Current and universal scaling in anomalous transport

Anomalous transport in tilted periodic potentials is investigated within the framework of the fractional Fokker-Planck dynamics and the underlying continuous time random walk. The analytical solution for the stationary, anomalous current is obtained in closed form. We derive a universal scaling law for anomalous diffusion occurring in tilted periodic potentials. This scaling relation is corroborated with precise numerical studies covering wide parameter regimes and different shapes for the periodic potential, being either symmetric or ratchetlike.

Figure 1

(Color online) The scaled nonlinear mobility $k_{B}T{\mu }_{\alpha }\left(F\right)∕{\kappa }_{\alpha }$ is depicted for the case of a tilted cosine potential versus $F∕F_{\mathrm{cr}}$. The numerics for different temperatures $T$ and different $\alpha$ values, varying between 0.1–1, fits the analytic prediction (16) (continuous lines) within the statistical errors.

Figure 2

(Color online) Universal scaling: Asymptotic values of the ratio $⟨\delta x^2\left(t\right)⟩∕{⟨x\left(t\right)⟩}^2$ as a function of the parameter $\alpha$ for anomalous diffusion. All the points corresponding to the same $\alpha$ but different values of $F$ and $T$ match the function given in Eq. (13) (solid line) within the statistical errors. We use three different temperatures: $T=0.01$ with the bias $F$ ranging between 0.9–2.0; correspondingly, $T=0.1$, $F=0.7–2.0$, and $T=0.5$, $F=0.4–2.0$. The open triangles correspond to the cosine potential $U_1\left(x\right)$, the filled squares to the double-hump potential $U_2\left(x\right)$, and the open circles to the asymmetric ratchet potential $U_3\left(x\right)$; see the text.