Gap probability and full counting statistics in the one-dimensional one-component plasma

Abstract

Abstract We consider the 1d one-component plasma in thermal equilibrium, consisting of N equally charged particles on a line, with pairwise Coulomb repulsion and confined by an external harmonic potential. We study two observables: (i) the distribution of the gap between two consecutive particles in the bulk and (ii) the distribution of the number of particles N I in a fixed interval I = [−L, +L] inside the bulk, the so-called full-counting-statistics (FCS). For both observables, we compute, for large N, the distribution of the typical as well as atypical large fluctuations. We show that the distribution of the typical fluctuations of the gap g is described by the scaling form P gap,bulk ( g , N ) ∼ N H α ( g N ) , where α is the interaction coupling and the scaling function H α (z) is computed explicitly. It has a faster than Gaussian tail for large z: H α ( z ) ∼ e − z 3 / ( 96 α ) as z → ∞. Similarly, for the FCS, we show that the distribution of the typical fluctuations of N I is described by the scaling form P FCS ( N I , N ) ∼ 2 α U α [ 2 α ( N I − N ¯ I ) ] , where N ¯ I = L N / ( 2 α ) is the average value of N I and the scaling function U α (z) is obtained explicitly. For both observables, we show that the probability of large fluctuations is described by large deviations forms with respective rate functions that we compute explicitly. Our numerical Monte-Carlo simulations are in good agreement with our analytical predictions.