A statistical description of a system can be made in terms of the distribution
function 
[11, 12, 13]
where
is defined such that 
is
the number
of molecules at time t positioned between 
and
which have
velocities in the range 
.
Consider a gas in which an external force 
acts
and assume initially that no collisions
take place between the gas molecules. In
time 
the velocity 
of any molecule will
change to 
and its
position will change to 
. Thus the number of molecules
is
equal to the number of molecules
, that is to say,
If, however, collisions do occur between the molecules there will
be a net difference between the number of molecules
and
the number of molecules
.
This can be written [11]
where 
is the collision operator. This gives the following
equation describing the evolution of the distribution function:
Dividing equation (2.2) by 
and letting 
gives the Boltzmann equation [11, 12]
where
The fluid density 
, velocity 
and internal
energy e can be found from the distribution function f as follows
[11]:
where m is the molecular mass and 
is the
peculiar velocity 
, the particle velocity with respect to
the fluid flow. The internal energy can be shown [11] to be
where T is the temperature and 
is Boltzmann's constant.
Any solution of Boltzmann's equation (2.3) requires that an
expression is found for the collision operator .
Without knowing the form of
there are however several properties which can be deduced. If the collision
is to conserve
mass, momentum and energy it is required that
The terms 
, i=0, ..., 4 where 
, 
,
, 
and 
are frequently called the
elementary collision invariants since 
[12]. Any linear combination
of the terms is also a collision invariant.
