Supersymmetry and Fokker-Planck dynamics in periodic potentials

Recently, the Fokker-Planck dynamics of particles in periodic potentials $\pm V$, have been investigated by using the matrix continued fraction method. It was found that the two periodic potentials, one being bistable and the other metastable give the same diffusion coefficient in the overdamped limit. We show that this result naturally follows from the fact that the considered potentials in the corresponding Schr\"{o}dinger equation form supersymmetric partners. We show that these differing potentials ${\pm}V$ also exhibit symmetry in current and diffusion coefficients: $J_{+}(F)=-J_{-}(-F)$ and $D_{+}(F)=D_{-}(-F)$ in the presence of a constant applied force F. Moreover, we show numerically that the transport properties in these potentials are related even in the presence of oscillating drive.