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

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 $N$.