  We introduce a lattice gas implementation that is based on coarse-graining a
Molecular Dynamics (MD) simulation. Such a lattice gas is similar to standard
lattice gases, but its collision operator is informed by an underlying MD
simulation. This can be considered an optimal lattice gas implementation
because it allows for the representation of any system that can be simulated
with MD. We show here that equilibrium behavior of the popular lattice
Boltzmann algorithm is consistent with this optimal lattice gas. This
comparison allows us to make a more accurate identification of the expressions
for temperature and pressure in lattice Boltzmann simulations which turn out to
be related not only to the physical temperature and pressure but also to the
lattice discretization. We show that for any spatial discretization we need to
choose a particular temporal discretization to recover the lattice Boltzmann
equilibrium.
