M ARC P IRLOT
Variational principle, conjugate convex functions and
“the equivalence of ensembles”
Annales scientifiques de l’Université de Clermont-Ferrand 2, tome 87, sérieProbabilités et applications, no4 (1985), p. 57-68
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Marc PIRLOT
Sumnary
In classical
statisti6al
mechanics on alattice,
we use thetheory
ofconjugate
convex functions to prove a(conjugate)
variationalprinciple
for the
entropy
and togive
aquite general explanation
of the"equivalence
of ensembles".1. Introduction and notations
In this paper we show how the abstract
theory
ofconju- gate
convex functions(generalizing
theLegendre
transforma-tion)
can be used to formulate andanalyse completely
theproblem
of"equi val ence
of ensembles" in Statistical Mecha- nics on a lattice.We show that the results obtained
by
Lanford[ 6]
are not subordinated to thepossibility
ofconstructing
athermodynamic potential (e.g.
Helmholtz’free
energy)
as a.thermodynamic
limit of localquantities
but thatthey
are of a verygeneral
nature. Ourapproach
is close to thephysicists’one.
Moreover, the results established here for
compact spins
onZ"
extendwithout
difficulty
to any model for which a vari ati onalprinciple
is avai-1 able.
The notion of
conjugate
convex functions wasalready
used or mentionedin
[ 7] , [ 9]
andby Yightman
in thepreface
of[ 5]
but results like our theo-rems 2 and 3 were not
derived
there.In § 2,
we remark that the usual vari ati onalprinciple
ex-are
conjugate
convex functions. Fromthat,
follows immedia-tely
the"conjugate"
variationalprinciple
for theentropy (Ruelle
and Israel gave much morecomplicated proofs :
see[5]).
The third
paragraph provides
akey. for understanding
theequivalence
of the ensembles. It is proven under a mild con-dition,
that the statesgiving
a fixed mean energy to some po- tentials and the value of the free energy(or
moregeneral
thermodynamic potentials) corresponding
to that mean energy,are
equilibrium
states(’i.e. tangent
tothe pressure).
Mathe-matically,
one shows that a"parti al " conjugate
of the pres-sure on a
subspace
of the interactions space is related to theentropy
in theexpected
way.Finally,
weonly
considerpartial conjugates
of the pres-sure on a finite dimensional
subspace
on which the pressure isstrictly
convex. Weinvestigate
the mild condition men-tioned above, draw some
physically
relevant consequences about the.shape
of the.thermodynamic potentials, give
aprecise
for-mulation of the
equivalence
of the ensembles and mention aversion of Gibbs Phase rule.
We consider the
general
modelof compact spins
onZ".
Let 00 be a
compact
metric the boreliana-algebra
o 0
on Qo and Ào
aprobability
on(Q0, F0).
The space of
configurations
= with theproduct topol ogy, i s compact
andmetrisable ;
thea-algebra (Ç
is theproduct
of thea-algebras (F
on each copy .For A a finite
part
ofZv, s2(A) : _ (S~ ) ,
is the pro-duct
a-algebra
onQ(A)
andÀA
theproduct
meas.ure on(n(A),
We denoteby
the restriction of aconfigu-
ration w G Q to A.
An interaction
potential
is afamily
of continuousfunctions
w~ : Q(X) -
R, indexedby
the finiteparts
ofZ".
§£is
the Banachspace of
transl ati on. invariantpotentials
with the norm
Another space of interest is
I,
thespace
of invariantpotentials
normedby
I
One defines the usual
thermodynamic
functions asfollows,
the limits
being
takenalong
sequences of finite boxes A ten-ding
toinfinity
in the sense of Van Hove(see [5]
forproofs).
For
11 E ë,I,
the set oftranslation
invariantprobabilities
on(Q, 6’),
one definesentropy s(p) by :
s(p) =1 if
thisexpression
makes sense- oo othe rwi se
Entropy
is an affine u.s.c.functi on ; - ~ s(p)
0.If p
and p E B, the mean energye(p,
the af-fine continuous function on (b defined
by :
The pressure P is a
continuous (Lipschitzian)
convexfunction defined for any w
in 8 by :
THEOREM 1.
(Variational principle, [ 5] ~.
