On the Cocycle Structure of the Boltzmann Distribution

TL;DR

This paper introduces a novel derivation of the Boltzmann distribution using cocycle structures, offering clearer insights into its dependence on energy levels and temperature without traditional Lagrange multipliers.

Contribution

It presents a new derivation method based on cocycle structures, enhancing understanding of the Boltzmann distribution's dependence on energy levels and temperature.

Findings

New derivation of Boltzmann distribution from cocycle structure

Provides transparent understanding of temperature dependence

Avoids traditional Lagrange multiplier method

Abstract

Based on a cocycle structure, we identify a new derivation of the Boltzmann distribution for finite energy-level systems from the maximal entropy principle (MEP). Our approach does not rely on the method of the Lagrange multiplier, and it provides a more transparent way to understand the dependence on the energy levels of the temperature $T = 1/\beta$ for the equilibrium distribution. Finally, we make two curious observations associated with our derivations.