Fractional Fokker-Planck dynamics: Numerical algorithm and simulations

Anomalous transport in a tilted periodic potential is investigated numerically within the framework of the fractional Fokker-Planck dynamics via the underlying continuous-time random walk. An efficient numerical algorithm is developed which is applicable for an arbitrary potential. This algorithm is then applied to investigate the fractional current and the corresponding nonlinear mobility in different washboard potentials. Normal and fractional diffusion are compared through their time evolution of the probability density in state space. Moreover, we discuss the stationary probability density of the fractional current values.

Figure 1

(Color online) Sketch of the numerical algorithm: after a random waiting time $\tau$ the particle jumps from the current position $x^{\left(n\right)}$ to the position $x^{\left(n\right)}+\mathrm{\Delta }x$ or $x^{\left(n\right)}-\mathrm{\Delta }x$. The process is reiterated until $t^{\left(n\right)}⩾t_{\mathrm{final}}$. The numerical measurements are performed after constant time intervals at times $t_m^{*}$. The solid circles represent the events that are used for the computation of the physical quantities (see text for an explanation).

Figure 2

(Color online) Rescaled mobility ${\mu }_{\alpha }\left(F\right){\eta }_{\alpha }$ for $F∕F_{\mathrm{cr}}=1$, as a function of $\alpha$. After the same rescaled simulation time $t\approx 200$ (see Sec. 3 for the definition of the rescaled units), the distance from the asymptotic limit predicted by the Stratonovich formula (solid line) of the mobility values corresponding to the Pareto residence time density (squares) increases significantly for $\alpha >0.9$, with respect to those of the Mittag-Leffler residence time density (circles).

Figure 3

(Color online). Dimensionless subcurrent $v_{\alpha }\left(F\right)∕(F_{\mathrm{cr}}∕{\eta }_{\alpha })$ and nonlinear mobility ${\mu }_{\alpha }\left(F\right){\eta }_{\alpha }$ for the case of the cosine substrate potential, cf. Eq. (20) (depicted in the inset) vs $F∕F_{\mathrm{cr}}$. Numerical values corresponding to different temperatures $T$ and fractional exponents $\alpha ∊[0.1,1]$ (symbols) fit the analytic predictions from Eq. (17) (solid lines).

Figure 4

(Color online) Same as in Fig. 3, for the double-hump potential, cf. Eq. (21).

Figure 5

(Color online). Same as in Fig. 3, for the ratchet potential, cf. Eq. (22). Here also negative tilting is studied.

Figure 6

Comparison between the time evolution of the probability density $P(x,t)$ in the case of normal diffusion (above) and anomalous diffusion with $\alpha =0.5$ (below) in a tilted cosine potential at various evolution times $t$. This setup has been calculated for the following parameter set: $T=0.1$, $F∕F_{\mathrm{cr}}=1$, and a cosine potential. Dotted lines at $t=0$ represent the initial conditions $P(x,0)=\delta \left(x\right)$; i.e., all particles start out from the same position $x=0$.

Figure 7

(Color online). Normalized theoretical stationary, reduced probability density ${\widehat{P}}_{\mathrm{st}}\left(x\right)$ for $F∕F_{\mathrm{cr}}=1$, $T=0.1$, and $\alpha =0.5$, computed from Eqs. (25, A9) (solid line) and corresponding numerical data (circles) at time $t=5$: left $y$ axis. Also the underlying potential $U\left(x\right)=\cos \left(x\right)-Fx$ is depicted: right $y$ axis.

Figure 8

(Color online) The time evolution for normal (above) and anomalous (below) diffusion of the reduced probability density $\widehat{P}(x,t)$ within the first period $x∊[0,L)$ defined according to Eq. (24). In this example, the potential is $U\left(x\right)=\cos \left(x\right)-Fx$, with $F∕F_{\mathrm{cr}}=1$, the temperature is $T=0.1$, and the anomalous diffusion process corresponds to $\alpha =0.5$. The curve labels 1, 2, and 3 correspond to increasing values of time: ${10}^{-3}$, $3\cdot {10}^{-3}$ and $9\cdot {10}^{-3}$ for the normal process, and ${10}^{-4}$ and $1.5\cdot {10}^{-3}$ and ${10}^{-2}$ for the anomalous case. The solid lines (labelled as 4 to indicate the time sequence) represent the theoretical stationary solution defined by Eq. (25) and practically coincide with the asymptotic numerical density.

Figure 9

(Color online) Anomalous (dashed line) and normal (solid line) probability densities of the (sub)velocity, computed according to Eq. (26) at $t=1000$ in rescaled time units. The potential, tilt, temperature, and $\alpha$ value are the same as in Fig. 8. The arrows point to the corresponding $y$ axes. Despite the very different shapes, the two probability densities possess the same average value as given by the (fractional) Stratonovich formula (17), indicated with the vertical dotted line.