On the geometrical representation of classical statistical mechanics

Abstract

In this work, a geometrical representation of equilibrium and near equilibrium classical statistical mechanics is proposed. Within this formalism the equilibrium thermodynamic states are mapped on Euclidian vectors on a manifold of spherical symmetry. This manifold of equilibrium states can be considered as a Gauss map of the parametric representation of Gibbs classical statistical mechanics at equilibrium. Most importantly, within the proposed representation, out of equilibrium thermodynamic states, can be described by a triplet consisting of an ‘infinitesimal volume’ of the points on our manifold, a Euclidian vector that points on the equilibrium manifold and a Euclidian vector on the tangent space of the equilibrium manifold. Finally in this work we discuss the relation of the proposed representation to the pioneer work of Ruppeiner and Weinhold at the limit of equilibrium, along with the notion of K–L divergence and its relation to the second law of thermodynamics.