Pressure from the partition function

Definition (canonical pressure).
In the canonical ensemble , the pressure is defined thermodynamically by

where $F$ is the statistical free energy . Using $F=-k_B T\ln Z$ with the canonical partition function $Z(\beta,V,N)$, this becomes the standard partition-function formula

This identity is one of the central instances of obtaining observables from log Z .

Grand canonical form.
In the grand canonical ensemble , with grand partition function $\Xi(\beta,\mu,V)$,

For homogeneous systems in the thermodynamic limit , $\ln \Xi$ is extensive in $V$, and one commonly writes the equivalent scaling relation

Physical interpretation.
Pressure measures the reversible work required to change volume: $dW_{\text{rev}} = -p\,dV$. The partition function $Z$ (or $\Xi$) encodes how the weighted count of microstates changes when the accessible configuration space expands or contracts. Differentiating $\ln Z$ with respect to $V$ extracts precisely that response, producing the macroscopic force per area. This statistical definition is consistent with thermodynamic pressure .

Example (ideal gas).
For a classical ideal gas, $Z(\beta,V,N)$ is proportional to $V^N$, so $\ln Z = N\ln V + \text{(terms independent of }V)$, and the formula above yields

Lattice systems note.
On a lattice in a finite region finite box $\Lambda$, “volume” is $|\Lambda|$. One often defines the (dimensionless) pressure/free-energy density as $\lim_{|\Lambda|\to\infty} |\Lambda|^{-1}\ln Z_\Lambda$, which is the lattice analogue of the continuum scaling behind the formulas above.