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Thermodynamic Limit for Mean-Field Spin Models

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A. Bianchi, P. Contucci, C. Giardina'

Mathematical Physics · Submitted 11 Nov 2003

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim For all these cases the existence of the thermodynamic limit by subadditivity and boundedness follows.

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arXiv IDmath-ph/0311017 (version 2)
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CitationA. Bianchi, C. Giardina', P. Contucci. "Thermodynamic Limit for Mean-Field Spin Models." arXiv:math-ph/0311017 (2003). https://pith.science/paper/BJH7NRAC
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,128 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.15 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.13 · compares with prior results 0.08 · mentions code or data availability 0.01.

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arXiv listing abstract v2. Results and the main claim are underlined; .

If the Boltzmann-Gibbs state $\omega_N$ of a mean-field $N$-particle system with Hamiltonian $H_N$ verifies the condition $$ \omega_N(H_N) \ge \omega_N(H_{N_1}+H_{N_2}) $$ 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.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 If the Boltzmann-Gibbs state $\omega_N$ of a mean-field $N$-particle system with Hamiltonian $H_N$ verifies the condition $$ \omega_N(H_N) \ge \omega_N(H_{N_1}+H_{N_2}) $$ for every decomposition $N_1+N_2=N$, then its free energy density increases with $N$. result 0.88 0.39 0.31 asserted 1.00
s1 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. contribution 0.81 0.94 0.17 asserted 1.00
s2 For all these cases the existence of the thermodynamic limit by subadditivity and boundedness follows. result 0.65 0.79 0.39 asserted 1.00

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Original abstract

If the Boltzmann-Gibbs state $\omega_N$ of a mean-field $N$-particle system with Hamiltonian $H_N$ verifies the condition $$ \omega_N(H_N) \ge \omega_N(H_{N_1}+H_{N_2}) $$ 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.

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