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

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 $s=(1\mathrm{/}N){\sum }_{i=1}^{N}f(x_i)$ 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 $\mathrm{V}\mathrm{a}\mathrm{r}(s)\approx b\mathrm{/}{N}^3$ for large $N$. We provide an exact compact formula for the constant $b=1\mathrm{/}(4\alpha ){\int }_{-2\alpha }^{2\alpha }[f^{\mathrm{\prime }}(x){]}^{2}dx$. In addition, we also calculate the full large deviation function characterising the tails of the full distribution $\mathcal{P}(s,N)$ for several different examples of $f(x)$. Our analytical predictions are confirmed by numerical simulations.