Statistical Ensemble (mathematical Physics) - Ensembles of Classical Mechanical Systems

Ensembles of Classical Mechanical Systems

For an ensemble of a classical mechanical system, one considers the phase space of the given system. A collection of elements from the ensemble can be viewed as a swarm of representative points in the phase space. The statistical properties of the ensemble then depend on a chosen probability measure on the phase space. If a region A of the phase space has larger measure than region B, then a system chosen at random from the ensemble is more likely to be in a microstate belonging to A than B. The choice of this measure is dictated by the specific details of the system and the assumptions one makes about the ensemble in general. For example, the phase space measure of the microcanonical ensemble (see below) is different from that of the canonical ensemble. The normalizing factor of the probability measure is referred to as the partition function of the ensemble. Physically, the partition function encodes the underlying physical structure of the system.

When the measure is time-independent, the ensemble is said to be stationary.