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

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