.
Phys. 9 (1973) 51.
T)
M.SUZUKi, Phys.
8)
R.B.Griffiths, in Phase Transitions and Critical' Phenomena
Letters A (in press).
ed. C .Domb and M.. S .Green
(Acado Press"
1972) and references'
cited therein. D.Ruell~,Statistical Mechanics:
Resul ts (VI.A.B~nj amin, Inc.,
and J. L. Lebowi tz,
Rigorous
New York, 1969). M. EoFisher
Comm. Ma th. Phys. 19 (]970) 25].
9)N.G.Van Kampen, Can. J.Physo 39 (196]).55];
mental Problems in Statistical' Mechanics
and in
Funda-
(E.G.D.Cohen, edo'
Amsterdam, ]962).
10)
van Hove, Physica 21 (]955) f)]7;
J J)M. Suzuki, Int. J .Magnet ism
K.Matsuo, private communication.
]3)
H.Kubo, J.Phvs.
]4)
,
1 (] 97])
]2 )
'
.
N.Bourbaki,
1
Soc. Japan -]7 (1962) 1100.
1
I.
.
I
Elements De Mathematique Fonctions-D Dne
K.Tomita and II.Tomita,
K.Tc:lmita~
123.
.
Va ria b I e He e I I e <II e r man n,
If))
23 (]957) 441.
Par is, 196]).
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T.Ohta and IJ.Tomita, ibid 52 (]974) 737.
~C53-
~~;fd~tt
16)
H.Mori, Prog. Theor. Phys. 52 (1974) 1133.
17)
ReJ eGlauber,
18)
M.Suzuki andR.Ktibo,
]9)
B. U.Felderhof,
J
.Math~
Phys. 4 Cl9(3) 294.
J .Phys. Soc. Japan 24 (1968) 51.
Fep. Mathe Phys. 1 (J97J) 2]5;
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B.U.Felderh()f and lVI.Suzuki, Physica 56 (1971) 1.3.
-C54-