Thermodynamic Limit for Mean-Field Spin Models

0.13 MB·English

by  A. Bianchi

Thermodynamic Limit for
Mean-Field Spin Models
A. Bianchi, P. Contucci, C. Giardin`a
4 Dipartimento di Matematica
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0 Universita` di Bologna, 40127 Bologna, Italy
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n {bianchi,contucci,giardina}@dm.unibo.it
a
J

8

2
v
7 Abstract
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1 If the Boltzmann-Gibbs state ωN of a mean-field N-particle system
1
3 with Hamiltonian HN verifies the condition
0
/
h ω (H ) ≥ ω (H +H )
p N N N N1 N2
-
h
t for every decomposition N1 + N2 = N, then its free energy density
a
m increases with N. We prove such a condition for a wide class of spin
:
v
models which includes the Curie-Weiss model, its p-spin generaliza-
i
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tions (for both even and odd p), its random field version and also the
r
a
finite pattern Hopfield model. For all these cases the existence of the
thermodynamic limit by subadditivity and boundedness follows.
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1 Introduction
The rigorous theory of the thermodynamic limit which already in the six-
ties was a well established part of equilibrium statistical mechanics [Ru]
received recently a new impulse thanks to the treatment of the Sherrington-
Kirkpatrick [MPV] model of the mean field spin glass done by Guerra and
Toninelli [GuTo]. Moreover in a sequel work [Gu] it became clear that a good
control of the limit, especially when it is obtained by monotonicity through
subadditivity arguments, may lead to sharp bounds for the model and carries
important informations well beyond the existence of the limit itself. In this
paper we build a theory of thermodynamic limit which apply to a family
of cases including both random and non-random mean field models like the
Curie Weiss model [Ba], its p-spin generalizations, its random field version
[MP], and also the finite pattern Hopfield model [Ho]. Due to the explicit
size dependence of the local interactions we stress that mean field models do
not fall into the class for which standard techinques [Ru] can be applyed to
prove the existence of the thermodynamic limit. Moreover even in the Curie
Weiss model in which the exact solution is available it is interesting to obtain
the existence of the thermodynamic quantities without exploiting the exact
solution (see [EN, CGI]).
With respect to the theory relative to the random case [CDGG] we use
here a different interpolation technique which works pointwise with respect
to the disorder. The novelty of our approach relies on the fact that while
in the previous case the condition for the existence of the limit is given
in terms of a suitably deformed quenched measure, in the class of models
we treat here we are able to give a condition with a direct thermodynamic
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meaning: the Bolztmann-Gibbs state for a large system provides a good
approximation for the subsystems. The fact that our existence condition is
fully independent from the interpolation parameter relies on the convexity of
the interpolating functional, a property still under investigation for the spin
glass models [CG1, CG2].
2 Definitions and Results.
We consider a system of N sites: {1,2,...,N}, to each site we associate a
spin variable σ taking values in {±1}. A spin configuration is specified
i
by the sequence σ = {σ ,σ ,...,σ } and we denote the sets of all spin
1 2 N
configurations by Σ = {±1}N. We will study models defined by a mean
N
field Hamiltonian, i.e. for a given bounded function g : [−1,1] → R,
H (σ) = −Ng(m ) (1)
N N
where
N
1
m (σ) = σ . (2)
N i
N
i=1
X
