desirable to apply the canonical ensemble to systems that are in direct contact with the heat bath, since it is that contact that ensures the equilibrium. In practical situations, the use of the canonical ensemble is usually justified either 1) by assuming that the contact is mechanically weak, or 2) by incorporating a suitable part of the heat bath connection into the system under analysis, so that the connection's mechanical influence on the system is modeled within the system. When the total energy is fixed but the internal state of the system is otherwise unknown, the appropriate description is not the canonical ensemble but the microcanonical ensemble. For systems where the particle number is variable (due to contact with a particle reservoir), the correct description is the grand canonical ensemble. In statistical physics textbooks for interacting particle systems the three ensembles are assumed to be thermodynamically equivalent: the fluctuations of macroscopic quantities around their average value become small and, as the number of particles tends to infinity, they tend to vanish. In the latter limit, called the thermodynamic limit, the average constraints effectively become hard constraints. The assumption of ensemble equivalence dates back to Gibbs and has been verified for some models of physical systems with short-range interactions and subject to a small number of macroscopic constraints. Despite the fact that many textbooks still convey the message that ensemble equivalence holds for all physical systems, over the last decades various examples of physical systems have been found for which breaking of ensemble equivalence occurs. Properties Free energy, ensemble averages, and exact differentials The partial derivatives of the function give important canonical ensemble average quantities: the average pressure is the Gibbs entropy is the partial derivative is approximately related to chemical potential, although the concept of chemical equilibrium does not exactly apply to canonical ensembles of small systems. and the average energy is Exact differential: From the above expressions, it can be seen that the function , for a given , has the exact differential First law of thermodynamics: Substituting the above relationship for into the exact differential of , an equation similar to the first law of thermodynamics is found, except with average signs on some of the quantities: Energy fluctuations: The energy in the system has uncertainty in the canonical ensemble. The variance of the energy is Example ensembles Boltzmann distribution (separable system) If a system described by a