Frisch et al. [8] use a probabilistic approach which is traditional in statistical mechanics. Thus they consider the mean population
the mean density
the mean mass current
and the mean velocity
Note that the mean density and mass current are defined to be the mean quantities per site, and not per unit area as they normally are in the real world. Frisch et al. [8] have found a steady state equilibrium solution for the mean population,
where f is the Fermi-Dirac function
is the equilibrium mean population
and h and are Lagrange multipliers which depend
on the mean density and the mean velocity
through
equations (3.9) - (3.12).
Note that although the solution is independent of the transition
probabilities Frisch's proof requires that
A satisfies semi-detailed balance: equation (3.8).
The Fermi-Dirac distribution for the equilibrium state of the mean population
is obtained because there is an exclusion principle applied to the
lattice gas model: only one particle is allowed on any link at any time-step.
Explicit solutions for h and are only known for special cases, such as when where is clearly a solution. Expressions for h and can be found in the limit of small by expanding equation (3.13) about the zero velocity case to give
where
and
This restricts us to using speeds u which are small compared to
, the speed of sound in the model. Given a solution for
it is not possible to use a
Galilean transformation to find the solution for non-zero
because the solution found here only applies in an
inertial frame in which the underlying grid is at rest. This lack of
Galilean invariance is an important feature of the model
and can lead to problems when the model is implemented.
It can also be shown [22] that the lattice gas model obeys an H-theorem. This states that the mean population must tend towards the equilibrium mean population .