[Paper Review] Thermodynamic Limit for Mean-Field Spin Models

Alessandra Bianchi, Pierluigi Contucci+1 more|TU/e Research Portal|Nov 11, 2003

Theoretical and Computational Physics8 references3 citations

TL;DR

This paper establishes the existence of the thermodynamic limit for a broad class of mean-field spin models—such as Curie-Weiss, p-spin, random field, and finite-pattern Hopfield models—by proving that if the Boltzmann-Gibbs state satisfies a subsystem energy dominance condition, the free energy density increases monotonically with system size, ensuring convergence via subadditivity and boundedness.

ABSTRACT

If the Boltzmann-Gibbs state $ω_N$ of a mean-field $N$-particle system with Hamiltonian $H_N$ verifies the condition $$ ω_N(H_N) \ge ω_N(H_{N_1}+H_{N_2}) $$ for every decomposition $N_1+N_2=N$, then its free energy density increases with $N$. We prove such a condition for a wide class of spin models which includes the Curie-Weiss model, its p-spin generalizations (for both even and odd p), its random field version and also the finite pattern Hopfield model. For all these cases the existence of the thermodynamic limit by subadditivity and boundedness follows.

Motivation & Objective

• To establish the existence of the thermodynamic limit for a wide class of mean-field spin models where standard techniques fail due to explicit size dependence.
• To provide a condition with direct thermodynamic meaning—namely, that the full system's energy is dominated by its subsystems—rather than relying on abstract deformation techniques.
• To extend the applicability of subadditivity arguments to both quenched and annealed models, including disordered systems like the random field Curie-Weiss and finite-pattern Hopfield models.
• To demonstrate that the thermodynamic limit exists without relying on exact solvability, even in cases where closed-form solutions are available.
• To generalize the result from polynomial interactions to continuous functions via the Stone-Weierstrass theorem, broadening the class of applicable models.

Proposed method

• Introduces an interpolating Hamiltonian $ H_N(t) = tH_N + (1-t)(H_{N_1} + H_{N_2}) $ for $ t \in [0,1] $, linking the full system to its subsystems.
• Uses the first derivative of the free energy density $ \alpha_N(t) $ to derive subadditivity: if $ \alpha_N'(t) \leq 0 $, then $ \alpha_N \leq \frac{N_1}{N}\alpha_{N_1} + \frac{N_2}{N}\alpha_{N_2} $.
• Proves the second derivative $ \alpha_N''(t) \geq 0 $ via Jensen’s inequality on the variance of $ H_N - H_{N_1} - H_{N_2} $, ensuring convexity of the interpolating functional.
• Establishes the key condition $ \omega_N(H_N) \geq \omega_N(H_{N_1} + H_{N_2}) $ as sufficient for monotonic increase of $ \alpha_N $, implying convergence.
• Applies the condition to models with Hamiltonians of the form $ H_N = -N g(m_N) $, where $ g $ is bounded and convex in the magnetization $ m_N $.
• Uses the Stone-Weierstrass theorem to extend results from polynomial $ g $ to continuous $ g $, ensuring broad applicability.

Experimental results

Research questions

• RQ1 Under what conditions does the free energy density converge in the thermodynamic limit for mean-field spin systems?
• RQ2 Can the thermodynamic limit be proven without relying on exact solvability or complex interpolation techniques?
• RQ3 Does the condition $ \omega_N(H_N) \geq \omega_N(H_{N_1} + H_{N_2}) $ hold for models like the Curie-Weiss, p-spin, and Hopfield models?
• RQ4 Can subadditivity be established pointwise in disorder (e.g., for quenched random fields) and then averaged to yield the thermodynamic limit?
• RQ5 To what extent can the class of models with a thermodynamic limit be extended beyond polynomial interactions?

Key findings

• The condition $ \omega_N(H_N) \geq \omega_N(H_{N_1} + H_{N_2}) $ for all $ N_1 + N_2 = N $ implies that the free energy density $ \alpha_N $ is non-decreasing in $ N $, ensuring convergence via subadditivity.
• For the Curie-Weiss model with $ p $-spin interactions, the condition holds because $ g(m) = m^p $ is convex for even $ p $, and the result extends to odd $ p $ via symmetry and continuity.
• The random field Curie-Weiss model satisfies the condition pointwise in the disorder realization $ h $, since the effective function $ g(m^+, m^-) $ is convex and bounded.
• The finite-pattern Hopfield model satisfies the condition because the magnetization $ m_N^\mu $ is linear in subsystems and $ g = \sum (m_N^\mu)^2 $ is convex in each $ m_N^\mu $.
• The thermodynamic limit $ \lim_{N \to \infty} \alpha_N = \inf_N \alpha_N $ exists for all models where $ g $ is continuous on $ [-1,1]^k $, as shown by the Stone-Weierstrass extension.
• The method applies to both ferromagnetic and antiferromagnetic interactions, and to mixed models, as long as the interaction function $ g $ is convex and bounded.