Indian Journal of Pure & App li ed Ph ys ics
Vol. 37. September 1999. pp . 647-651
Non-equilibrium ensembles and thermodynamic functions
*A Maghari & B Haghi ghi
Department of Chemi stry. Universit y of Tehran. Tehran. Iran
Recei ved 2 Jun e 1997: revised 28 M ay 1998: accepted 10 June 1999
A IlIJll -equili hrium ense mble is formul ated and th e part iti on fun ction of a system in non -equilibrium transient state is
derived hy a meth od based upon th e max imi zation of the Boltzman n's entropy co nstrain ed by non linear Boltzmannkineti c equation . We co nsici er a logica l th eorem which ex presses that th e characteri stic fun cti on in a non-equ ilihri um
ensemhle has a form simil ar to th e equilibrium one. Th e connecti on hetween th e N-part iele to one-parti cl e non-equilibrium
p;lrtiti on functi on is oh tain ed and th e ti me dependent thermodyn amic fun cti ons as we ll as tran sport ki neti c CCle lTi cient s ca n
th en he G;l lcu lated.
the class of th e loca l integral s o f moti on on w hi ch the
stati sti cal operator can depend . He max imi zes an
The co nce pt or an equilibrium cla ss ica l
entropy co nstrained by th e Fouri er transform s of the
ensembl e in pha se space wa s introd uced by JW
co nservati on laws o f energy. momentum, and the
Gi bbs'. O n the basis o f th e prin cipl e of equal a priori
parti cle number.
pro/)({bilily, a ce rtain number o f equilibrium
On th e oth er hand several auth ors max imi ze the
ense mbles, can be co nstructed . ft is show n that
normali za ti on co nstant, th at i s, th e partiti on non-equilibrium entropy'rI .I.2.1. 27.2"-II, and recentl y a non-
I Introduction
function (Zustands summe from the German word
mea ns "su tl1 over stat es") co rres pond i ng to the
equilibrium ensembl e co ntain s all the informati on
necess ary ror th e cal culati on o f th e th ermodynami c
quantiti es. Hence. th e ce ntral prob lem o f equilibrium
statistical mechanics is reduced to the co mputat ion o f
the partiti on function.
Several authors try fo r co nstru ctin g th e
systemati c meth ods parall el to equilibrium stati sti ca l
ensembl e in order to stud y th e systems which are not
in e quilibrium ~7 T hese meth ods naturall y open th e
ways for th e stud y or the time evoluti on of th e
th ermody nami c quantiti es. From thi s ense mbl e, one
can defin e not th e th ermodynami c state but rath er th e
time evo luti on of any non-equilibriutl1 macros tate.
Z ubarev' proposed a meth od w hi ch he ca ll ed th e
'n on-equilihrium stati sti cal ensembl e' in order to
ex tend th e idea or G ibb sian stati sti ca l ensembl e to
include th c non-equ ili brium case. In thi s meth od a
non-eq uilibriull1 stati sti cal operator (N ESO) was
constru cted ror stati onary process by generali zati on
': '1'0 whom correspondencc shou ld hc addresseci . e- lll;lil :
Illagh;l ri @kha yam.ut.ac. ir
equilibrium stati sti ca l ensembl e for a class ica l case
was formulated by ma x imi zin g a time integrated
entropy co nstrained by Li ou vi li e equati on. T he
reason for thi s procedure is that th e co nser vati on of
energy, momentum, and pa rti cle numbers are
co nsequence of Li ouvill e's equati on. Thi s approach.
however, has two diffi culties , despite o f very vas t
properti es o f Li ouville equati on that have been
studi ed '-l 22 : (i) requires th e so luti on o f Li ouvi lle
equ ati on w hi ch in vo l ves 6N variabl es in th e N-body
prob lem ( ii ) The non-equilibrium partiti on f uncti on
drops to th e time independent partiti on functi on,
because Gibbs entropy is an integral o f moti on.
In th e present wo rk , we describe a new
formulati on of th e mcth od in th e theory o r nonequilibrium ense mbl e int rod uced by Sobouti 'vv ith the
purpose or makin g hi s idea morc appli c;tbl e and
co mputabl e ror a non-equilibrium sys tem. W e deri ve
an ex pres sion for one-parti c le non-equilibriutll
partiti on fun cti on by max imizin g the B o ltzmann ' s
entropy co nstrain ed by th e Bolt zmann kin etic
equati on and the co nncc ti on between N -parti c le to
one-parti cle partiti on fun cti on is ex hibited in terms o f
th e U rse ll run cti ons. Thi s prov id es a moti vati on for
INDI AN J PURE APPL PHY S. VOL 37 . SEPTEM BER 1999
648
relatin g non-eg ui I ibriu m th ermodynamic fun cti on
and one particl e partiti on fun cti on. Essenti al to th e
approach tak en here is th e noti on that the general
form of th e characteri sti c fun cti on in a non- w here
equilibrium stati sti ca l ensembl e is th e sa me as to th e
equilibrium one.
