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

$Q_{NpT}=\frac{\beta p}{{\mathrm{\Lambda }}^{3}N!}\underset{0}{\overset{\mathrm{\infty }}{\int }}dV{V}^N\exp [-\beta pV]\int d(R^{*}{)}^{3N}\exp [-\beta U(V,(R^{*}{)}^{3N})]$

where

• $V$ is the Volume:

• $\beta :=\frac{1}{k_{B}T}$;

• $\mathrm{\Lambda }$ is the de Broglie thermal wavelength

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

• $U$ 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