The Boltzmann equation detailed above describes the evolution of the
distribution function f of a fluid. The fluid density, momentum and energy
can then be found from the distribution function by considering the
appropriate integral. In theory this
appears straightforward, however in practice it can be difficult because
of the complicated form of the collision term 
. A large amount of
the detail of the two-body interaction, which is contained in the Boltzmann
collision operator, is unlikely to influence significantly the values of the
macroscopic quantities. It is therefor assumed
[12] that 
can be replaced
by a simplified collision operator which retains only the qualitative and average properties of the actual collision operator. Any replacement
collision function must satisfy the conservation of mass, momentum and
energy expressed by equation (2.7).
Such an
operator is based on the idea of a single relaxation time and can be written
[6]
where 
is the
local Maxwell-Boltzmann equilibrium distribution, given by
equation (2.34), and 
is the `relaxation time' which is
of the order of the time between collisions. This model is frequently
called the BGK model after Bhatnagar, Gross and Krook [6] who
first introduced it.