Boltzmann’s mean entropy formula for unbounded spin systems

Abstract

In this paper, we rigorously prove Boltzmann’s entropy formula [Formula: see text] log [Formula: see text] 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 [Formula: see text]. Notably, the lattice [Formula: see text] 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.