Examination of Boltzmann's H-Function: Dimensionality and Interaction Sensitivity Dependence, and a comment on his H-Theorem

TL;DR

This study investigates Boltzmann's H-function through molecular dynamics simulations and analytic theory, revealing its sensitivity to interaction potential and dimensionality, and providing a new analytic expression for H(t).

Contribution

It provides the first closed-form analytic expression for H(t) and explores its dependence on system dimensionality and interaction potential.

Findings

H(t) is highly sensitive to potential and dimensionality.

Relaxation of H(t) is longer in 1D than in 3D.

Derived a new analytic expression for H(t).

Abstract

Boltzmann's H-Theorem, formulated 150 years ago in terms of H-function that also bears his name, is one of the most celebrated theorems of science and paved the way for the development of nonequilibrium statistical mechanics. Nevertheless, quantitative studies of the H-function, denoted by H(t), in realistic systems are relatively scarce because of the difficulty of obtaining the time-dependent momentum distribution analytically. Also, the earlier attempts proceeded through the solution of Boltzmann's kinetic equation, which was hard. Here we investigate, by direct molecular dynamics simulations and analytic theory, the time dependence of H(t). We probe the sensitivity of nonequilibrium relaxation to interaction potential and dimensionality by using the H-function H(t). We evaluate H(t) for three different potentials in all three dimensions and find that it exhibits surprisingly strong sensitivity to these factors. The relaxation of H(t) is long in 1D, but short in 3D. We obtain, for the first time, a closed-form analytic expression for H(t) using the solution of the Fokker-Planck equation for the velocity space probability distribution and compare its predictions with the simulation results. Interestingly, H(t) is found to exhibit linear response when vastly different initial nonequilibrium conditions are employed. The oft-quoted relation of H-function with Clausius's entropy theorem is discussed.