Boltzmann Distribution from Invariance of Coarse-Graining-Scale and Energy-Shift
Abstract
We present a concise derivation of the Boltzmann form for single-particle energy distributions in classical many-body Hamiltonian systems. The derivation relies on two physical facts: coarse-graining-scale invariance of the empirical distribution and invariance under a uniform shift of the energy zero. These conditions uniquely yield the Boltzmann factor, whose parameter is fixed by the mean energy per particle. For separable Hamiltonians, the equilibrium weight factorizes into kinetic and configurational contributions sharing the same parameter, identified from the kinetic part as the inverse kinetic temperature. The principle extends to any physical quantity with a stationary distribution and translational invariance. It is illustrated in a one-dimensional diatomic hard-core gas and a nonlinear lattice chain, where it predicts velocity, energy, spacing, collision-time, and pressure-dependent displacement distributions in agreement with simulations. The lattice model further shows how harmonic elasticity, anharmonic corrections, internal pressure, and thermal expansion emerge from the same exponential equilibrium weights. Finally, the relationships among different ensembles are briefly discussed.
Versions
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BibTeX
@misc{fu2026boltzmann,
title = {{Boltzmann Distribution from Invariance of Coarse-Graining-Scale and Energy-Shift}},
author = {Weicheng Fu and Yisen Wang and Yong Zhang and Hong Zhao},
howpublished = {arXiv},
year = {2026},
month = may,
doi = {10.48550/arxiv.2605.25987},
eprint = {2605.25987},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.25987},
}
RIS
TY - UNPB
TI - Boltzmann Distribution from Invariance of Coarse-Graining-Scale and Energy-Shift
AU - Weicheng Fu
AU - Yisen Wang
AU - Yong Zhang
AU - Hong Zhao
PY - 2026
DA - 2026/05/25
PB - arXiv
DO - 10.48550/arxiv.2605.25987
UR - https://arxiv.org/abs/2605.25987
AB - We present a concise derivation of the Boltzmann form for single-particle energy distributions in classical many-body Hamiltonian systems. The derivation relies on two physical facts: coarse-graining-scale invariance of the empirical distribution and invariance under a uniform shift of the energy zero. These conditions uniquely yield the Boltzmann factor, whose parameter is fixed by the mean energy per particle. For separable Hamiltonians, the equilibrium weight factorizes into kinetic and configurational contributions sharing the same parameter, identified from the kinetic part as the inverse kinetic temperature. The principle extends to any physical quantity with a stationary distribution and translational invariance. It is illustrated in a one-dimensional diatomic hard-core gas and a nonlinear lattice chain, where it predicts velocity, energy, spacing, collision-time, and pressure-dependent displacement distributions in agreement with simulations. The lattice model further shows how harmonic elasticity, anharmonic corrections, internal pressure, and thermal expansion emerge from the same exponential equilibrium weights. Finally, the relationships among different ensembles are briefly discussed.
ER -