Isothermal–isobaric partition function

The isothermal–isobaric partition function $\Delta(\beta,p,N)$ is the normalization constant of the isothermal–isobaric (NPT) ensemble , which describes equilibrium at fixed inverse temperature $\beta$, fixed pressure $p$, and fixed particle number $N$, with fluctuating volume.

It is naturally expressed in terms of the canonical partition function $Z(\beta,V,N)$ by integrating over all volumes with the pressure weight:

Mathematically, $\Delta$ is a Laplace transform of $Z$ with respect to $V$.

Physical meaning of the weight

In the NPT ensemble, microstates carry Boltzmann weight for the combination $H+pV$, where $H$ is the system Hamiltonian . The factor $e^{-\beta pV}$ expresses mechanical equilibrium with a pressure reservoir: larger volumes are penalized by the work term $pV$ in the effective energy.

Gibbs free energy from Δ

The partition function $\Delta$ generates the Gibbs free energy :

where $k_B$ is the Boltzmann constant and $\beta = 1/(k_B T)$. This matches the thermodynamic fact that $G$ is the natural potential at fixed $(T,p,N)$, i.e. after Legendre transforming the Helmholtz free energy with respect to volume.

In the thermodynamic limit , $G$ is extensive and one often uses $g = \lim_{N\to\infty} G/N$.

Generating formulas for volume statistics

The logarithm of $\Delta$ generates equilibrium averages (see observables from log partition functions ), in particular for the volume:

Mean volume

Volume fluctuations

These relations connect volume fluctuations to mechanical response, providing a fluctuation–response expression for the isothermal compressibility (compare susceptibilities in statistical mechanics ).

Relation to other ensembles

The NPT normalization $\Delta$ is to the NPT ensemble what $Z$ is to the canonical ensemble and what $\Xi$ (see grand partition function ) is to the grand-canonical ensemble : each partition function both normalizes the corresponding equilibrium measure and generates the appropriate thermodynamic potential via $-k_B T\ln(\cdot)$.