Isothermal-isobaric ensemble

Ensemble variables:

• N is the number of particles
• p is the pressure
• T is the temperature

The classical partition function, for a one-component atomic system in 3-dimensional space, is given by

$Q_{NpT}={\frac {\beta p}{\Lambda ^{3N}N!}}\int _{0}^{\infty }dVV^{N}\exp \left[-\beta pV\right]\int d(R^{*})^{3N}\exp \left[-\beta U\left(V,(R^{*})^{3N}\right)\right]$

where

• $\left.V\right.$ is the Volume:

• $\beta :={\frac {1}{k_{B}T}}$, where $k_{B}$ is the Boltzmann constant

• $\left.\Lambda \right.$ is the de Broglie thermal wavelength

• $\left(R^{*}\right)^{3N}$ represent the reduced position coordinates of the particles; i.e. $\int d(R^{*})^{3N}=1$

• $\left.U\right.$ is the potential energy, which is a function of the coordinates (or of the volume and the reduced coordinates)

References

1. D. Frenkel and B. Smit, "Understanding Molecular Simulation: From Alogrithms to Applications", Academic Press