An exact formula for the variance of linear statistics in the one-dimensional jellium mode

TL;DR

This paper derives an exact formula for the variance of linear statistics in a one-dimensional jellium model with Coulomb interactions, showing fluctuations are Gaussian with a specific variance scaling.

Contribution

It provides the first exact compact formula for the variance of linear statistics in the 1D jellium model with Coulomb interactions, including large deviation functions.

Findings

Fluctuations are Gaussian with variance ~1/(4αN^3) times an integral of (f')^2.

Derived an exact formula for the variance constant b.

Confirmed analytical results with numerical simulations.

Abstract

We consider the jellium model of $N$ particles on a line confined in an external harmonic potential and with a pairwise one-dimensional Coulomb repulsion of strength $\alpha > 0$. Using a Coulomb gas method, we study the statistics of $s = (1/N) \sum_{i=1}^N f(x_i)$ where $f(x)$, in principle, is an arbitrary smooth function. While the mean of $s$ is easy to compute, the variance is nontrivial due to the long-range Coulomb interactions. In this paper we demonstrate that the fluctuations around this mean are Gaussian with a variance ${\rm Var}(s) \approx b/N^3$ for large $N$. We provide an exact compact formula for the constant $b = 1/(4\alpha) \int_{-2 \alpha}^{2\alpha} [f'(x)]^2\, dx$. In addition, we also calculate the full large deviation function characterising the tails of the full distribution ${\cal P}(s,N)$ for several different examples of $f(x)$. Our analytical predictions are confirmed by numerical simulations.