Definition 1 For each N and a given inverse temperature β we introduce
the partition function
Z = e−βHN(σ), (3)
N
σX∈ΣN
the free energy density (and the auxiliary function α )
N
1
−βf = lnZ = α , (4)
N N N
N
and, for a generic observable O(σ), the Boltzmann-Gibbs state
O(σ)e−βHN(σ)
ω (O) = σ∈ΣN . (5)
N
Z
P N
3
Remark 1 Obviously from (3) and (4), one has that if g and g′ are two
functions from [−1,1] to R such that
kg −g′k := sup{|g(x)−g′(x)| : −1 ≤ x ≤ 1} (6)
is bounded, then one has
|α −α′ | ≤ βkg −g′k (7)
N N
for every N.
We can now state our main result:
THEOREM 1 Let H (σ) be a mean field Hamiltonian (see eq. (1), (2)).
N
If for every
partition of the set {1,2,...,N} into {1,2,...,N } and {N +1,...,N} with
1 1
N = N +N and
1 2
H = H (σ ,...,σ ) , H = H (σ ,...,σ ) , (8)
N1 N1 1 N1 N2 N2 N1+1 N
the condition
ω (H ) ≥ ω (H +H ) , (9)
N N N N1 N2
is verified, then the thermodynamic limit exists in the sense:
lim α = infα = α. (10)
N N
N→∞ N
3 Proof
Definition 2 Let us define the interpolating Hamiltonian as a function of
the parameter t ∈ [0,1]:
H (t) = tH +(1−t)[H +H ] , (11)
N N N1 N2
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and consider its relative partition function Z (t), free energy density f (t)
N N
and Boltzmann state ω .
N,t
Theinterpolationmethodthatwearegoingtouseisbasedonthesigncontrol
for boththefirst and second derivative ofα (t). More precisely thefollowing
N
holds:
Lemma 1 Let H be the mean field Hamiltonian and H (t) its relative
N N
interpolation.
If
d
α (t) ≤ 0 (12)
N
dt
for all t ∈ [0,1], then
N N
1 2
α ≤ α + α , (13)
N N N1 N N2
for each decomposition N = N +N .
1 2
Proof: trivially follows from the fundamental theorem of calculus and from
the observation that definition (11) implies:
Z (1) = Z , (14)
N N
α (1) = α , (15)
N N
Z (0) = Z Z , (16)
N N1 N2
and
N N
1 2
α (0) = α + α . (17)
N N N1 N N2
(cid:3)
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Lemma 2 Computing the t derivative of α (t), we get:
N
d 1 β e−βHN(t)
α′(t) = logZ (t) = − [H −H −H ]
dtN N N N N1 N2 Z (t)
N
σX∈ΣN
β
= − ω [H −H −H ]. (18)
N N,t N N1 N2
Lemma 3 The second derivative of α (t) is positive:
N
d2
α′′ (t) = α (t) ≥ 0, (19)
N dt2 N
Proof: a direct computation gives
d β
α′′ (t) = − ω [H −H −H ]
N dt N N,t N N1 N2
(cid:18) (cid:19)
β2
= ω (H −H −H )2 −ω2 [H −H −H ] . (20)
N N,t N N1 N2 N,t N N1 N2
(cid:16) (cid:17)
(cid:2) (cid:3)
From Jensen’s inequality applied to the convex function x 7→ x2, it follows
that α′′ (t) ≥ 0. (cid:3)
N
We are now able to prove the statement of Theorem 1.
Proof of THEOREM 1. From Lemma (2) we notice that the hypothesis
(9)
ω (H ) ≥ ω (H +H )
N N N N1 N2
is equivalent to the condition α′ (1) ≤ 0. On the other hand from Lemma
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(3) it follows that α′ (t) is an increasing function of t. This means that
N
the determination of the sign of α′ (t) can be in general established by the
N
evaluation of the sign in the extremes of the interval [0,1]. In particular we
have:
α′ (1) ≤ 0 =⇒ α′ (t) ≤ 0, ∀t ∈ [0,1] (21)
N N
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Using now Lemma (1), the subadditivity property (13) holds for α and
N
then, by standard arguments [Ru],
lim α = infα (22)
N N
N→∞ N
The existence of thermodynamic limit finally follows from boundedness of
the function g in Eq. (1). Indeed, calling K the maximum of g(x) on the
interval [−1,1], we have
1 1
α = ln eβNg(mN) ≥ lneβNK = βK . (23)
N
N N
σX∈ΣN
4 Applications
In this Section we identify a class of mean field models for which the hy-
potheses of our theorem are verified. Specifically these will be all models
such that the function g of formula (1) is convex or polynomial.
Corollary 1 Let the Hamiltonian be of the form
H (σ) = −Ng(m ) (24)
