mikrocanonical

Mikrocanonical, or microcanonical, refers to a statistical ensemble used in statistical mechanics to describe an isolated system with fixed energy E, volume V, and particle number N. The term mikrocanonical is sometimes used in non-English literature, but the standard English term is microcanonical; the concept is also called the isolated ensemble.

In classical mechanics, the microcanonical ensemble assigns equal a priori probability to all microstates lying on the energy shell H(p, q) = E, within a small energy window. In quantum mechanics, the corresponding description uses a density operator proportional to the projector onto the subspace of states with energy E, effectively counting states within a narrow energy range. The number of available states in that range is denoted Ω(E, V, N), and the Boltzmann entropy is S = kB log Ω.

Observables are computed as ensemble averages over the microcanonical distribution. For an observable A, the average is ⟨A⟩ = (1/Ω) Tr[ A δ(H − E) ] in the quantum formulation, or the equivalent phase-space average in the classical case. The temperature is defined by 1/T = ∂S/∂E at fixed V and N. Because energy is fixed by construction, energy fluctuations are absent in the microcanonical ensemble, while other quantities may fluctuate.

Relation to other ensembles: the microcanonical ensemble is the isolated-system counterpart to the canonical (fixed T, fixed V, N) and grand canonical ensembles. In the thermodynamic limit, these ensembles are typically equivalent for many systems, though deviations can occur for small systems or those with long-range interactions. The microcanonical framework is especially natural for truly isolated systems, such as certain nuclear, atomic, or nanoscale clusters, or theoretical models emphasizing fixed energy constraints.