Thermodynamic bounds from Otto-Villani functional inequalities

The dissipation in the relaxation of an ensemble of conservative stochastic systems towards the equilibrium steady state is quantified by the free energy difference. Functional inequalities within the framework of [F. Otto and C. Villani, J. Funct. Anal. 173, 361 (2000)] are here revisited which connect the free energy dynamics and optimal transport, offering a geometric perspective on the instantaneous speed of relaxation in the presence of potential barriers. These are illustrated with numerical relaxation experiments on Landau-Ginzburg potentials. The same inequalities hold for relaxation toward nonequilibrium steady states once the potential is replaced by the pseudo-potential defined by the stationary density.