On the Cocycle Structure of the Boltzmann Distribution
Chuan-Tsung Chan, Chan-Yi Chang, Zhong-Tang Wu
TL;DR
The paper tackles deriving the canonical Boltzmann distribution from the maximal entropy principle without resorting to Lagrange multipliers. It introduces a geometric, vector-space formulation that decomposes the probability vector into a low-dimensional subspace spanned by $I$ and the energy vector, plus an orthogonal complement, and reduces the MEP to a finite-dimensional optimization. The critical point is shown to yield the Boltzmann form $p_k = (1/Z) e^{-eta e_k}$ with $Z = \sum_k e^{-eta e_k}$ and parameters $\alpha=\ln Z$, $\beta$ determined by energy levels through a cocycle condition $p_j^{e_{lk}} p_k^{e_{jl}} p_l^{e_{kj}}=1$. This cocycle structure extends to general $N$-level systems and is underpinned by a Hessian-based maximality argument; the paper notes potential connections to number theory and discusses extensions to grand-canonical ensembles, including questions about quantum statistics.
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 $T = 1/β$ for the equilibrium distribution. Finally, we make two curious observations associated with our derivations.