Covariant Equilibrium Statistical Mechanics

TL;DR

This paper develops a covariant equilibrium statistical mechanics framework using Poincare-invariant constrained Hamiltonian dynamics, applicable to classical gases, and derives thermodynamic quantities with a focus on the classical Boltzmann gas.

Contribution

It introduces a covariant approach to equilibrium statistical mechanics based on an extended phase space and invariant time constraints, applicable to classical gases.

Findings

Derived the canonical partition function for a classical monatomic gas.

Formulated the Liouville equation and equilibrium conditions in a covariant manner.

Compared the new approach with existing methods for the perfect gas.

Abstract

A manifest covariant equilibrium statistical mechanics is constructed starting with a 8N dimensional extended phase space which is reduced to the 6N physical degrees of freedom using the Poincare-invariant constrained Hamiltonian dynamics describing the micro-dynamics of the system. The reduction of the extended phase space is initiated forcing the particles on energy shell and fixing their individual time coordinates with help of invariant time constraints. The Liouville equation and the equilibrium condition are formulated in respect to the scalar global evolution parameter which is introduced by the time fixation conditions. The applicability of the developed approach is shown for both, the perfect gas as well as the real gas. As a simple application the canonical partition integral of the monatomic perfect gas is calculated and compared with other approaches. Furthermore, thermodynamical quantities are derived. All considerations are shrinked on the classical Boltzmann gas composed of massive particles and hence quantum effects are discarded.