On the origin of the Boltzmann distribution

TL;DR

This paper provides a new characterization of Boltzmann distributions in statistical mechanics, showing they are uniquely defined by an independence property for uncoupled systems, using algebraic structures.

Contribution

It introduces a novel mathematical characterization of Boltzmann distributions based on independence, connecting statistical mechanics with algebraic properties of probability measures.

Findings

Unique characterization of Boltzmann distributions

Connection between statistical mechanics and algebraic structures

Mathematical proof involving semi-group endomorphisms

Abstract

The family of Boltzmann distributions is used in statistical mechanics to describe the distribution of states in systems with a given temperature. We give a novel characterization of this family as the unique one satisfying independence for uncoupled systems. The theorem boils down to a statement about endomorphisms of the convolution semi-group of finitely supported probability measures on the natural numbers, or, alternatively, about endomorphisms of the multiplicative semi-group of polynomials with non-negative coefficients.