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# Isothermal-isobaric ensemble

The isothermal-isobaric ensemble has the following 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

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

where

• $\left. V \right.$ is the Volume:
• $\beta := \frac{1}{k_B T}$, where $k_B$ is the Boltzmann constant
• $\left( R^* \right)^{3N}$ 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)