An H-Theorem for the Lattice Boltzmann Approach to Hydrodynamics

– 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. Introduction. – The lattice Boltzmann approach is a method for the simulation of hydrodynamic flow that was originally developed as a model to directly simulate the statistical average densities of lattice gas models. However, deriving the collision term for the lattice Boltzmann model from a lattice gas collision term unnecessarily restricts the Boltzmann model. Early lattice Boltzmann methods also suffered from the exclusion principle (i.e., there can be at most one particle at a given site), leading to an anomalous prefactor in the Navier Stokes equation that breaks Galilean invariance [1]. This constraint was removed in the linearized lattice Boltzmann model first introduced by Higuera and co-workers [2], where it was observed that the collision operator can be linearized around a local equilibrium and need not correspond to the detailed choice of collision rules of the lattice gas automata, provided the operator conserves mass and momentum. A further simplification was introduced by Qian, d’Humières and Lallemand [3], who proposed using the Bhatnagar-Gross-Krook (BGK) approximation [4] for the collision term in the lattice Boltzmann method. This approximation writes the collision operator as a function of the difference between the value of the distribution function and the equilibrium distribution function. For a recent review on the lattice Boltzmann method see [5]. Another interpretation of the lattice Boltzmann approach is as a discretized version of the continuum Boltzmann equation. The microscopic derivation of an H-theorem has been given by Boltzmann for the famous Boltzmann equation (see [6]). An H-theorem states that a functional can be defined which is a strictly decreasing function in time. For the continuous Typeset using EURO-TEX 2 EUROPHYSICS LETTERS Boltzmann equation this is the famous H-functional H(t) = ∫