Constructing the canonical partition function

The canonical ensemble describes a system at fixed temperature $T$, volume $V$, and particle number $N$ (see canonical ensemble ). Its normalization constant is the canonical partition function, which also acts as a generating function for thermodynamic quantities.

Setup (microstates and energy)

A classical microstate is a point in classical phase space , and the energy of a microstate $x$ is given by the Hamiltonian function $H(x)$. Integration over microstates uses the phase-space volume element $d\Gamma(x)$ (which may include conventions such as $h^{-dN}$ and $1/N!$ for identical particles).

Definition (classical canonical partition function)

Fix $(N,V)$ and inverse temperature $\beta$ (see inverse temperature ). The canonical partition function is

where $\Gamma_{N,V}$ denotes the accessible phase space at particle number $N$ and volume $V$.

The associated canonical probability measure on microstates is

so $Z_N$ is precisely the normalizing constant that turns the Boltzmann weight into a probability distribution. Expectations with respect to this measure are ensemble averages .

Quantum version (trace form)

For a quantum system with Hamiltonian operator $\hat H$ on the $N$-particle Hilbert space (and suitable boundary conditions implementing $V$), the canonical partition function is

Density-of-states representation

If $g_N(E;V)$ is the density of states (so that $g_N(E;V)\,dE$ counts microstates with energy in $[E,E+dE]$ in an appropriate sense), then

i.e. $Z_N$ is the Laplace transform of the density of states. This ties the canonical construction to microcanonical objects such as the microcanonical shell and microcanonical measure .

Physical interpretation

• $e^{-\beta H(x)}$ penalizes high-energy microstates at low temperature: larger $\beta$ concentrates the measure on lower-energy regions of phase space.
• $Z_N(\beta,V)$ encodes “how many” energetically accessible microstates exist once weighted by temperature; it is the central object from which the free energy is constructed .
• The thermodynamic temperature is related to $\beta$ by $\beta = 1/(k_B T)$, where $k_B$ is the Boltzmann constant .