On the classical two-dimensional one-component Coulomb plasma

Abstract

We study some aspects of the equilibrium statistical mechanics of a classical two-dimensional one-component Coulomb plasma (the interaction is logarithmic). The free energy is calculated exactly, for one special value of the temperature. At low temperature, the particles form a triangular lattice, which is stable ; its vibration modes are well-behaved longitudinal plasmons and transverse phonons. The self-consistent harmonic approximation is studied ; it has a solution for the transverse sound velocity at low temperature only, a strong indication that the solid phase is unstable at high temperature. The free énergies obtained in different ways are compared. It is very likely that the model has a solid-fluid phase transition

The crystallization conjecture: a review

“…This is not surprising, since screening effects should make the effective potential be integrable at infinity, but probably not decay faster than 1/r 2 . Numerical simulations indicate that, in dimension d ≥ 2, there exists a critical temperature T c > 0 such that, if T > T c , then Jellium is not crystallized [42,125,191,103,9,64,8,228,69,43,30]. The link between a trapped Coulomb gas in the mean-field limit and the positive-temperature Jellium problem was recently studied in [150,149], similarly to what we discussed in (43).…”

Large Deviations of Radial Statistics in the Two-Dimensional One-Component Plasma

“…Dyson's gas or 2D Coulomb gas [4,17,[23][24][25][26][27]30,38,40,52,60]. The 2D-OCP consists of N identical classical point-like particles, each carrying a charge q (one species of particle) on a two-dimensional domain.…”

“…Recall that, as for the more familiar Coulomb interaction in dimension 3, at electrostatic equilibrium any excess of charge in a conductor is localized on the surface. 4 To our knowledge this idea has been proposed for Hermitian matrix models in [41, Theorem 2.2] generalizing a previous result [7,Theorem 6.2] for the rate function of the largest eigenvalue of GOE matrices. Later, by The presence of one particle outside the unit disk corresponds to a spontaneous symmetry breaking in the system: the problem is symmetric under rotations U (1) in the complex plane, however the plasma configuration with the split-off of a single particle is necessarily no longer symmetric.…”

Particles in a magnetic field and plasma analogies: doubly periodic boundary conditions

“…One notes that the corresponding free energy per unit volume in the thermodynamic limit is precisely that known for the plasma system at Γ = 2 in a disk [1], and also there is no surface free energy in keeping with the system being doubly periodic.…”