arxiv:2407.21194v1 [math-ph] 30 Jul 2024 Page 1 Lectures on Coulomb and Riesz Gases
Sylvia Serfaty arxiv:2407.21194v1 [math-ph] 30 Jul 2024 Page 2 Page 3 Contents Preface 5 …

We prove that for α≥ 1, among 2d unit density lattices, min L∑ P∈ L (| P| 2− β) e− π α| P| 2
is achieved at hexagonal lattice for β≤ 1 2 π α and does not exist for β> 1 2 π α⁠. Here the …

We consider a two-dimensional analogue of Jacobi theta functions and prove that, among
all lattices $\Lambda\subset\mathbb {R}^ 2$ with fixed density, the minimal value is …

We consider two-dimensional zero-temperature systems of N particles to which we
associate an energy of the form EV (X):= ∑ _ 1\leqq i< j\leqq NV (| X (i)-X (j)|), EV (X):=∑ 1≦ …

We investigate the local and global optimality of the triangular, square, simple cubic, face-
centered-cubic (fcc) and body-centered-cubic (bcc) lattices and the hexagonal-close …

We present two families of lattice theta functions accompanying the family of lattice theta
functions studied by Montgomery in [H. Montgomery, Minimal theta functions. Glasgow …

Consider the energy per particle on the lattice given by $\min_ {\Lambda}\sum_ {\mathbb
{P}\in\Lambda}\left|\mathbb {P}\right|^ 4 e^{-\pi\alpha\left|\mathbb {P}\right|^ 2} $, where …

We consider pairwise interaction energies and we investigate their minimizers among
lattices with prescribed minimal vectors (length and coordination number), ie the one …

The Riesz potential fs (r)= r− s f_s(r)=r^-s is known to be an important building block of many
interactions, including Lennard‐Jones–type potentials fn, m LJ (r):= ar− n− br− m …

Let $\zeta (s, z)=\sum_ {(m, n)\in\mathbb {Z}^ 2\backslash\{0\}}\frac {(\Im (z))^ s}{| mz+
n|^{2s}} $ be the Eisenstein series/Epstein Zeta function. Motivated by widely used Lennard …