Canonical ensemble

The canonical ensemble models a system that can exchange energy with a heat bath, so its temperature is fixed while its energy fluctuates.

Let $\Gamma$ be a classical phase space with Liouville volume element $d\Gamma$, and let $H(x)$ be the Hamiltonian . Fix an inverse temperature inverse temperature $\beta>0$ (equivalently a temperature $T$ via $\beta = 1/(k_B T)$ with Boltzmann's constant $k_B$).

The canonical probability density on $\Gamma$ is

where $Z(\beta)$ is the canonical partition function .

(Quantum version: replace the phase-space integral by a trace, $Z(\beta)=\mathrm{Tr}\,e^{-\beta \hat H}$, and $\rho_\beta$ by the Gibbs density matrix.)

Ensemble averages

For an observable $A(x)$, the ensemble average in the canonical ensemble is

Thermodynamic potentials from $Z$

The canonical Helmholtz free energy is obtained from the partition function via the statistical free energy

This link is the starting point for constructing free energy from the partition function .

Energy and fluctuations

Canonical energy moments are derivatives of $\ln Z(\beta)$ (see observables from log partition functions ):

Energy fluctuations control the heat capacity at fixed volume (in classical $N,V$ settings), encoded by specific heat as an energy fluctuation .

Physical interpretation and construction

The canonical distribution weights microstates by the Boltzmann factor $e^{-\beta H}$: lower-energy states are exponentially favored, with the strength set by temperature. It can be derived either by weak coupling of the system to a large bath or by the variational principle of maximum entropy at fixed mean energy .

In the thermodynamic limit , canonical and microcanonical predictions often agree for bulk observables (ensemble equivalence), formalized by constructing the canonical ensemble from the microcanonical one .