Propagation of chaos and Poincar\'e inequalities for a system of particles interacting through their cdf

In the particular case of a concave flux function, we are interested in the long time behaviour of the nonlinear process associated to the one-dimensional viscous scalar conservation law. We also consider the particle system obtained by remplacing the cumulative distribution function in the drift coefficient of this nonlinear process by the empirical cdf. We first obtain trajectorial propagation of chaos result. Then, Poincar\'e inequalities are used to get explicit estimates concerning the long time behaviour of both the nonlinear process and the particle system.