A higher Boltzmann distribution

TL;DR

This paper generalizes the classical Boltzmann distribution to higher dimensions using combinatorial Hodge theory on CW complexes, providing explicit formulas and connecting to higher Kirchhoff and Matrix-Tree theorems.

Contribution

It introduces a higher-dimensional Boltzmann distribution on CW complexes, extending classical concepts via combinatorial Hodge theory and explicit formulas.

Findings

Defined higher Boltzmann distributions as (d-1)-cycles on CW complexes.

Provided explicit formulas for these higher distributions.

Connected the theory to higher Kirchhoff Network and Matrix-Tree theorems.

Abstract

We characterize the classical Boltzmann distribution as the unique solution to a certain combinatorial Hodge theory problem in homological degree zero on a finite graph. By substituting for the graph a CW complex and a degree d, we are able to define, by direct analogy, a higher dimensional Boltzmann distribution as a certain (d-1)-cycle on the real cellular chain complex which is characterized by appropriate constraints. We then give an explicit formula for this cycle. We explain how this circle of ideas relates to the authors' Higher Kirchhoff Network Theorem. We also give an improved version of the authors' Higher Matrix-Tree Theorems.