where g ( x ) is the Coulomb kernel or Green's function of the Laplace equation in d dimensions, so g ( x ) = { − − log ⁡ ⁡ | x | if d = 2 , 1 ( d − − 2 ) | x | d − − 2 if d > 2. The free energy due to these interactions is then (proportional to) F = ∑ ∑ i ≠ ≠ j V i j , and the partition function is given by integrating over different configurations, that is, the positions of the charged particles.