We review what is known, unknown, and expected about the mathematical properties of Coulomb and Riesz gases. Those describe infinite configurations of points in 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.
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For a smooth domain [for instance, satisfying (63)], there are on the order of ℓd−1R points located at a distance R from the boundary. Those see a bounded potential. For the N + o(N) particles inside, at a distance , the potential induced by the particles outside is of the order O(Rd−s) by Lemma 9. Thus, the energy shift is of order ℓd−1R + ℓdRd−s. After optimizing over R, this provides the claimed .
512.
In the physics literature, the name “jellium” is often employed for electrons (which are quantum with spin), whereas the “one-component plasma” is mainly used for classical particles as considered in the present article.
514.
The particles have no spin and one should use the magnetic Laplacian. This is also equivalent to a Fermi system in a harmonic trap rotating at the largest possible speed.307
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