ForThe supremum
--all equilibrium
states2.
Con j ugate
variationalprinciple
Let 6 be
a Banach space(in
fact alocally
convex lineartopological
space isenough)
and6£*
its dual. Let6E*
be en-dowed the
weak-topology
andd’with
the weaktopology.
. Wi ththose
topologies, B and B*
are weak duals of each other(see
e.g.
theor. 1,
p.112 ~ .
There i s a one to one corres-pondence
between the setr(~)
ofweakly
closed~.1. s. c. ~ ,
pro-per
(not i denti cal ly equal
to +~),
convex functions on I andof
weakly closed~ proper,
convex functions on0.
Let fbelong
one defines a functionf~
on0152> ~
by :
f*
is called theconjugate
of f. As is well-known([3], [8]),
f~’ belongs
toT($*)
andReci procal ly, starting
with g we defineand due to t,he weak
duality between B and B*,
it ispossible
to prove, in the same way as for
f * , that g~ belongs
toand .
Let us define the
weak *
lower semi continuous, proper,convex function t
(a) on 6*,
asif there is othe rwi se
The . def i ni ti on is
unambiguous
as twodifferent p
cannotyield
the same a.
By
the direct variationalprinciple (theor. 1) :
:As P is the
conjugate
of t, P is inF-(8)
andOne sees that t
(a)
is finite iff a isP-bounded,
i.e. ifthere exists a constant c in R such that for any ~
in 6 :
c
The
following
threeproperties
are all immediate conse- quences of whatprecedes. Corollary
2.1 is theorem I I .3.4i n [ 51
andcorollary
2.2implies
theorem 11.1.2 in[ 5] .
THEOREM 2 : The
conjugate
of P i s the functi on t : :othe rwi se
For any
6 theconjugate + . is
ta - a(e).
COROLLARY
2.1 (Conjugate variat-ional principle)
For everyp
in ~~ :
’ ’
COROLLARY
2.2 q eB*
is P-bounded iff there exists u inCOROLLARY
2.2 cc is P-bounded iff there_in EI
such that = -
e(p, «.0)
for ands(p)
is finiteiff
t (a)
is finite.3. Partial
conjugates
of thepressure
Let
b’
be a linearsubspace of 19
For instance(Z’
isfinite dimensional and
generated by
theindependent components
=
(¡PI’
...,&~), ~i
in 8.physically
relevant situations in the modelwhere 0 0
={0,1}
are the :a)
canonicaldescription :
n =1, ¡PI
= i ifIxi
= 1and wX = 1
~0
otherwisebLmicrocanonical description :
n =2, ~.
is the same asin the canonical
description and ¡P2
describes the interactions between theparticles.
The
partial conjugate
of P on 6 + with ein 01,
isthe convex, 1 s.c. -function
F(e, a’), a’ ’G defined by :
The supremum
defining
F is notalways
attained(see
theor. 4below, for the case dim
6’
isfinite).
THEOREM
3.a’)
inf[ t a - a 8 ;
I=all
and the i nfimum i s reached for
ever a’
I i n’
Proof . It i s clear
( by
theor.1 )
thatF(0, a’ )
t(a) - a(0)
for any a
agreeing
with a’ on~I. Suppose
first that a’is
P(o
+.)-bounded
on6’ ,
i.e. there exists c E R such thatP(6 + p) > a’(p) -
c for every p in B’ . Thisimpl ies F(0, a’ )
is finite for any in
t’ :
The second member i s convex, conti nuous
in f
so thatby
Hahn-Banach’s theorem, there is a
E B* agreeing
with a’on ’
andsatisfying
the sameinequality
as a’ but for any win 6.
So,a is P-bounded with constant
F( e , a’)
+a(e)
and :This proves
the
theorem when a’ ispee
+.)-bounded
on 6)’.If it is not,
F(e, a’)
is infinite and noa E-= 0 agreeing with
a’ I on
6’
is P-bounded so that t(a)
is also infinite for thosea
(theor.
2 and cor.2.2).
