Coulomb and Riesz gases: The known and the unknown

Abstract

We review what is known, unknown, and expected about the mathematical properties of Coulomb and Riesz gases. Those describe infinite configurations of points in [Formula: see text] interacting with the Riesz potential ±| x|− s (respectively, −log | x| for s = 0). Our presentation follows the standard point of view of statistical mechanics, but we also mention how these systems arise in other important situations (e.g., in random matrix theory). The main question addressed in this Review is how to properly define the associated infinite point process and characterize it using some (renormalized) equilibrium equation. This is largely open in the long range case s < d. For the convenience of the reader, we give the detail of what is known in the short range case s > d. Finally, we discuss phase transitions and mention what is expected on physical grounds.

Three‐dimensional lattice ground states for Riesz and Lennard‐Jones–type energies

“…For instance, in the Coulomb case $$s=d-2$$ (or $$f(r)=-\log r$$ in dimension $$d=2$$), the Wigner Conjecture for Jellium 24 states that electrons embedded in an uniform background of positive charges must crystallize on a triangular and a body‐centered‐cubic (BCC) lattice in dimensions 2 and 3, respectively (see also Refs. 2, 25). This is also called the Abrikosov Conjecture 26 or Vortices Conjecture 27 in the two‐dimensional setting related to the vortices in the Ginzburg‐Landau theory of superconductors of type II.…”

“…An equivalent problem can be stated in higher dimension and for general Riesz interaction. 21,25,[28][29][30][31][32] The general Riesz energy and its related minimization problem also appear in the theory of random point configurations 33 as well as approximation theory 34 and number theory. 23,35,36 On the other hand, Lennard-Jones potentials have been introduced by Mie 37 and popularized by Jones 38 in its classical (𝑛, 𝑚) = (12,6) form, that is, when the interaction is of Van der Waals type ∼𝑏𝑟 −6 for large 𝑟, initially for studying gas argon.…”

Improved Lieb–Oxford bound on the indirect and exchange energies

“…where ζ is the Epstein Zeta function of this lattice [BL15,Lew22]. The constraint that the UEG must have a uniform density is not so easy to handle and the upper bound…”

Partition functions of determinantal and Pfaffian Coulomb gases with radially symmetric potentials

“…Here, Q : C → R is called the confining/external potential that satisfies suitable potential theoretic conditions. We refer to [28,44,49] and references therein for recent developments of two-dimensional Coulomb gases. Contrary to (1.1), the configurational canonical Coulomb gas ensemble in the upper-half plane [39] (cf.…”