Riemannian thermo-statistics geometry

TL;DR

This paper introduces a Riemannian geometric framework for classical statistical mechanics, interpreting thermodynamic quantities as geometric notions and using curvature to measure statistical dependence among macroscopic observables.

Contribution

It extends Ruppeiner geometry by formulating a Riemannian structure on the manifold of macroscopic observables, linking curvature to irreducible statistical dependence.

Findings

Entropy and thermodynamic quantities are geometric invariants.

Statistical curvature scalar characterizes irreducible dependence.

Analogy established with Einstein's General Relativity.

Abstract

It is developed a Riemannian reformulation of classical statistical mechanics for systems in thermodynamic equilibrium, which arises as a natural extension of Ruppeiner geometry of thermodynamics. The present proposal leads to interpret entropy $\mathcal{S}_{g}(I|\theta)$ and all its associated thermo-statistical quantities as purely geometric notions derived from the Riemannian structure on the manifold of macroscopic observables $\mathcal{M}_{\theta}$ (existence of a distance $ds^{2}=g_{ij}(I|\theta)dI^{i}dI^{j}$ between macroscopic configurations $I$ and $I+dI$). Moreover, the concept of statistical curvature scalar $R(I|\theta)$ arises as an invariant measure to characterize the existence of an \textit{irreducible statistical dependence} among the macroscopic observables $I$ for a given value of control parameters $\theta$. This feature evidences a certain analogy with Einstein General Relativity, where the spacetime curvature $R(\mathbf{r},t)$ distinguishes the geometric nature of gravitation and the reducible character inertial forces with an appropriate selection of the reference frame.