In this work a
geometrical representation of equilibrium and near equilibrium statistical
mechanics is proposed. Using a formalism consistent with the Bra-Ket notation
and the definition of inner product as a Lebasque integral, we describe the
macroscopic equilibrium states in classical statistical mechanics by “properly
transformed probability Euclidian vectors” that point on a manifold of
spherical symmetry. Furthermore, any macroscopic thermodynamic state “close” to
equilibrium is described by a triplet that represent the “infinitesimal volume”
of the points, the Euclidian probability vector at equilibrium that points on a
hypersphere of equilibrium thermodynamic state and a Euclidian vector a vector
on the tangent bundle of the hypersphere. The necessary and sufficient
condition for such representation is expressed as an invertibility condition on
the proposed transformation. Finally, the relation of the proposed geometric
representation, to similar approaches introduced under the context of differential
geometry, information geometry, and finally the Ruppeiner and the Weinhold
geometries, is discussed. It turns out that in the case of thermodynamic
equilibrium, the proposed representation can be considered as a Gauss map of a
parametric representation of statistical mechanics.
Reconsidering the Concept of Equilibrium in Classical Statistical Mechanics
