Modeling of subdiffusion in space-time-dependent force fields beyond the fractional Fokker-Planck equation

Aleksander Weron, Marcin Magdziarz, and Karina Weron

Phys. Rev. E 77, 036704 – Published 20 March 2008

Abstract

In this paper we attack the challenging problem of modeling subdiffusion with an arbitrary space-time-dependent driving. Our method is based on a combination of the Langevin-type dynamics with subordination techniques. For the case of a purely time-dependent force, we recover the death of linear response and field-induced dispersion—two significant physical properties well-known from the studies based on the fractional Fokker-Planck equation. However, our approach allows us to study subdiffusive dynamics without referring to this equation.

Received 3 December 2007

DOI:https://doi.org/10.1103/PhysRevE.77.036704

Authors & Affiliations

Aleksander Weron^* and Marcin Magdziarz^†

Hugo Steinhaus Center, Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wybrzeze Wyspianskiego 27, 50-370 Wroclaw, Poland

Karina Weron^‡

Institute of Physics, Wroclaw University of Technology, Wybrzeze Wyspianskiego 27, 50-370 Wroclaw, Poland

^*aleksander.weron@pwr.wroc.pl
^†marcin.magdziarz@pwr.wroc.pl
^‡karina.weron@pwr.wroc.pl

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 77, Iss. 3 — March 2008

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
(Color online) An exemplary sample path of the standard Brownian particle biased by the dichotomously alternating force (6). After each time unit the sign of the force changes, switching the motion of the particle. The particle either oscillates around zero or moves left/right. Note that only in the case of standard Brownian diffusion $τ = t$ . The parameters are $c = 10^{2}$ and $η = K = 1$ .Reuse & Permissions
Figure 2
(Color online) An exemplary sample path of the process $\tilde{Y} (t) = \tilde{X} (S_{t})$ with the dichotomously alternating force (6). The constant intervals of the trajectory are typical for subdiffusion. However, the shape of the trajectory indicates that the force $F$ varies in random time, which is physically not acceptable. We observe the changes of particle motion in some random time points and not after each time unit, cf. Fig. 1. This is the consequence of the subordination procedure. The parameters are $α = 0.9$ , $c = 10^{2}$ , and $η = K = 1$ .Reuse & Permissions
Figure 3
(Color online) An exemplary trajectory of the process $\hat{Y} (t) = \hat{X} (S_{t})$ describing subdiffusion in a space-time-dependent potential with the dichotomously alternating force $F (x, t)$ given by (6). The shape of the trajectory confirms that the force $F$ varies in the real time $t$ , since, by (4), $U (S_{t}) = t$ . Note that the change of the particle motion is observed after each time unit, cf. Figs. 1, 2. The constant intervals of the trajectory are typical for subdiffusion and represent the heavy-tailed rests of the test particle. The parameters are $α = 0.9$ , $c = 10^{2}$ , and $η = K = 1$ .Reuse & Permissions
Figure 4
(Color online) Quantile lines (10%, 20%, …, 90%) corresponding to the process $\tilde{Y} (t) = \tilde{X} (S_{t})$ in the presence of the dichotomously alternating force (6). The lowest line is the 10%-quantile line, the second line from the bottom corresponds to the 20%-quantile line, etc. The shape of the lines indicates that the force does not switch after each time unit, thus $F$ does not vary in the real time $t$ . This is the consequence of the random time change after subordination. The parameters as in Fig. 2. The quantile lines were estimated by Monte-Carlo methods [4, 5].Reuse & Permissions
Figure 5
(Color online) Quantile lines (10%, 20%, …, 90%) corresponding to the process $\hat{Y} (t) = \hat{X} (S_{t})$ in the presence of the dichotomously alternating force (6). The lowest line is the 10%-quantile line, the second line from the bottom corresponds to the 20%-quantile line, etc. The shape of the lines confirms that the force $F$ varies in the real time $t$ . The change of their shape is observed after each time unit, cf. Fig. 4. The parameters as in Fig. 3. The quantile lines were estimated by Monte Carlo methods [4, 5].Reuse & Permissions
Figure 6
(Color online) The first two moments of the process $\hat{Y} (t) = \hat{X} (S_{t})$ with periodic rectangular driving force, evaluated using expressions (14, 15). In the top panel the mean particle position stagnates. This effect is termed “death of linear response” [2]. The linear growth of ${⟨ {\hat{X}}^{2} (S_{t}) ⟩}^{1 ∕ α}$ in the bottom panel means that the asymptotic growth of the field-dependent second moment is given by $t^{α}$ . This is the manifestation of the effect called “field-induced dispersion” [2, 3]. The parameters are $α = 0.4$ , $η = f_{0} = 1$ , and $K = 1 ∕ 2$ .Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics