Ranked diffusion, delta Bose gas and Burgers equation

We study the diffusion of $N$ particles in one dimension interacting via a drift proportional to their rank. In the attractive case (self-gravitating gas) a mapping to the Lieb Liniger quantum model allows to obtain stationary time correlations, return probabilities and the decay rate to the stationary state. The rank field obeys a Burgers equation, which we analyze. It allows to obtain the stationary density at large $N$ in an external potential $V(x)$ (in the repulsive case). In the attractive case the decay rate to the steady state is found to depend on the initial condition if its spatial decay is slow enough. Coulomb gas methods allow to study the final equilibrium at large $N$.