Consider a system with Hamiltonian 
. Let 
and 
be specific components of the phase space vector. The classical
virial theorem states that
where the average is taken with respect to a microcanonical ensemble.
To prove the theorem, start with the definition of the average:
where the fact that 
has been used. Also, the
N and V dependence of the partition function have been suppressed.
Note that the above average can be written as
However, writing
allows the average to be expressed as
The first integral in the brackets is obtained by integrating the
total derivative with respect to over the phase space variable
. This leaves an integral that must be performed over all other
variables at the boundary of phase space where H=E, as indicated
by the surface element 
. But the
integrand involves the factor H-E, so this integral will vanish.
This leaves:
where 
is the partition function of the uniform ensemble.
Recalling that
we have
which proves the theorem.
Example:
and i=j. The virial theorem says that
Thus, at equilibrium, the kinetic energy of each particle must be kT/2. By summing both sides over all the particles, we obtain a well know result
