An H-Theorem for the Lattice Boltzmann Approach to Hydrodynamics

TL;DR

This paper investigates the conditions under which lattice Boltzmann models obey an H-theorem, proving that while standard models do not, alternative models can be constructed to satisfy this property, impacting their theoretical understanding.

Contribution

The paper demonstrates that standard nine-velocity lattice Boltzmann models do not obey an H-theorem, but provides conditions and constructions for models that do.

Findings

Standard nine-velocity models lack an H-theorem.

Models satisfying an H-theorem can be constructed.

On a lattice, decreasing an H-functional does not guarantee a unique ground state.

Abstract

The lattice Boltzmann equation can be viewed as a discretization of the continuous Boltzmann equation. Because of this connection it has long been speculated that lattice Boltzmann algorithms might obey an H-theorem. In this letter we prove that usual nine-velocity models do not obey an H-theorem but models that do obey an H-theorem can be constructed. We consider the general conditions a lattice Boltzmann scheme must satisfy in order to obey an H-theorem and show why on a lattice, unlike the continuous case, dynamics that decrease an H-functional do not necessarily lead to a unique ground state.