# Canonical ensemble

Variables:

• Number of Particles, $N$
• Volume, $V$
• Temperature, $T$

## Partition Function

The partition function, $Q$, for a system of $N$ identical particles each of mass $m$ is given by

$Q_{NVT}=\frac{1}{N!h^{3N}}\iint d{\mathbf p}^N d{\mathbf r}^N \exp \left[ - \frac{H({\mathbf p}^N,{\mathbf r}^N)}{k_B T}\right]$

where $h$ is Planck's constant, $T$ is the temperature, $k_B$ is the Boltzmann constant and $H(p^N, r^N)$ is the Hamiltonian corresponding to the total energy of the system. For a classical one-component system in a three-dimensional space, $Q_{NVT}$, is given by:

$Q_{NVT} = \frac{V^N}{N! \Lambda^{3N} } \int d (R^*)^{3N} \exp \left[ - \beta U \left( V, (R^*)^{3N} \right) \right] ~~~~~~~~~~ \left( \frac{V}{N\Lambda^3} \gg 1 \right)$

where:

• $\beta := \frac{1}{k_B T}$, with $k_B$ being the Boltzmann constant, and T the temperature.
• $U$ is the potential energy, which depends on the coordinates of the particles (and on the interaction model)
• $\left( R^*\right)^{3N}$ represent the 3N position coordinates of the particles (reduced with the system size): i.e. $\int d (R^*)^{3N} = 1$