The Enskog equation for confined elastic hard spheres

TL;DR

This paper derives a generalized kinetic equation for confined elastic hard spheres, extending the Enskog equation to account for arbitrary wall shapes and moderate densities, and analyzes its hydrodynamic implications.

Contribution

It introduces a new kinetic equation that incorporates confinement effects for elastic hard spheres and disks, extending the classical Enskog equation.

Findings

Derivation of a generalized Enskog equation for confined systems

Identification of a Lyapunov functional that decays over time

Establishment of hydrodynamic balance equations with collisional contributions

Abstract

A kinetic equation for a system of elastic hard spheres or disks confined by a hard wall of arbitrary shape is derived. It is a generalization of the modified Enskog equation in which the effects of the confinement are taken into account and it is supposed to be valid up to moderate densities. From the equation, balance equations for the hydrodynamic fields are derived, identifying the collisional transfer contributions to the pressure tensor and heat flux. A Lyapunov functional, $\mathcal{H}[f]$, is identified. For any solution of the kinetic equation, $\mathcal{H}$ decays monotonically in time until the system reaches the inhomogeneous equilibrium distribution, that is a Maxwellian distribution with a the density field consistent with equilibrium statistical mechanics.