Isothermal-isobaric ensemble

Variables:

• N (Number of particles)
• p (Pressure)
• T (Temperature)
• V (Volume)

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

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

where

• $\beta ={\frac {1}{k_{B}T}}$;

• $\Lambda$ is the de Broglie wavelength

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

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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