Microcanonical

Microcanonical describes an ensemble or description of an isolated physical system with fixed energy, volume, and particle number. In the classical formulation, all microstates consistent with the fixed energy H(p, q) = E, inside the given volume and particle number, are assumed to be equally probable; in practice a narrow energy window ΔE is often used to define the microcanonical shell. In quantum mechanics, the microcanonical ensemble is represented by a density operator proportional to the projector onto the subspace of energy eigenstates with E ≤ H ≤ E + ΔE.

Entropy in the microcanonical framework is given by Boltzmann’s principle: S = k_B ln Ω(E, V, N), where Ω is the phase-space volume (classical) or the number of quantum states within the specified energy window. Thermodynamic quantities arise from derivatives of S. The temperature is defined by 1/T = ∂S/∂E at fixed V and N, and pressure is related by p = T(∂S/∂V)_{E,N}. These relations connect the microcanonical description to observable macroscopic properties.

The microcanonical ensemble is appropriate for isolated systems with no exchange of energy with an environment. It contrasts with the canonical ensemble, where temperature is fixed and energy fluctuates, and with the grand canonical ensemble, where particle exchange and chemical potential are also controlled. In the thermodynamic limit, the microcanonical, canonical, and grand canonical ensembles typically give equivalent predictions for macroscopic observables, though equivalence can fail for small systems or systems with long-range interactions.

Applications include foundational derivations of entropy, analyses of isolated systems, and discussions of ergodicity and phase transitions within a rigorous statistical framework.