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 ${\mathbb{R}}^d$ interacting with the Riesz potential $\pm |x{|}^{-s}$ (resp. $-\log |x|$ 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 the article 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 $s>d$. In the last part we discuss phase transitions and mention what is expected on physical grounds.

Final version to appear in J. Math. Phys. in the special collection of papers honoring Freeman Dyson

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Cited by in corpus (11)

• Dynamical fluctuations in the Riesz gas
• Improved Lieb-Oxford bound on the indirect and exchange energies
• Gap probability and full counting statistics in the one dimensional one-component plasma
• Partition functions of determinantal and Pfaffian Coulomb gases with radially symmetric potentials
• An exact formula for the variance of linear statistics in the one-dimensional jellium mode
• Three-dimensional lattice ground states for Riesz and Lennard-Jones type energies
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• Dirac cones for a mean-field model of graphene
• Classical Density Functional Theory: Representability and Universal Bounds
• Grand-Canonical Optimal Transport
• Homogeneous Electron Liquid in Arbitrary Dimensions: Exchange and Correlation Using the Singwi-Tosi-Land-Sjölander Approach