MPEJ Volume 9, No. 6, 15 pp.
Received: Nov 17, 2003. Revised: Jan 8, 2004. Accepted: Jan 23, 2004.
A. Bianchi, P. Contucci, C. Giardina
Thermodynamic Limit for Mean-Field Spin Models
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) >=
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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