Canonical ensemble
A system at fixed temperature, volume and particle number, free to exchange energy with a heat bath.
Definition
The canonical ensemble describes a system held at constant temperature T, volume V and particle number N by thermal contact with a large reservoir. Energy is not conserved by the system alone — it flows freely across the contact and fluctuates — but the temperature is fixed. The probability of each microstate is the Boltzmann weight exp(−E_i/k_BT)/Z, and the natural thermodynamic potential is the Helmholtz free energy F = −k_BT ln Z.
It is the ensemble of everyday laboratory thermodynamics, because most experiments are done at a controlled temperature rather than at a controlled energy. Energy fluctuations in the canonical ensemble are real but small: their relative size scales as 1/√N, so for macroscopic samples the energy is sharp to one part in 10¹² and the canonical and microcanonical descriptions coincide.
Gibbs introduced the canonical ensemble alongside the microcanonical and grand canonical ensembles as one of three standard ways to specify what is held fixed and what is left free, choosing whichever makes the partition function easiest to evaluate.