Consider two systems (1 and 2) in thermal contact such that
and the total Hamiltonian is just 
Since system 2 is infinitely large compared to system 1, it acts as an infinite heat reservoir that keeps system 1 at a constant temperature T without gaining or losing an appreciable amount of heat, itself. Thus, system 1 is maintained at canonical conditions, N,V,T.
The full partition function 
for the combined system is
the microcanonical partition function
Now, we define the distribution function, 
of the phase space
variables of system 1 as
Taking the natural log of both sides, we have
Since 
, it follows that 
, and we may
expand the above expression about 
. To linear order, the expression becomes
where, in the last line, the differentiation with respect to 
is replaced
by differentiation with respect to E.
Note that
where T is the common temperature of the two systems. Using these two facts, we obtain
Thus, the distribution function of the canonical ensemble is
The prefactor 
is an irrelevant constant that can be
disregarded as it will not affect any physical properties.
The normalization of the distribution function is the integral:
where Q(N,V,T) is the canonical partition function. It is convenient to
define an inverse temperature 
.
Q(N,V,T) is the canonical partition function.
As in the microcanonical case, we add in the ad hoc quantum corrections
to the classical result to give
The thermodynamic relations are thus,
To see that this must be the definition of A(N,V,T), recall the definition of A:
But we saw that
Substituting this in gives
or, noting that
it follows that
This is a simple differential equation that can be solved for A. We will show that the solution is
Note that
Substituting in gives, therefore
so this form of A satisfies the differential equation.
Other thermodynamics follow:
