Suppose the potential energy of a gas of point charges with positions and free to move on the infinite line , is given by
The probability density of the positions when the gas is in thermodynamic equilibrium is:
where is the Boltzmann constant, the temperature and a constant. Then the partition function (with ) is given by
For charges free to move on a circular wire of radius ,
and the partition function is given by
For further information see Mehta (2004).