Entropy and curvature variations from effective potentials

TL;DR

This paper investigates the relationship between classical Boltzmann entropy and quantum or thermal entropy using the Jacobi metric, focusing on conditions where these entropies coincide near potential energy critical points.

Contribution

It introduces a method to compare classical and quantum/thermal entropies via effective potentials and analyzes their equivalence under perturbation theory.

Findings

Classical and quantum entropies are equal near non-degenerate critical points under certain conditions.

The approach uses the Jacobi metric to relate geometric properties to thermodynamic quantities.

First-order perturbation theory reveals specific criteria for entropy equivalence.

Abstract

By using the Jacobi metric of the configuration space, and assuming ergodicity, we calculate the Boltzmann entropy $S$ of a finite-dimensional system around a non-degenerate critical point of its potential energy $V$. We compare $S$ with the entropy of a quantum or thermal system with effective potential $V_{eff}$. We examine conditions, up to first order in perturbation theory, under which these entropies are equal.