Isothermal–isobaric (NPT) ensemble

The isothermal–isobaric ensemble (often called the NPT ensemble) describes a thermodynamic system in contact with reservoirs that fix the temperature (see temperature ) and the pressure (see pressure ), while the particle number $N$ is held fixed. In this setting the volume $V$ is not fixed; it is a fluctuating quantity determined by mechanical equilibrium with the pressure reservoir.

Microscopically, one convenient classical description takes microstates (see microstate ) as a pair consisting of a point $x$ in phase space and a volume parameter $V>0$. The energy of a microstate is given by a Hamiltonian function $H(x;V)$, and integration over microstates uses the phase-space volume element at fixed $V$ together with an outer integration over $V$.

Probability weight

Let $\beta$ denote the inverse temperature (see inverse temperature ), with $\beta = 1/(k_B T)$ where $k_B$ is the Boltzmann constant . In the NPT ensemble, the unnormalized weight of a microstate $(x,V)$ is proportional to

so the probability density has the schematic form

where $dx$ denotes the phase-space measure at fixed $V$.

The combination $H+pV$ is the enthalpy-like quantity that plays the role of “effective energy” when pressure is imposed rather than volume.

Normalization and thermodynamic potential

The normalization constant of the NPT distribution is the isothermal–isobaric partition function $\Delta(\beta,p,N)$. It can be expressed as a Laplace transform of the canonical partition function $Z(\beta,V,N)$:

In equilibrium statistical mechanics, $\Delta$ generates the Gibbs free energy through

up to conventions that become irrelevant in the thermodynamic limit . This matches the thermodynamic Legendre structure: NPT is the natural ensemble for $G$ (compare with the canonical ensemble which is natural for the Helmholtz free energy ).

Averages and fluctuations

Ensemble averages are defined as in any ensemble average : for an observable $A(x,V)$,

Derivatives of $\ln\Delta$ generate moments of $V$:

In particular, the isothermal compressibility can be read from volume fluctuations:

These relations are instances of general fluctuation–response identities (see fluctuation formulas from log partition functions ).