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 $d{s}^2=g_{ij}(I|\theta )d{I}^{i}d{I}^j$ 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.

arXiv commentNo figures, 19 pages, iopart style (latex)