Allowed and forbidden regimes of entropy balance in lattice Boltzmann collisions

We study the possibility of modifying collisions in the lattice Boltzmann method to keep the proper entropy balance. We demonstrate that in the space of distributions operated on by lattice Boltzmann methods which respect a Boltzmann type H theorem, there exists a vicinity of the equilibrium where collisions with entropy balance are possible and, at the same time, there exists a region of nonequilibrium distributions where such collisions are impossible. In particular, for a strictly concave and uniformly bounded entropy function with positive equilibria, we show that proper entropy balance is always possible sufficiently close to the local equilibrium and it is impossible sufficiently far from it, where additional dissipation has to appear. We also present some nonclassical entropies that do not share this concern. The cases where the distribution enters the region far from equilibrium typically occur in flows with low viscosity and/or high Mach number flows and in simulations on coarse grids.