Combinatorial origins of the canonical ensemble

TL;DR

This paper provides an exact combinatorial analysis of the Darwin-Fowler method, revealing its non-asymptotic behavior and connections to Lah and Stirling numbers, which enhances understanding of statistical physics models.

Contribution

It introduces a combinatorial approach using Bell polynomials to solve the Darwin-Fowler method exactly, moving beyond asymptotic approximations.

Findings

Exact solutions for the Darwin-Fowler method using Bell polynomials.

Identification of relationships with Lah and Stirling numbers.

Insights into non-asymptotic behavior of statistical physics models.

Abstract

The Darwin-Fowler method in combination with the steepest descent approach is a common tool in the asymptotic description of many models arising from statistical physics. In this work, we focus rather on the non-asymptotic behavior of the Darwin-Fowler procedure. By using a combinatorial approach based on Bell polynomials, we solve it exactly. Due to that approach, we also show relationships of typical models with combinatorial Lah and Stirling numbers.