  We consider Fokker-Planck equations with tilted periodic potential in the
subcritical regime and characterize the spatio-temporal dynamics of the partial
masses in the limit of vanishing diffusion. Our convergence proof relies on
suitably defined substitute masses and bounds the approximation error using the
energy-dissipation relation of the underlying Wasserstein gradient structure.
In the appendix we also discuss the case of an asymmetric double-well potential
and derive the corresponding limit dynamics in an elementary way.
