Canonical partition function

The canonical partition function $Z(\beta,V,N)$ is the normalization constant of the canonical ensemble describing equilibrium at fixed inverse temperature $\beta$ (see inverse temperature ), fixed volume $V$, and fixed particle number $N$.

For a system with discrete energy levels $\{E_s\}$ (indexed by microstates $s$), it is defined by

For a classical system with Hamiltonian $H(x)$ on phase space , one writes schematically

where $dx$ denotes the phase-space volume element at fixed $V$ (often with conventional prefactors such as $1/N!$ and powers of Planck’s constant to make $Z$ dimensionless).

In either case, the canonical probability of a microstate is

Connection to density of states

If $\Omega(E;V,N)$ denotes the density of states (or microcanonical level density), then $Z$ is its Laplace transform:

This is the basic bridge between canonical equilibrium and the microcanonical description (see also canonical-from-microcanonical construction ).

Thermodynamic potential: Helmholtz free energy

The canonical partition function generates the Helmholtz free energy :

where $k_B$ is the Boltzmann constant and $\beta = 1/(k_B T)$. In the thermodynamic limit , $F$ becomes extensive and is the natural potential at fixed $(T,V,N)$ (compare with statistical free energy ).

Key derivative formulas (energy, pressure, fluctuations)

The logarithm of $Z$ is a generating function for equilibrium averages (see observables from log partition functions ). In particular:

Mean energy

Energy fluctuations and heat capacity

This is the canonical fluctuation formula behind specific heat from fluctuations .

Pressure from $Z$ The mechanical pressure can be obtained from volume dependence of $Z$:

as summarized in pressure from the partition function .

These identities show how $Z$ simultaneously normalizes the ensemble and encodes the response of the equilibrium state to changes in control parameters.