  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.
