ALEX 2018 Workshop: Abstracts

Mass transport in Fokker-Planck equations with tilted periodic potential

Michael Herrmann${}^{\mathrm{(}\mathrm{1}\mathrm{)}}$ and Barbara Niethammer${}^{\mathrm{(}\mathrm{2}\mathrm{)}}$,

(1) Technische Universität Carolo-Wilhelmina zu Braunschweig (Germany)

(2) Hausdorff Center for Mathematics, Universität Bonn (Germany)

We consider the Fokker-Planck equation

$$\tau {\partial }_t\rho ={\nu }^2{\partial }_p^2\rho +{\partial }_p\left((H^{\prime }(p)-\sigma )\rho \right),$$ (1)

with small parameters $\tau$ and $\nu$, where $p$ denotes an internal scalar state variable. It describes the evolution of the probability density $\rho =\rho (t,p)$ of a particle that undergoes a random walk under the influence of the potential $H$ and a force term $\sigma$. Here we are interested in the case that $H$ is smooth and periodic, and $\sigma$ is fixed, such that the effective potential $H_{\mathrm{eff}}(p)=H(p)-\sigma$ is tilted, but still has local minima that represent metastable traps for the particles. Our goal is to derive a simple equation for the dynamics in the limit of vanishing $\nu$ and (appropriately chosen) $\tau$.

Since $\nu$ is small, $\rho$ develops narrow peaks located at the local minima of $H_{\mathrm{eff}}$, but since $\nu >0$ the peaks still exchange mass on the Kramers time scale $c_1\exp \left(-\frac{c_2}{{\nu }^2}\right)$, where $c_1$ and $c_2$ depend on $H$ and $\sigma$. We present a simple approach [HN18] how to derive rigorously in the limit $\nu \to 0$, with $\tau =c_1\exp \left(-\frac{c_2}{{\nu }^2}\right)$, the effective limit dynamics for the mass exchange between the local wells.

Our result is closely related to, and also applies to, the case of potentials with two wells that have been studied in [AMP, HN11, PSV10] for symmetric potentials. One advantage of our approach is that it also applies to the case of asymmetric energy landscapes.

Acknowledgments: The authors are partially supported by the DFG through the CRC 1060 The mathematics of emergent effects.