Boltzmann's equation describes the evolution of molecules in a rare gas.
If no external forces are present ( 
) then,
after a long time, the gas should reach an equilibrium state. This can
be seen [14, 11] by considering the function
Differentiating equation (2.19) with respect to time we get
Substituting 
from equation (2.3), with
,
and using
equation (2.16) we get
Now, following [14],
consider the first term on the right hand side of
equation (2.21). Since 
and 
are
independent variables we can write 
as
and using Gauss's
theorem we can write
where 
is the surface enclosing the volume
. Now
so the
first term on the right-hand side
of equation (2.21) vanishes. This leaves
Now consider the reverse of the collision: particles with velocity
and 
colliding
and moving off with velocities 
and
. For this collision we have
since 
.
Summing equations (2.23) and (2.24) and dividing by two gives
Changing the dummy variables 
and 
we get
Finally summing equations (2.25) and (2.26) and dividing by two gives
Now 
and all the other
terms in the integrand of equation (2.27) are positive so
This means that H can never increase and is known as Boltzmann's
H-theorem. It can also be shown that H is bounded below [11].
We know that 
converges because
the total energy of the molecules must converge. Thus either H converges or
more rapidly than 
as
. In the later case
more rapidly than 
which implies
that H converges. Since H can never increase but tends to a finite
limit the finite limit must correspond to 
. This is
only possible, see equation (2.27), if
This condition is known as detailed balance [11] and can be expressed equivalently as
Thus if 
is the equilibrium distribution then
is a collision invariant and so must be of the form
where 
are the collision invariants defined below
equation (2.7) and 
are constants. This can be
re-written [11]
where 
and 
. With
where 
we can write
This is the Maxwell distribution function for a gas [11, 12, 14] and it describes the equilibrium state of the distribution function f. The form of the constants can be found by substituting equation (2.33) into equation (2.5) to give the more common form of the Maxwell distribution function [11, 12]
The H-theorem states that the distribution function f must tend
towards its equilibrium state .
The entropy S(t) of the system (which
is a non-decreasing function of time) is given by [11, 14]
