  The molecular dynamics lattice gas method maps a molecular dynamics
simulation onto a lattice gas using a coarse-graining procedure. This is a
novel fundamental approach to derive the lattice Boltzmann method by taking a
Boltzmann average over the molecular dynamics lattice gas. A key property of
the lattice Boltzmann method is the equilibrium distribution function, which
was originally derived by assuming that the particle displacements in the
molecular dynamics simulation are Boltzmann distributed. However, we recently
discovered that a single Gaussian distribution function is not sufficient to
describe the particle displacements in a broad transition regime between free
particles and particles undergoing many collisions in one time step. In a
recent publication, we proposed a Poisson weighted sum of Gaussians which shows
better agreement with the molecular dynamics data. We derive a lattice
Boltzmann equilibrium distribution function from the Poisson weighted sum of
Gaussians model and compare it to a measured equilibrium distribution function
from molecular dynamics data and to an analytical approximation of the
equilibrium distribution function from a single Gaussian probability
distribution function.
