DOI: 10.1007/s10955-005-5958-9 Journal of Statistical Physics, Vol. 121, Nos. 1/2, October 2005 (© 2005) Nonexistence of H Theorem for Some Lattice Boltzmann Models Wen-An Yong 1 and Li-Shi Luo 2 Received October 12, 2004; accepted April 21, 2005 In this paper, we provide a set of sufficient conditions under which a lattice Boltzmann model does not admit an H theorem. By verifying the conditions, we prove that a number of existing lattice Boltzmann models does not admit an H theorem. These models include D2Q6, D2Q9 and D3Q15 athermal mod- els, and D2Q16 and D3Q40 thermal (energy-conserving) models. The proof does not require the equilibria to be polynomials. KEY WORDS: Lattice Boltzmann equation; H -theorem. 1. INTRODUCTION The lattice Boltzmann equation (LBE) (1, 2) has emerged as an effective method for computational fluid dynamics (CFD) (cf. a recent review (3) and refs. therein). The most notable feature of the lattice Boltzmann equation is its direct connection to a discretization of the Boltzmann equation, (4, 5) rather than to discretizations of the Navier–Stokes equations. The kinetic origin of the LBE method immediately leads to the question whether or not the H theorem associated with the Boltzmann equation is preserved in the lattice Boltzmann equation, after the drastic approximations made to derive it. (4, 5) The H theorem has many important ramifications and is directly related to the stability of the LBE method. Therefore it has been a subject of considerable research interest (cf. refs. 6–10 and refer- ences therein). It seems to be intuitive that the LBE models with relax- ation collision operators and polynomial equilibria do not admit an 1 IWR, Universit¨ at Heidelberg, Im Neuenheimer Feld 294, 69120 Heidelberg, Germany; e-mail: yong.wen-an@iwr.uni-heidelberg.de 2 Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA 23529, USA; e-mail: lluo@odu.edu 91 0022-4715/05/1000-0091/0 © 2005 Springer Science+Business Media, Inc.