COROLLARY 3 .1 I f the re i s cp E
6~’
s uch th atF(8, a’
- P 8
+~p ,
then there exists atleast one probability sati sfyi ng -
=a’ s~ for all w
ing’
and such thatAny
suchprobability p
is anequilibrium
state for theoten-
tial e + ~p.From the
physical point
ofview,
it is a nuisance that the supremumdefining F(e, a’)
is ingeneral
attained forseveral
potentials
p of ~’ . It is due to the fact that P is notstrictly
convexon (P.) (nor on 65’).
From the results stated in
[10],
one infers that P wi ll bestrictly
convex on a linearsubspace D of B (or
on e+2,
0 G I) if i
contains at most onepotential
of each class ofphysically equivalent potentials (for
adefinition,
see[ 10] ~,
For instance in the case where
n
=~0,1} ,
P isstrictly
con-vex on the space of lattice gas
potentials ([ 4] ~ .
4.
Consequences
a)
In thesequel ,
we consider asubspace 2
on which P isstrictly
convex and which isgenerated by
a(~1’
1... 9 *n)
of nindependent potentials of 9.
A linear form an
, _ n _ _given by
avector u of !R"
such thatu
= -a(y). By
theorem 3, we have :
- - n
E EI eI
o0Denote
by
the sety)
eRn; 11 E
This is
precisely
the set ofpoints u
ofRn
on whichF ( e, u)
is finite
(by
cor.2.2)
for all ein ~.
As P isstrictly
convex on 0
+I , by [ 8] ,
theor.26.3, F(6, . )
isessentially
smooth, that is :1)
intK(*)
is notempty ;
2) F( 6, . )
is differentiablethroughout
intK(~)
3)
the norm ofgrad F(e, u)
tends toinfinity
asu
tendsto the
boundary
ofThe
gradient
ofF(e, "U-)
is the vectorv
ofRn
such thatr /
THEOREM 4. The supremum
defining F e, u
isexactly
attainedon the interior of the convex bounded set
K(y).
Proof. In view of the remarks made
before,
itonly
remains toprove that if
u belongs
to theboundary
ofK{1P),
the supremummay not be attained.
Suppose
on thecontrary
that the supre-mum is reached for some
v E Rn.
Thereexists y E Rn
such thatY. u
= mi n[y.z ; K(&)] ;
defines asubgradient
of thefunction
P(k):
=P(o + v.~,
+ kF.§),k 9 R,
in k = 0. AsP(k)
is
strictly
convex and finite for all k in ~, its left andright
derivativesP’ (k)
andP+ (k)
areincreasing
functions.So,
there is apoint
k where -Fl§ P’ (k)
Po(k).
But
P~ (k) = -
for a certainz
inK(V) ,
becauseequilibrium
states are invariant
probabilities
with finiteentropy.
Thisis in contradiction with the
hypothesis
ony.
COROLLARY 4.1.
If p
is s a translation invariant Gibbs state, then i s an i nterioroint
ofProof. Let p be an invariant Gibbs state associated to
then
defines asubgradient
to P in 6 +~.$
for e = ~p -~.$
and the supremum
defining F(6, e(p, ~))
is attained.Remark. In the case where P 0 is a finite set and À the nor-
2013201320132013 o
_
- 0
I
I .
malized
counting
measure, the setK(y)
iscompact compact
ands(p)
is a bounded function.b)
According
to theorem 3, a function likeF(0, u)
may beinterpreted
as athermodynamic potential (see [ 1] ).
For sim-licity
suppose n = 1 sothat v
=(y ) .
If there is aphase
transition in 0 and an
order parameter
for that tran-sition,
then thegraph
ofF(e, ul) presents
a flat(horizon-
tal) part.
To compare with thephysical
usual formalism, see for instance[ 2] :
the extreme valuesof, u
on the flatpart
of
F( 6, ul)
coincide with the two minima of the function usedby
thephysicists.
. In fact, this
provides
a betterdescription
than theusual one, as can be seen with the
v i-u1 diagrams (e.g. p-V diagrams, ...) relating
twoconjugate parameters (intensive
vi,
extensiveul) :
Maxwell’sequal
area rule has not to beinvoked as the flat
part
isnaturally
obtained. Thediagram
is therepresentation
d Fin’(uI’
1v ) -axi s
i of the set ofcouples (Ui , -
1 which are thecouples minimizing
1
F(e, ul) + v. u1
or,by conjugation, P(o + v- *i) + v ul-
c) Equivalence of ensembles
A
possible interpretation
of the "ensembles" and their"equi valence"
is thefollowing.
