Discrete Boltzmann distributions via multisets and their coefficients

TL;DR

This paper explores the combinatorial foundations of Boltzmann distributions using multisets with fixed sums, providing a new mathematical framework that generalizes binomial coefficients and links to physical models.

Contribution

It introduces a novel multiset-based reconstruction of Boltzmann distributions, generalizing classical coefficients and connecting combinatorics with statistical physics.

Findings

General description of N-nomial coefficients from multisets

Reconstruction of Boltzmann distributions using combinatorial methods

Linking combinatorics to physical explanations in statistical mechanics

Abstract

This paper investigates the combinatorics that gives rise to the Boltzmann probability distribution. Despite being one of the most important distributions in physics and other fields of science, the mathematics of the underlying model of particles at different energy levels is underexplored. This paper gives a reconstruction, using multisets with fixed sums as mathematical representations. Counting (the coefficients of) such multisets gives a general description of binomial, trinomial, quadrinomial etc.\ coefficients, here called N-nomials. These coefficients give rise to multiple discrete Boltzmann distributions that are linked to explanations in the physics literature.