ABSTRACT: We analyze the evolution of the free energy of interacting gases along an optimal path for transporting at minimal cost one arbitrary probabibilty density into another. This leads to an inequality between the relative total energy -- internal, potential and interactive -- of two arbitrary probability densities, their Wasserstein distance, their barycenters and their entropy production functional.

This inequality is remarkably encompassing as it implies most known geometrical -- Gaussian and Euclidean -- inequalities as well as new ones, while allowing a direct and unified way for computing best constants and extremals: Sobolev, Gagliardo-Nirenberg, Log Sobolev, HWI, Transport, Concentration, Poincare, etc... As expected, such inequalities also lead to exponential rates of convergence to equilibria for solutions of Fokker-Planck and McKean-Vlasov type equations. The proposed inequality also leads to a remarkable correspondence between ground state solutions of certain quasilinear (or semi-linear) equations and stationary solutions of (non-linear) Fokker-Planck type equations.