Full counting statistics of 1d short-range Riesz gases in confinement

Abstract

We investigate the full counting statistics (FCS) of a harmonically confined 1d short-range Riesz gas consisting of $N$ particles in equilibrium at finite temperature. The particles interact with each other through a repulsive power-law interaction with an exponent $k>1$ which includes the Calogero-Moser model for $k=2$. We examine the probability distribution of the number of particles in a finite domain $[-W,W]$ called number distribution, denoted by $\mathcal{N}(W,N)$. We analyze the probability distribution of $\mathcal{N}(W,N)$ and show that it exhibits a large deviation form for large $N$ characterised by a speed $N^{\frac{3k+2}{k+2}}$ and by a large deviation function of the fraction $c=\mathcal{N}(W,N)/N$ of the particles inside the domain and $W$. We show that the density profiles that create the large deviations display interesting shape transitions as one varies $c$ and $W$. This is manifested by a third-order phase transition exhibited by the large deviation function that has discontinuous third derivatives. Monte-Carlo (MC) simulations show good agreement with our analytical expressions for the corresponding density profiles. We find that the typical fluctuations of $\mathcal{N}(W,N)$, obtained from our field theoretic calculations are Gaussian distributed with a variance that scales as $N^{{\nu }_k}$, with ${\nu }_k=(2-k)/(2+k)$. We also present some numerical findings on the mean and the variance. Furthermore, we adapt our formalism to study the index distribution (where the domain is semi-infinite $(-\infty ,W])$, linear statistics (the variance), thermodynamic pressure and bulk modulus.

36 pages, 7 figures

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