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

We investigate the full counting statistics 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 N(W,N) . We analyze the probability distribution of N(W,N) and show that it exhibits a large deviation form for large N characterized by a speed N3k+2/k+2 and by a large deviation function (LDF) of the fraction c=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 LDF that has discontinuous third derivatives. Monte-Carlo simulations based on Metropolis-Hashtings (MH) algorithm show good agreement with our analytical expressions for the corresponding density profiles. We find that the typical fluctuations of 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 (-infinity,W]) , linear statistics (the variance), thermodynamic pressure and bulk modulus.