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

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

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:

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

where:

• $\mathrm{\Lambda }$ 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)

• ${(R^{*})}^{3N}$ 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