N N
with g : [−1,1] → R a bounded convex function. Then the thermodynamic
limit of the free energy exists.
Proof: For a given σ ∈ Σ and for every decomposition N = N + N we
N 1 2
define the quantities
1 N1 1 N
m (σ) = σ m (σ) = σ , (25)
N1 N i N2 N i
1 2
Xi=1 i=XN1+1
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so that the total magnetization is a convex linear combination of the two:
N N
1 2
m = m + m . (26)
N N N1 N N2
Using this definitions the hypothesis (9) of Theorem (1) is verified:
N N
1 2
ω (H −H −H ) = −Nω g(m )− g(m )− g(m ) ≥ 0
N N N1 N2 N N N N1 N N2
(cid:18) (cid:19)
(27)
where the last inequality follows from convexity of g. (cid:3)
Remark 2 The previous Corollary can be obviously generalized to the case
where the function g is a convex bounded function of many variables, each
of them fulfilling the property (25).
Corollary 2 Let the Hamiltonian be of the form
H (σ) = −Ng(m ), (28)
N N
with g : [−1,1] → R a polynomial function of degree n ∈ N. Then the
thermodynamic limit exists.
Proof: First of all we consider the case g(x) = xk (the generalization will
follow in a simple way) with associated Hamiltonian
N
1
H (σ) = −Nmk = − σ σ ···σ . (29)
N N Nk−1 i1 i2 ik
i1,i2X,...,ik=1
By splitting the summation into two pieces, the first containing the summa-
tion with indexes all different among themselves, the second containing the
remaining terms, we have
∗
1
H (σ) = − σ ···σ + σ ···σ (30)
N Nk−1 i1 ik i1 ik
" #
i16=X...6=ik i1X,...,ik
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∗
where the second summation includes all terms with at least two equal
indexes. A simple computationPshows that
∗
1
σ ···σ = O(1) (31)
Nk−1 i1 ik
i1X,...,ik
Defining now the model with Hamiltonian
N
1
H˜ (σ) = − σ σ ···σ , (32)
N (N −1)(N −2)···(N −k +1) i1 i2 ik
i16=i26=X...6=ik=1
it follows that
(N −1)(N −2)···(N −k +1)
H (σ) = H˜ (σ)+O(1)
N Nk−1 N
= H˜ (σ)+O(1) (33)
N
Using Remark 1 one has that the two models H and H˜ have the same
N N
˜
thermodynamic limit (if any). On the other hand for the model H we have
N
1
ω (H˜ ) = − ω (σ σ ···σ )
N N (N −1)(N −2)···(N −k +1) N i1 i2 ik
i16=iX26=...6=ik
= −Nω (σ σ ···σ ) (34)
N 1 2 k
wherethelastequalityfollowsfrompermutationinvariance(ω (σ σ ···σ )
N i1 i2 ik
does not depend on the choice of the indexes). Analogously one can repeat
the same computation for H˜ and H˜ in the state ω . By permutation
N1 N2 N
symmetry, this yields that Hypothesis (9) of Theorem (1) is verified as an
equality
˜ ˜ ˜
ω H −H −H = 0, (35)
N N N1 N2
(cid:16) (cid:17)
˜
implying the existence of thermodynamic limit for the model H , and so for
N
the model H .
N
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Since we have proved the Corollary for g(x) = xk, the case of a generic
polynomial function of degree n
n
g(x) = a xk (36)
k
k=0
X
(cid:3)
is treated by the same argument, applied to each monomial of the sum.
Corollary 3 Using Remark 1 and the Stone-Weierstrass theorem, the ther-
modynamic limit of α exists if, instead of g being polynomial, g is merely
N
continuous up to the boundary of [−1,1].
4.1 Examples
1. The Curie-Weiss models.
For every integer p, with p < N, consider the model defined by
N
1
H (σ) := − σ σ ···σ (37)
N Np−1 i1 i2 ip
i1,i2X,...,ip=1
which represents the generalized Curie Weiss model with p-spin inter-
action. The standard Curie-Weiss model corresponds to the case p = 2.
From eq. (2) the previous Hamiltonian can be written as
H = −Nmp (38)
N N
and the existence of the thermodynamic limit is then implied by Corol-
lary (2). Moreover, the same result holds for any linear combination
(see Eq.(36)) of generalized Curie Weiss models with p-spin interaction,
both ferromagnetic and antiferromagnetic.
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