Noncommutative spaces and covariant formulation of statistical mechanics

TL;DR

This paper develops a covariant formalism for statistical mechanics based on symplectic geometry, linking phase space topology and measure to thermodynamic properties, with applications to noncommutative Snyder space-time.

Contribution

It introduces a covariant symplectic approach to statistical mechanics, connecting topology and local structure of phase space to thermodynamics, applied to noncommutative Snyder space-time.

Findings

Topology relates to total microstates

Invariant measure depends on local symplectic form

Thermodynamics of harmonic oscillator analyzed

Abstract

We study the statistical mechanics of a general Hamiltonian system in the context of symplectic structure of the corresponding phase space. This covariant formalism reveals some interesting correspondences between properties of the phase space and the associated statistical physics. While topology, as a global property, turns out to be related to the total number of microstates, the invariant measure which assigns {\it a priori} probability distribution over the microstates, is determined by the local form of the symplectic structure. As an example of a model for which the phase space has a nontrivial topology, we apply our formulation on the Snyder noncommutative space-time with de Sitter four-momentum space and analyze the results. Finally, in the framework of such a setup, we examine our formalism by studying the thermodynamical properties of a harmonic oscillator system.