Research PaperNo Access

Boltzmann’s mean entropy formula for unbounded spin systems

Department of Mathematics, Hokkaido University, Kita-ku, Kita-10jou Nishi-8 choume, Sapporo, Japan

E-mail Address: kajisa.takahiro.n7@elms.hokudai.ac.jp

https://doi.org/10.1142/S0129055X24500211Cited by:0 (Source: Crossref)

Abstract

In this paper, we rigorously prove Boltzmann’s entropy formula $S = k$ $S = k$ log $W$ $W$ in classical unbounded spin systems. By restricting our consideration to negatively interacting super-stable potentials, we prove the existence of pressure and the concavity of micro-canonical entropy corresponding to log $W$ $W$ . Notably, the lattice $φ_{ν}^{4}$ $φ_{ν}^{4}$ model becomes a feasible example through the exploitation of the physical equivalence of potentials. Our proof is mostly based on a cut-off technique for configuration spaces that enables us to lift-up the results for bounded systems. This approach is independent of the existence of Gibbs states and, moreover, holds a clear mathematical conciseness.

Keywords:

AMSC: 22E46, 53C35, 57S20

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