Geometry of the basic statistical physics mapping

TL;DR

This paper explores the geometric structure of statistical physics models, analyzing hypersurfaces' curvature and entropy, including ideal and non-ideal cases, phase transition singularities, and tropical limits.

Contribution

It introduces a geometric framework for understanding statistical physics models, including curvature and entropy calculations, and examines singularities and tropical limits.

Findings

Derived metric, curvature, and entropy for statistical hypersurfaces

Analyzed singularities related to phase transitions

Discussed tropical and double scaling limits

Abstract

Geometry of hypersurfaces defined by the relation which generalizes classical formula for free energy in terms of microstates is studied. Induced metric, Riemann curvature tensor, Gauss-Kronecker curvature and associated entropy are calculated. Special class of ideal statistical hypersurfaces is analyzed in details. Non-ideal hypersurfaces and their singularities similar to those of the phase transitions are considered. Tropical limit of statistical hypersurfaces and double scaling tropical limit are discussed too.