The dissipation in the relaxation of an ensemble of conservative stochastic systems toward 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. , 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.