Download On Extensive Properties of Probability Distribution Functions in Non

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.
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