Out of equilibrium dynamics of repulsive ranked diffusions: the expanding crystal

TL;DR

This paper analyzes the non-equilibrium dynamics of repulsive ranked diffusions, revealing an exact solution for particle distributions, convergence to an expanding crystal, and detailed fluctuation analysis using advanced mathematical models.

Contribution

It provides an exact formula for particle distributions in a non-equilibrium setting and links the dynamics to an expanding crystal analogy, including fluctuation and correlation analysis.

Findings

System converges to a linearly expanding crystal at large time

Fluctuations around the crystal are Gaussian to leading order

Deviations from Gaussian behavior show long-range correlations

Abstract

We study the non-equilibrium Langevin dynamics of $N$ particles in one dimension with Coulomb repulsive linear interactions. This is a dynamical version of the so-called jellium model (without confinement) also known as ranked diffusion. Using a mapping to the Lieb-Liniger model of quantum bosons, we obtain an exact formula for the joint distribution of the positions of the $N$ particles at time $t$, all starting from the origin. A saddle point analysis shows that the system converges at large time to a linearly expanding crystal. Properly rescaled, this dynamical state resembles the equilibrium crystal in a time dependent effective quadratic potential. This analogy allows to study the fluctuations around the perfect crystal, which, to leading order, are Gaussian. There are however deviations from this Gaussian behavior, which embody long-range correlations of purely dynamical origin, characterized by the higher order cumulants of, e.g., the gaps between the particles, that we calculate exactly. We complement these results using a recent approach by one of us in terms of a noisy Burgers equation. In the large $N$ limit, the mean density of the gas can be obtained at any time from the solution of a deterministic viscous Burgers equation. This approach provides a quantitative description of the dense regime at shorter times. Our predictions are in good agreement with numerical simulations for finite and large $N$.