# Canonical ensemble

Variables:

• Number of Particles, ${\displaystyle N}$
• Volume, ${\displaystyle V}$
• Temperature, ${\displaystyle T}$

## Partition Function

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

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

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

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

where:

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