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:

• $\Lambda$ is the de Broglie thermal wavelength (depends on the temperature)

• $\beta :={\frac {1}{k_{B}T}}$, 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. $\int d(R^{*})^{3N}=1$

See also

• Ideal gas partition function