Boltzmann-Gibbs thermal equilibrium distribution for classical systems and Newton law: A computational discussion

TL;DR

This paper uses numerical methods to compare classical Hamiltonian dynamics with the Boltzmann-Gibbs distribution, demonstrating their connection at intermediate energies and discussing discrepancies at higher energies.

Contribution

It provides a computational framework to directly compare Newtonian dynamics with statistical distributions in classical systems.

Findings

Boltzmann-Gibbs distribution aligns with Newtonian dynamics at intermediate energies

Partial agreement observed between time and ensemble averages at higher energies

Numerical approach applicable to paradigmatic first-neighbor models

Abstract

We implement a general numerical calculation that allows for a direct comparison between nonlinear Hamiltonian dynamics and the Boltzmann-Gibbs canonical distribution in Gibbs $\Gamma$-space. Using paradigmatic first-neighbor models, namely, the inertial XY ferromagnet and the Fermi-Pasta-Ulam $\beta$-model, we show that at intermediate energies the Boltzmann-Gibbs equilibrium distribution is a consequence of Newton second law (${\mathbf F}=m{\mathbf a}$). At higher energies we discuss partial agreement between time and ensemble averages.