An equilibrium ensemble is one for which
there is no explicit time-dependence in the phase
space distribution function, 
. In this case, Liouville's
equation reduces to
which implies that 
must be a pure function of the Hamiltonian
The specific form that 
has depends on the specific
details of the ensemble.
The integral over the phase space distribution function plays a special role in statistical mechanics:
It is known as the partition function and is equal to the number of members in the ensemble. That is, it is equal to the number of microstates that all give rise to a given set of macroscopic observables. Thus, it is the quantity from which all thermodynamic properties are derived.
If a measurement of a macroscopic observable 
is made, then the
value obtained will be the ensemble average:
Eqs. (1) and (2) are the central results of ensemble theory, since they determine all thermodynamic and other observable quantities.
