Higher-dimensional Coulomb gases and renormalized energy functionals (Q2790838)

In this article the authors investigate the equilibrium properties of a classical Coulomb gas: a system of \(n\) classical charged particles living in the full space of dimension \(d\geq 2,\) interacting via Coulomb forces and confined by an external electrostatic potential \(V\). They considered in the mean-field regime where the number \(n\) of particles is large and the pair-interaction strength scales as the inverse of \(n\). The authors study the ground states of the system as well is statistical mechanics when temperature is added. By a suitable splitting of the Hamiltonian NEWLINE\[NEWLINE H_n(x_1,\ldots,x_n)=\sum_{i\neq j}w(x_i-x_j)+n\sum_{i=1}^nV(x_i)NEWLINE\]NEWLINE where \(x_1,x_2,\ldots,x_n\) is the positions of the particles NEWLINE\[NEWLINE w(x)=\begin{cases} \frac{1}{|x|^{d-2}} &\text{ if }d\geq 3 \\ -\log|x| &\text{ if }d = 2, \end{cases}NEWLINE\]NEWLINE is a multiple of the Coulomb potential in dimensions \(d\geq 2,\) the authors extract the next-to-leading-order term in the ground state energy beyond the mean-field limit. Also, is considered the equilibrium properties of the system in the regime \(n\rightarrow \infty,\) that is, on the large-particle-number asymptotic of the ground state and the Gibbs state at given temperature.