Suppose the potential energy of a gas of $n$ point charges with positions ${x}_{1},{x}_{2},\mathrm{\dots},{x}_{n}$ and free to move on the infinite line $$, is given by

5.20.1 | $$ | ||

The probability density of the positions when the gas is in thermodynamic equilibrium is:

5.20.2 | $$P({x}_{1},\mathrm{\dots},{x}_{n})=C\mathrm{exp}\left(-W/(kT)\right),$$ | ||

where $k$ is the Boltzmann constant, $T$ the temperature and $C$ a constant. Then the partition function (with $\beta =1/(kT)$) is given by

5.20.3 | $$\begin{array}{ll}{\psi}_{n}(\beta )& ={\int}_{{\mathbb{R}}^{n}}{\mathrm{e}}^{-\beta W}dx\\ & ={(2\pi )}^{n/2}{\beta}^{-(n/2)-(\beta n(n-1)/4)}{\left(\mathrm{\Gamma}\left(1+\frac{1}{2}\beta \right)\right)}^{-n}\prod _{j=1}^{n}\mathrm{\Gamma}\left(1+\frac{1}{2}j\beta \right).\end{array}$$ | ||

See (5.14.6).

For $n$ charges free to move on a circular wire of radius $1$,

5.20.4 | $$ | ||

and the partition function is given by

5.20.5 | $${\psi}_{n}(\beta )=\frac{1}{{(2\pi )}^{n}}{\int}_{{[-\pi ,\pi ]}^{n}}{\mathrm{e}}^{-\beta W}d{\theta}_{1}\mathrm{\cdots}d{\theta}_{n}=\mathrm{\Gamma}\left(1+\frac{1}{2}n\beta \right){(\mathrm{\Gamma}\left(1+\frac{1}{2}\beta \right))}^{-n}.$$ | ||

See (5.14.7).

For further information see Mehta (2004).