On the Cocycle Structure of the Boltzmann Distribution

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 $T=1\mathrm{/}\beta$ for the equilibrium distribution. Finally, we make two curious observations associated with our derivations.