Microcanonical ensemble

The microcanonical ensemble describes an isolated thermodynamic system with fixed total energy $E$, particle number $N$, and volume $V$.

Classical formulation

The ensemble assigns equal probability to all microstates in a thin energy shell:

where $\Omega$ is the density of states and $H$ is the Hamiltonian .

Microcanonical entropy

The Boltzmann entropy is

where $\Omega$ counts the number of accessible microstates (or phase space volume).

Thermodynamic quantities

Temperature and pressure emerge from entropy derivatives:

Relation to other ensembles

In the thermodynamic limit , the microcanonical ensemble is equivalent to the canonical and grand canonical ensembles for typical observables.