We are going present a multi-scale analysis to derive log-Sobolev inequalities for interacting particle systems. This leads to a generalised Bakry-Emery criterion which can be applied to non-convex potentials. The approach relies on a Hamilton-Jacobi equation know as Polchinski equation. If time allows, we will explain how this multi-scale criterion allows to build Lipschitz transport maps. This talk is based on the survey arXiv:2307.07619 in collaboration with R. Bauerschmidt and B. Dagallier.