Define the function (
Coulomb kernel, or
Riesz kernel)\begin{align} g_s(x) = \begin{cases}-\log|x| & \text{ if } s = 0, \\ \frac{1}{s|x|^{s}} & \text{ if } s \neq 0 \end{cases} \end{align}The setup starts with considering N charged particles in \mathbb{R}^d with positions \mathbf{r}_i and charges q_i. From
electrostatics, the pairwise potential energy between particles labelled by indices i,j is (up to scale factor) V_{ij} = q_iq_jg_{d-2}(|\mathbf{r}_i - \mathbf{r}_j|),where g_{d-2}(x) is the
Coulomb kernel or
Green's function of the
Laplace equation in d dimensions. The
free energy due to these interactions is then (proportional to) F = \sum_{i \neq j} V_{ij}, and the
partition function is given by integrating over different configurations, that is, the positions of the charged particles. More generally, any choice of s \in [d-2, d) makes sense. This general case is called
Riesz gas, of which the Coulomb gas is a special case. The naming comes from the fact that the Riesz kernel is the Green's function of the
fractional Laplacian, which can be defined using the
Riesz potential. Specifically,(-\Delta)^{\frac{d-s}2} g_s = c_{d, s} \delta_0where c_{d, s} = \begin{cases}\frac{2^{d-s} \pi^{d / 2} \Gamma\left(\frac{d-s}{2}\right)}{\Gamma\left(\frac{s}{2}\right)} & \text { for } s>\max (0, d-2) \\ \frac{2 \pi^{d / 2}}{\Gamma\left(\frac{d}{2}\right)}=\left|\mathbb{S}^{d-1}\right| & \text { if } s=d-2>0 \\ 2 \pi & \text { if } s=0, d=1 \text { or } d=2\end{cases} == Names ==