Consider bringing two systems into thermal contact. By thermal contact, we
mean that the systems can only exchange heat. Thus, they do not exchange
particles, and there is no potential coupling between the systems.
In this case, if system 1 has a phase space vector 
and
system 2 has a phase space vector 
, then the total Hamiltonian
can be written as
Furthermore, let system 1 have 
particles in a volume 
and
system 2 have 
particles in a volume 
. The total
particle number N and volume V are 
and 
.
The entropy of each system is given by
The partition functions are given by
However, it can be shown that the total partition function can be written as
where C' is an overall constant independent of the energy. Note that the dependence of the partition functions on the volume and particle number has been suppressed for clarity.
Now imagine expressing the integral over energies in the above expression as a Riemann sum:
where 
is the small energy interval (which we will allow to
go to 0) and 
. The reason for writing the integral this way
is to make use of a powerful theorem on sums with large numbers of
terms.
Consider a sum of the form
where 
for all 
. Let 
be the largest
of all the 's. Clearly, then
Thus, we have the inequality
or
This gives upper and lower bounds on the value of 
.
Now suppose that 
. Then the above inequality
implies that
This would be the case, for example, if 
.
In this case, the value of the sum is given to a very good
approximation by the value of its maximum term.
Why should this theorem apply to the sum expression for 
?
Consider the case of a system of free particles 
,
i.e., no potential. Then the expression for the partition function is
since the particle integrations are restricted only the volume of the container.
Thus, the terms in the sum vary exponentially with N. But the number
of terms in the sum P also varies like N since and
, since E is extensive. Thus, the terms in the sum under
consideration obey the conditions for the application of the theorem.
Let the maximum term in the sum be characterized by energies
and 
. Then, according to the above
analysis,
Since , 
.
But , while 
. Since 
, the above
expression becomes, to a good approximation
Thus, apart from constants, the entropy is approximately additive:
Finally, in order to compute the temperature of each system, we make a small
variation in the energy 
. But since
, 
. Also, this variation is
made such that the total entropy S and energy E remain constant. Thus,
we obtain
from which it is clear that 
, the expected condition for thermal
equilibrium.
It is important to point out that the entropy S(N,V,E) defined
via the microcanonical partition function is not the only entropy
that satisfies the properties of additivity and equality of
temperatures at thermal equilibrium. Consider an ensemble
defined by the condition that the Hamiltonian, 
is less
than a certain energy E. This is known as the uniform ensemble
and its partition function, denoted 
is defined by
where 
is the Heaviside step function. Clearly, it is
related to the microcanonical partition function by
Although we will not prove it, the entropy 
defined
from the uniform ensemble partition function via
is also approximately additive and will yield the condition for
two systems in thermal contact. In fact, it differs from S(N,V,E)
by a constant of order 
so that one can also define the thermodynamics
in terms of . In particular, the temperature is given by
