Mass transport in Fokker-Planck equations with tilted periodic potential

TL;DR

This paper analyzes the mass transport behavior in Fokker-Planck equations with tilted periodic potentials, providing a rigorous limit description as diffusion vanishes, and extends results to asymmetric double-well potentials.

Contribution

It offers a novel convergence proof for the mass dynamics in the subcritical regime using Wasserstein gradient structures and introduces elementary derivations for asymmetric potentials.

Findings

Convergence of partial masses in the vanishing diffusion limit

Characterization of limit dynamics for tilted periodic potentials

Extension to asymmetric double-well potentials

Abstract

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.