F
2 Non-Equilibrium Ensemble Method
In 1877
functi on as1'
Bo ltzmann
introd uced
...(6)
= k, j, lnf, - 1..(1)1, - y(p, .q,.r{ ;:; - J(j;. f,))
... (7)
the entropy
and usin g the Euler-Lagran ge equati on in ca lculu s of
va ri ati on
51! =-k/J ffdq ,dP J,( q " p ,,t) ln I ,
l
(q l, PI J ) +Nk /l ln(f./h: )
... ( I )
where E is th e intern al energy for parti cle, q and p are
pha se vari abl es, N is the parti c le number of th e
system, k Ba nd h stand t or th e Bo lt zmann and
Planck
co nstant s,
res pec ti ve ly .
Th e
functi on
f, (q " p " t) is onc- parti cle di stri buti on functi on
... (8)
th en we obtain
... (9)
dcfined as
·
N! J J1' (N )d I
d
w here A and ya re Lagrange multip liers. For a
.I l (q l ,PI, I ) = (N- I)! ....
1 2 ( 1 , . .... 1 N ···(2 ) co nve ni ent definiti ons
.. .( l Oa)
f ( I) is N -Parti cle di stributi on fun cti on. Th e
functi on f, (q l ,PI' t) is normali zed to N and sati sfi es
w here
... ( I Db)
th e Bo lt zmann kineti c equati on:
...(3)
the one- particle di strib uti on fun cti on then beco mes
... ( II )
...(4)
where H
I
is th e one-parti c le Ha milt oni an,
1.. . )
denotes a class ical Po isson bracket, J ( f, ' f , ) IS a
w here
q Il~ ( t ) is
non-equilibrium
one
parti cle
bin ary co lli sion term . In a time interva l ( t l , t 2 ) we partiti on f uncti on and pl ays a central ro le In nonequilibrium
stati sti ca l
th ermodynami cs.
On
ca n def ine a ti me integrated 9\- fun cti on.
substitutin g Eq.( II ) into th e integral in Eq.(3), we get
91= f :: 5/1(t) dt =-k" f :: d1
ff dq Idp I [l l I n I I - N In (c/ /7 ' )]
... (5)
To determin e G
It ca n be postul ated that 9\ - fun cti on is referred
dQ =0
as th e ac tio n integral and sati sfies the L agrangian
cit
forn llli ati on o f mechani cs . As a res ult , we must
max imi ze Eq.(S) sub ject to th e constraint s of Eqs (3) w here
and (4) . In oth er word , we can define a functi on
10
I'
one fin d
... ( 13)
64<)
MAGHARI & HAGHIGHI : THERMODYNAMIC FUNCTIONS
Q==G I +k 13 Tln(q ll ,, (t)/q,, )+
(
f J (II"/I ) cit
... ( 14)
II
q nc ( t)
A(t)
I
= exPll- ~ )
... (20)
and from Eq. (20) we find
On the other hand, it can be shown that the NParticl e di stribution function can be written as
( I)
qne
A(O)
= --k-
.. .(2 1)
qc
13
... ( 15)
where "-(0) sati sfies the following equati on
A(O) - k n
= -k J) Inq e
wh ere Qnc (t) is N-particle partition function and in
Now we can defin e a
both Eqs ( I I) and ( 15) ~ has its usual meanlll g. characteristic function as foll ow
47
Using th e Ursell functions, we ca n write
.. .(22)
non-equilibrium
... (23)
Thi s definition exhibits a logical notion since at
a limitation , when time goes to infinity, the nonexp(-BG; )ITU ~;)
equilibrium
partition fun ction goes to the equilibrium
11/ = 0 r ..\ IiI \
i.k Ef'
partition functi on and definiti on (23) beco mes an
He re s is a li stin g of free particl es, r is a listin g usual equati on in equilibrium statistical mec hani cs.
of pair partic les and U (i~ ) is Ursell function defined With simpl e manipulati on one ca n show
NI?
