Gap probability and full counting statistics in the one dimensional one-component plasma

TL;DR

This paper analyzes the distribution of gaps between particles and the number of particles in a fixed interval in a 1D one-component plasma, providing explicit scaling functions and large deviation rate functions for large particle numbers.

Contribution

It introduces explicit scaling functions and large deviation rate functions for gap and particle number distributions in the 1D OCP, including their asymptotic behaviors.

Findings

Scaling form for typical gap fluctuations with explicit function

Scaling form for full counting statistics with explicit function

Explicit large deviation rate functions for both observables

Abstract

We consider the $1d$ one-component plasma (OCP) in thermal equilibrium, consisting of $N$ equally charged particles on a line, with pairwise Coulomb repulsion and confined by an external harmonic potential. We study two observables: (i) the distribution of the gap between two consecutive particles in the bulk and (ii) the distribution of the number of particles $N_I$ in a fixed interval $I=[-L,+L]$ inside the bulk, the so-called full-counting-statistics (FCS). For both observables, we compute, for large $N$, the distribution of the typical as well as atypical large fluctuations. We show that the distribution of the typical fluctuations of the gap are described by the scaling form ${\cal P}_{\rm gap, bulk}(g,N) \sim N H_\alpha(g\,N)$, where $\alpha$ is the interaction coupling and the scaling function $H_\alpha(z)$ is computed explicitly. It has a faster than Gaussian tail for large $z$: $H_\alpha(z) \sim e^{-z^3/(96 \alpha)}$ as $z \to \infty$. Similarly, for the FCS, we show that the distribution of the typical fluctuations of $N_I$ is described by the scaling form ${\cal P}_{\rm FCS}(N_I,N) \sim 2\alpha \, U_\alpha[2 \alpha(N_I - \bar{N}_I)]$, where $\bar{N}_I = L\,N/(2 \alpha)$ is the average value of $N_I$ and the scaling function $U_\alpha(z)$ is obtained explicitly. For both observables, we show that the probability of large fluctuations are described by large deviations forms with respective rate functions that we compute explicitly. Our numerical Monte-Carlo simulations are in good agreement with our analytical predictions.