Statistical density of particles in one dimensional interaction and Jellium Model
Mohamed Bouali
TL;DR
This work analyzes a one-dimensional gas of $n$ charged particles with long-range interactions described by a generalized Riesz potential and an external confining potential $Q$, formulating a mean-field energy $E(\mu)$ and proving the existence and uniqueness of an equilibrium measure $\mu^*_\Sigma$ with compact support. It develops a generalized repulsion framework based on negative-definite kernels, deriving explicit equilibrium densities across regimes (including Ullman, Wigner, and generalized Marchenko–Pastur cases) and studying the positive-half-line and barrier variants. The paper then applies potential-theoretic and large-deviation techniques to show convergence of empirical measures to $\mu^*_{\Sigma,k}$ and to characterize the asymptotics of the partition function and extreme-value statistics, such as the distribution of the rightmost particle $x_{\max}$, which exhibits non-TW fluctuations. Overall, the results provide a unified, solvable framework linking Riesz gases, random-matrix-type densities, and extreme-value statistics with explicit closed-form densities and rate functions across diverse parameter choices.
Abstract
We study a one-dimensional gas of $n$ charged particles confined by a potential and interacting through the Riesz potential or a more general potential. In equilibrium, and for symmetric potential the particles arrange themselves symmetrically around the origin within a finite region. Various models will be studied by modifying both the confining potential and the interaction potential. Focusing on the statistical properties of the system, we analyze the position of the rightmost particle, $x_{\text{max}}$, and show that its typical fluctuations are described by a limiting distribution different from the Tracy-Widom distribution found in the one-dimensional log-gas. We also derive the large deviation functions governing the atypical fluctuations of $x_{\text{max}}$ far from its mean.