Consider that the fundamen- talproblem
of statistical mechanics is to determine the cou-p les (.p, ll), .p E ~ , C-
associated to agiven
situation.An extreme case is the
grand canonical description
where p iscompletely
known(up
totemperature,
chemicalpotential,
... :all intensive relevant
parameters
areknown) and p totally
un-known ;
the setj(w)
of associatedprobabilities
is the set ofGibbs states for w
(here
weonly
consider the set of invariant GibbsStates).
Agene ral i zati on
of the canonical ormi crocanoni cal
setting
is the case where oneonly
knows that theright potential
is a linear combination of somepotentials
(generating D)
with unknown coefficients(intensive parameters) ;
but on the otherhand,
one knowsthe
values ofthe extensive
parameters conjugated
to the unknown intensiveparameters,
i.e. one knows the meanIF of
the observables asso-ciated
to 1jJ
=(tP1" ... , ~, n ) ;
the+2 , u)
of. (inva-
riant)
states associated to theprobl em
iscomposed
of theprobabilities p E satisfying e(p, ~)
=u
ande(p, 6) -s(p)
= inf
[e(v, 0) - s(v) ; V
E2~, e(v,
=ul
The other ex-treme case
("absolute microcanomcaT")
would be acomplete knowledge
of p and a totalignorance
about p; it is what lies under theconjugate
variationalprinciple (cor. 2.1).
torollary
3.1 and theorem 4 allow toanalyse
the connec-tions
between G0(0
+v.w)
1.
If u
is an interiorpoint
ofK(T),
we have ingenral
0 +tD , ’u)
g10(
6+ forv conjugated
toG;
Both des-criptions
areequivalent
iff +~~))
=1 or, in
the pre-sence of
phase transition,
if allprobabilities
of4,(6+
give
the same value toe(., ~),
i.e. if there is no order pa- rameter for the. transition.2. If
5
is not in int no"grand
canonical"descrip-
tion can be
given butt ( 0+.8,-u)
is non-void whenu belongs
to the
boundary
ofd)
One can prove without
change
but in our much more gene- ralsetting,
the main esultof 71 ,
theorem 1 :it says
es-sentially
that iffi
= vectn }
contains a represen-tative set of order
parameters
for aphase
transition in 6(i.e.
two invariantequilibrium states
for 0 that don’t differare
equal),
then the number of purephases (ergodic equi-
librium
states)
for 6 is less than orequal
to n+1.REFERENCES
1.
BOCCARA, N.,
Lesprincipes de
lathermodynamique classi-
que,
Paris, P.U.F.,
Coll.Sup.
1968.2.
BOCCARA, N., Symétries brisées, Paris, Hermann,
1976.3.
EKELAND, I., TEMAN,
R. ,Analyse convexe
etproblèmes
va-riationnels, Paris, Dunod-Gauthier-Villars, 1974.
4.
GRIFFITHS, R.B., RUELLE, D.,
Strictconvexity ("continuity"
of the pressure in lattice
systems.
Com. math.phys. 23,
169-175(1971).
5.
ISRAEL, R.B., Convexity
in thetheory of
lattice gases, Princeton Series inPhysics,
Princeton Univ.Press,
1979.6. LANFORD
III, O.E., Entropy
andequilibrium
states in classical
statisticalmechanics,
in Statistical mechanics andmathematical problems,
Ed. A.Lenard,
Lect.notes in phys.
20, Springer 1972.
7.
MARTIN-LOF, A.,
Theequivalence
of ensembles and Gibbsphase
rule for classical latticesystems,
J. Stat.Phys.
20,
557-571(1979).
8.
ROCKAFELLAR, R.T.,
Convexanalysis,
Princeton Univ. Press 1970.9.
ROOS,
H.,Consequences
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a variationalprin- ciple
for the pressure and relatedresults, Reports
math.phys. 14, 147-160 (1978).
10.
RUELLE, D., Thermodynamic formalism, Addison-Wesley,
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YOSIDA,
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M. PIRLOT
Université de Mons Place Warocqug, 17, B
7000 MONS
Belgique
Requ en Mars 1985