Consider a thought experiment in which N particles are placed in a container
of volume V and allowed to evolve according to Hamilton's equations
of motion. The total energy 
is conserved. Moreover, the
number of particles N and volume V are considered to be fixed.
This constitutes a set of three thermodynamic variables N,V,E that
characterize the ensemble and can be varied to alter the conditions
of the experiment.
The evolution of this system in time generates a trajectory that samples
the constant energy hypersurface 
. All points on this surface
correspond to the same set of macroscopic observables.
Thus, by the
postulate of equal a priori probabilities, the corresponding ensemble,
called the microcanonical ensemble, should have a distribution function
that reflects the fact that all points on the constant energy
hypersurface are equally probable. Such a distribution function need only
reflect the fact that energy is conserved and can be written as
where 
is the Dirac delta function. The delta function has the
property that
for any function f(x).
Averaging over the microcanonical distribution function is equivalent to
computing the time average in our thought experiment. The microcanonical
partition function 
is given by
In Cartesian coordinates, this is equivalent to
where 
is a constant of proportionality. It is given by
Here h is a constant with units Energy 
Time,
and 
is a constant having units of energy.
The extra factor
of is needed because the 
function has units
of inverse energy. Such a constant has no effect at all on
any properties). Thus, is
dimensionless. The origin of is quantum mechanical in
nature (h turns out to be Planck's constant)
and must be put into the classical expression by hand.
Later, we will explore the effects of this constant on
thermodynamic properties of the ideal gas.
The microcanonical partition function function measures the number of microstates available to a system which evolves on the constant energy hypersurface. Boltzmann identified this quantity as the entropy, S of the system, which, for the microcanonical ensemble is a natural function of N, V and E:
Thus, Boltzmann's relation between , the
number of microstates and S(N,V,E) is
where k is Boltzmann's constant 1/k= 315773.218 Kelvin/Hartree. The importance of Boltzmann's relation is that it establishes a connection between the thermodynamic properties of a system and its microscopic details.
Recall the standard thermodynamic definition of entropy:
where an amount of heat dQ is assumed to be absorbed reversibly, i.e., along a thermodynamic path, by the system. The first law of thermodynamics states that the energy, E of the system is given by the sum of the heat absorbed by the system and the work done on the system in a thermodynamic process:
If the thermodynamic transformation of the system is carried reversibly,
i.e., along a thermodynamic path, then the first law will be valid for
the differential change in energy, dE due to absorption of a
differential amount of heat, 
and a differential amount
of work, dW done on the system:
The work done on the system can be in the form of compression/expansion work
at constant pressure, P, leading to a change, dV in the volume and/or the
insertion/deletion of particles from the system at constant chemical
potential, 
, leading to a change dN in the particle number. Thus, in general
(The above relation for the work is true only for a one-component system.
If there are M types of particles present, then the second term
must be generalized according to 
).
Then, using the fact that dQ = TdS, we have
or
But since S=S(N,V,E) is a natural function of N, V, and E, the differential, dS is also given by
Comparing these two expressions, we see that
Finally, using Boltzmann's relation between the entropy S and the
partition function 
, we obtain a prescription for
obtaining the thermodynamic properties of the system starting
from a microscopic, particle-based description of the system:
Of course, the ultimate test of Boltzmann's relation between entropy and the partition function is that the above relations correctly generate the known thermodynamic properties of a given system, e.g. the equation of state. We will soon see several examples in which this is, indeed, the case.