The random gas of hard spheres

TL;DR

This paper introduces a novel stochastic model for gas particle dynamics that aligns with the Enskog equation, providing a more accurate description of dense gases and their hydrodynamic limits, while maintaining entropy non-increase.

Contribution

It proposes a random process-based dynamical system for gas particles that approximates collision dynamics and derives a variant of the Enskog equation with new hydrodynamic effects.

Findings

The model's steady state distributions are explicitly computed.

The Kullback-Leibler entropy of the system is nonincreasing over time.

The derived Enskog equations include additional effects in hydrodynamic limits.

Abstract

The inconsistency between the time-reversible Liouville equation and time-irreversible Boltzmann equation has been pointed out long ago by Loschmidt. To avoid Loschmidt's objection, here we propose a new dynamical system to model the motion of atoms of gas, with their interactions triggered by a random point process. Despite being random, this model can approximate the collision dynamics of rigid spheres via adjustable parameters. We compute the exact statistical steady state of the system, and determine the form of its marginal distributions for a large number of spheres. We find that the Kullback-Leibler entropy (a generalization of the conventional Boltzmann entropy) of the full system of random gas spheres is a nonincreasing function of time. Unlike the conventional hard sphere model, the proposed random gas model results in a variant of the Enskog equation, which is known to be a more accurate model of dense gas than the Boltzmann equation. We examine the hydrodynamic limit of the derived Enskog equation for spheres of constant mass density, and find that the corresponding Enskog-Euler and Enskog-Navier-Stokes equations acquire additional effects in both the advective and viscous terms. In the dilute gas approximation, the Enskog equation simplifies to the Boltzmann equation, while the Enskog-Euler and Enskog-Navier-Stokes equations become the conventional Euler and Navier-Stokes equations.