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Higher-dimensional Coulomb gases and renormalized energy functionals. (English) Zbl 1338.82043
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 \[ H_n(x_1,\ldots,x_n)=\sum_{i\neq j}w(x_i-x_j)+n\sum_{i=1}^nV(x_i) \] where \(x_1,x_2,\ldots,x_n\) is the positions of the particles \[ w(x)=\begin{cases} \frac{1}{|x|^{d-2}} &\text{ if }d\geq 3 \\ -\log|x| &\text{ if }d = 2, \end{cases} \] 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.

MSC:
82C40 Kinetic theory of gases in time-dependent statistical mechanics
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