Pressure identity in the canonical ensemble

Statement

For a system in the canonical ensemble at inverse temperature $\beta$, with canonical partition function Z(β,V) depending on the volume parameter $V$, define the Helmholtz free energy $F(\beta,V)=-(1/\beta)\log Z(\beta,V)$.

Then the canonical pressure satisfies

Key hypotheses

• A well-defined dependence of the Hamiltonian and/or configuration domain on the volume parameter $V$, so that $Z(\beta,V)$ is differentiable in $V$.
• Interchange of $\partial/\partial V$ with the integral defining $Z(\beta,V)$ is justified.

Conclusion

• Pressure is a logarithmic derivative of the canonical partition function with respect to volume.
• Equivalently, pressure is the negative volume derivative of the Helmholtz free energy.

Cross-links to definitions

• Thermodynamic pressure
• Helmholtz free energy
• Canonical partition function
• Statistical free energy

Proof idea / significance

Start from $F=-(1/\beta)\log Z$. Differentiating in $V$ gives $-(\partial F/\partial V)_\beta = (1/\beta)\,(\partial_V \log Z)_\beta$. This identity is one of the main “thermodynamic observables from $Z$” formulas, paralleling the energy identity for $\langle H\rangle_\beta$.