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