... ( 16)
IIIT
as
A (t)
= A c + TA(O) bt
... (24)
u \~) == exp(-BG \~» ) -exp[-B ( G \ I ) +G~I » )] ... ( 17) where A e is equilibrium part of He lmholtz free
energy.
Co mbinations of Eqs ( II ), ( 15) and ( 16) with
Now define the flu ctu ati on ex In the following
Eq. (2) yie ld s th e connecti on between the N-particl e
mann
er
to one- particle non-equilibrium partition function ,
that is
ex = A(t) - A e = TA(O) bt
.. .(25)
I
Q(I)=-
'"
I
N!
N!
1
~c' v, dr
e- '
r
N'•
~ m! (N _ 211l)!2
Nt"
===
In
( j il l '
(I)
f
flux
II
r
termin ology
, we have
= eX =n co)
of
irrevers ibl e
.. .(26)
12 (.
... ( 18)
where
usual
th e rmod y nami cs4"4~
[U J
Ir
N -2 111
II I
th e
is the ele ment of phase space .
and th e force is defined as derivative of e ntropy
res pect to flu ctuations
force
= af,S = ~ ( aM I
ria
3 Calculation of Onsager Coefficients
a
()/
)
.. .(27 )
Co nsi J e r a sys te m is tak en from an equilibrium where
state at t = () to a non-equilibrium state. If the sys tem
.. .( 28)
deviates onl y sli ghtl y from equilibrium, th e nonequi Ii bri um one-part ic le partiti on function ca n be with So is equilibrium part of entropy. Then
expa nded in powers of small time Ot, and it ca n be
dropped after two terms as
... (29)
.
q nc ( t)
= q c + q '''' bt
(I)
From Eqs ( lOa) and ( II ), we ha ve
... ( 19)
Based upon aforementioned te rminology, one
phenomenological coefficient is defineJ as
65()
INDIA
T 2A? (0)
L===---
as B/at
J PURE APPL PHYS . VOL 37, SEPTEMBER 1999
Acknowledgment
.. .(30)
The financial support of the Research Council of
University of T ehran is acknow ledged.
4 Conclusion
By combining th e ph enomenological theory of
irreversibl e processes and th e re~ ult s of the kin eti c
th eory of irreversibl e processes obtained from th e
Bolt zmann-kinetic equation for a dilute gas, a noneq uilibrium ensembl e method has been fo rmul ated
fo r dilute gase~; as a parall el ex tension to the Gibbs
ensembl e
meth od
in
equi librium
stati sti ca l
mechanics. This meth od is di stinct from those of
McLennan" 1<1 , Zubarev' and Sobouti 13 . The main
di stin gui shin g feature is. the use of an irreversibl e
kinetic equati on (i .e . th e Boltzmann-kineti c equation)
in stead of the time reversa l in va riant Liouvill e
equation .
Non-equi li brium processes In th e sys tem
inherently require th e sy stem to interact with its
surroundings which perturb th e former by heat, mass ,
and momentum tran sfers (stresses). Therefore, a
microcanonica ! ensembl e is not an appreciate
ensemble to use f or non-equi librium form the
co nce ptual standpoint. Th e eq uilibrium so luti on of
th e Boltzmann-kineti c eq uati on namely, th e Maxwell
distributi on function. is a ca noni ca l di stributi on
function. and it reflects the fact that the Boltzmannkinetic th eory is suitab le for a ca nonica l ensembl e.
Based upon aforementi oned reason. we postul ate th at
in a non-equilibrium system, th e Helmholtz free
energy, which is a characte ri stic function , has a form
similar to equilibrium one IEq. (23) I· T hi s can be
seen if we exa mine th e Boltzmann co lli sion term
w hi ch describes interaction of th e system, a single
particle in thi s case, with th e rest of th e world which
is represent ed by an other parti cle ullcorrelated to the
oth er subunit s co mpri sing the rest of th e world . Thi s
res t or th e wor ld ma y be viewed as th e surroundings
of th e system. and in thi s interpretati on, we have a
canon ica l ense mble interacting w ith surroundings.
This interpretation IS co nsi stent w ith Maxwel l
distributi on functi on for th e equilibriulll so luti on of
the Boltzmann-kinetic eq uati on. It is log ica l to look
for the non-eq uilibrium di stributi on functio n for th e
B o lt z lll~lnnjiln sys tem in terms of non-equilibrium
canonical form as has been done in th e present nonequilib riulll ense mbl e meth od.
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