If the Boltzmann-Gibbs state omega_N of a mean-field N-particlesystem with Hamiltonian H_N verifies the condition omega_N(H_N) >=omega_N(H_{N_1}+H_{N_2}), for every decomposition N_1+N_2=N, then its freeenergy density increases with N. We prove such a condition for a wide class ofspin 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 patternHopfield model. For all these cases the existence of the thermodynamic limit bysubadditivity and boundedness follows.
Thermodynamic Limit for Mean-Field Spin Models / A., Bianchi; P., Contucci; Giardina', Cristian. - In: MPEJ. - ISSN 1086-6655. - ELETTRONICO. - 9:(2003), pp. 6-1-6-15.
Thermodynamic Limit for Mean-Field Spin Models
GIARDINA', Cristian
2003
Abstract
If the Boltzmann-Gibbs state omega_N of a mean-field N-particlesystem with Hamiltonian H_N verifies the condition omega_N(H_N) >=omega_N(H_{N_1}+H_{N_2}), for every decomposition N_1+N_2=N, then its freeenergy density increases with N. We prove such a condition for a wide class ofspin 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 patternHopfield model. For all these cases the existence of the thermodynamic limit bysubadditivity and boundedness follows.
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