Connecting lattice Boltzmann methods to underlying physical systems: Molecular Dynamics lattice Gas, Monte Carlo lattice Gas and their implications

Lattice Boltzmann simulations are becoming ever more popular for a large number of applications from high Reynolds number flows to multiphase flow in multicomponent mixtures. The method's succss has been credited to its relation to an underlying physical model which guarantees exact mass and momentum conservation. However, the relation back to physical systems has relied on matching appropriate dimensionless numbers, and the exact meaning of the lattice Boltzmann densities has remained obscure. By directly matching these densities to a coarse-grained grained Molecular Dynamics simulation we can show how these densities can be related to underlying physical quantities, and also where these direct identifications can become problematic.
One such problematic process is the ubiquitous over-relaxation process used in lattice Botzmann methods. We comment on how a scaling analysis of a lattice gas method can shed light on its justification.