The distribution of the energies of a collection of particles depends on the average energy, which depends on the temperature. The most probable distribution of the energies of a collection of particles depends on the temperature therefore. The Boltzmann distribution expresses this dependence.
The Boltzmann distribution relates the number of particles per state at in terms of the energies of the states:
where 
is the number of particles at energy
which has degeneracy
etc.
Proof: The goal is to find
that maximises
This expresses the number of ways that a particular distribution of molecules throughout a set of allowed energies can be achieved. Taking logs of this expression, and using Stirling's approximation
for large
gives
We differentiate with respect to general
obtaining
(1).
We set this derivative equal to zero. The
are subject to the constraint
is fixed, and the energy
over and above the zero point energy is also fixed.
Now use Lagrange multipliers to find the
is the function
(2). We differentiate with respect to eachforming
or
for each
Substitute from (1) to obtain
which rearranges to
We can eliminate the factor
by considering
at two different energies, obtaining
Consideration of (2) since
but
for one degree of freedom so we can write
and further analysis gives
hence