The collision operator 
can be further simplified
[43] by assuming that the particle distribution function relaxes
to its equilibrium state at a constant rate
which gives a collision term
A lattice Boltzmann equation with this collision operator is called a
lattice BGK equation because of its similarity to the
classical BGK Boltzmann operator [6]. With the introduction
of this single relaxation parameter and the comparison with the classical
Boltzmann equation the form of the equilibrium distribution was also compared.
In the classic model the equilibrium distribution is a
Maxwell-Boltzmann distribution [11], while in the earlier
lattice Boltzmann models, which have been considered to
be an evolution of the lattice gas models, a Fermi-Dirac distribution
is used. This originates in the constraint that only one particle is allowed
on each of the lattice links. This constraint was applied to simplify the
computation, it allowed the state of each link to be described by a Boolean
variable and limited the number of collisions which could take place. There
is no physical requirement for such a constraint. The single relaxation time
model described here is inspired by
the classical Boltzmann BGK model described in section 2.4.
Using the lattice BGK approach and
selecting an equilibrium distribution such that the correct macroscopic
equations are satisfied, rather than adopting the lattice gas equilibrium
distribution, the lattice Boltzmann model is found to satisfy (to second-order
in the velocity and the Knudsen number) the continuity equation and the
exact Navier-Stokes equation without the 
term
[44, 45, 46]. This will be shown in section 4.2.