Galilean-invariant multi-speed entropic lattice Boltzmann models

In recent work [Phys. Rev. E 68 (2003) 025103], it was shown that the requirement of Galilean invariance determined the form of the H function used in entropic lattice Boltzmann models for the incompressible Navier–Stokes equations in D dimensions. The form obtained was that of the Burg entropy for D=2, and the Tsallis entropy with q=1−2/D for D≠2. The conclusions obtained in that work were restricted to particles of a single-mass and speed on a Bravais lattice. In this work, we generalize the construction of such Galilean-invariant entropic lattice Boltzmann models by allowing for certain models with multiple masses and speeds. We show that the required H function for these models must be determined by solving a certain functional differential equation. Remarkably, the solutions to this equation also have the form of the Tsallis entropy, where q is determined by the solution to a certain transcendental equation, involving the dimension and symmetry properties of the lattice, as well as the masses and speeds of the particles.