Microcanonical Ensemble

For a completely isolated system (e.g. the universe) , (the number of particles), (the volume), (the energy) is assumed to be constant. The thermodynamics of an isolated system is described by the micro canonical ensemble. Generally, the micro-canonical ensemble has a discrete probability distribution and thus cannot have a partition function (which is defined in the context of continuous probability distribution). The probability of a micro-state is

where is the entropy. Thus, the most likely configuration is that configuration for which is maximized.

The partition function can be derived by the discrete probability distribution in terms of the Dirac Delta function, as so: