  We study the Fokker-Planck equation as the hydrodynamic limit of a stochastic
particle system on one hand and as a Wasserstein gradient flow on the other. We
write the rate functional, that characterizes the large deviations from the
hydrodynamic limit, in a way that the free energy appears explicitly. Next we
use this formulation via the contraction principle to prove that the discreet
time rate functional is asymptotically equivalent in the Gamma-convergence
sense to the functional derived from the Wasserstein gradient discretization
scheme.
