Riemannian thermo-statistics geometry

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.