On the equivalence of the microcanonical and the canonical ensembles: a geometrical approach

TL;DR

This paper introduces a geometrical approach to compare microcanonical and canonical ensembles, proposing an intermediate ensemble that bridges the two and deriving key distributions from this perspective.

Contribution

It presents a novel geometrical framework that unifies microcanonical and canonical ensembles, providing new insights into their equivalence in the thermodynamic limit.

Findings

The intermediate ensemble is shown to be equivalent to traditional ensembles in the thermodynamic limit.

Maxwellian and Boltzmann-Gibbs distributions are derived from the new formalism.

A new microcanonical derivation of the Boltzmann factor is provided.

Abstract

In this paper, we consider the volume enclosed by the microcanonical ensemble in phase space as a statistical ensemble. This can be interpreted as an intermediate image between the microcanonical and the canonical pictures. By maintaining the ergodic hypothesis over this ensemble, that is, the equiprobability of all its accessible states, the equivalence of this ensemble in the thermodynamic limit with the microcanonical and the canonical ensembles is suggested by means of geometrical arguments. The Maxwellian and the Boltzmann-Gibbs distributions are obtained from this formalism. In the appendix, the derivation of the Boltzmann factor from a new microcanonical image of the canonical ensemble is also given.