The Canonical Distribution without Thermodynamic Limit

TL;DR

This paper derives the canonical distribution without relying on the thermodynamic limit or the equal a priori probability postulate, using only extensivity and composition rules, with numerical demonstrations on molecules.

Contribution

It provides a novel derivation of the canonical distribution independent of the thermodynamic limit and traditional postulates.

Findings

Derived the canonical distribution without thermodynamic limit

Numerical results for free and oscillating molecules

Implications for statistical mechanics foundations

Abstract

We derive the continuous canonical distribution only by requiring the extensivity of the mean energy and the multiplicative probabilistic composition rule. The derivation is independent of the thermodynamic limit and moreover it does not use the usual equal a priori probability postulate. We numerically demonstrate the implications of our derivation for the free and oscillating molecules.