No abstract

In this paper, we study the probability distribution of the observable , with 1 ⩽ N′ ⩽ N and x 1 < x 2 <⋯< x N representing the ordered positions of N particles in a 1D one-component plasma, i.e. N harmonically confined charges on a line, with pairwise repulsive 1D Coulomb interaction |x i − x j |. This observable represents an example of a truncated linear statistics—here proportional to the center of mass of the N′ = κN (with 0 < κ ⩽ 1), rightmost particles. It interpolates between the position of the rightmost particle (in the limit κ → 0) and the full center of mass (in the limit κ → 1). We show that, for large N, s fluctuates around its mean ⟨s⟩ and the typical fluctuations are Gaussian, of width O(N −3/2). The atypical large fluctuations of s, for fixed κ, are instead described by a large deviation form , where the rate function ϕ κ (s) is computed analytically. We show that ϕ κ (s) takes different functional forms in five distinct regions in the (κ, s) plane separated by phase boundaries, thus leading to a rich phase diagram in the (κ, s) plane. Across all the phase boundaries the rate function ϕ κ (s) undergoes a third-order phase transition. This rate function is also evaluated numerically using a sophisticated importance sampling method, and we find a perfect agreement with our analytical predictions.

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.

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 $s = (1/N) \sum_{i=1}^N f(x_i)$ 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 ${\rm Var}(s) \approx b/N^3$ for large $N$. In this paper, we provide an exact compact formula for the constant $b = 1/(4\alpha) \int_{-2 \alpha}^{2\alpha} [f'(x)]^2\, dx$. In addition, we also calculate the full large deviation function characterising the tails of the full distribution ${\cal P}(s,N)$ for several different examples of $f(x)$. Our analytical predictions are confirmed by numerical simulations.