Towards an $H$-theorem for granular gases

TL;DR

This paper introduces a Lyapunov functional for dissipative granular gases that behaves similarly to the classical $H$-theorem, supported by multiple simulation methods, extending the concept of irreversibility to non-elastic systems.

Contribution

The authors construct a new functional $\\mathcal{H}$ that acts as a Lyapunov functional for dissipative granular gases, generalizing the $H$-theorem to non-elastic collisions.

Findings

The functional $\\mathcal{H}$ is positive and non-increasing during evolution.

Simulations using spectral, DSMC, and Molecular Dynamics support the functional's properties.

The approach applies to both driven and unforced granular gases.

Abstract

The $H$-theorem, originally derived at the level of Boltzmann non-linear kinetic equation for a dilute gas undergoing elastic collisions, strongly constrains the velocity distribution of the gas to evolve irreversibly towards equilibrium. As such, the theorem could not be generalized to account for dissipative systems: the conservative nature of collisions is an essential ingredient in the standard derivation. For a dissipative gas of grains, we construct here a simple functional $\mathcal H$ related to the original $H$, that can be qualified as a Lyapunov functional. It is positive, and results backed by three independent simulation approaches (a deterministic spectral method, the stochastic Direct Simulation Monte Carlo technique, and Molecular Dynamics) indicate that it is also non-increasing. Both driven and unforced cases are investigated.