Canonical ensemble

If you have a classical Hamiltonian system and you assume it is in thermal equilibrium with a large heat bath at temperature $T$ (i.e. it may exchange energy with the bath, coupling is weak and the bath is so large that its temperature is unchanged), then the description of that system is the canonical ensemble, i.e., the distribution on phase space given by the Boltzmann distribution

Important caveats / distinctions:

• If the system is isolated (fixed total energy) then thermal equilibrium is described by the microcanonical ensemble, not the canonical one.
• If the system also exchanges particles, use the grand canonical ensemble (factors $e^{-\beta (E-\mu N)}$).

Here we are a well-known partition function, that let us compute the entropy, energy and Helmholtz free energy.