Continuous-time random walks and Fokker-Planck equation in expanding media

F. Le Vot and S. B. Yuste

Phys. Rev. E 98, 042117 – Published 8 October 2018

Abstract

We consider a separable continuous-time random walk model for describing normal as well as anomalous diffusion of particles subjected to an external force when these particles diffuse in a uniformly expanding (or contracting) medium. A general equation that relates the probability distribution function (pdf) of finding a particle at a given position and time to the single-step jump length and waiting time pdfs is provided. The equation takes the form of a generalized Fokker-Planck equation when the jump length pdf of the particle has a finite variance. This generalized equation becomes a fractional Fokker-Planck equation in the case of a heavy-tailed waiting time pdf. These equations allow us to study the relationship between expansion, diffusion, and external force. We establish the conditions under which the dominant contribution to transport stems from the diffusive transport rather than from the drift due to the medium expansion. We find that anomalous diffusion processes under a constant external force in an expanding medium described by means of our continuous-time random walk model violate the generalized Einstein relation and lead to propagators that are qualitatively different from the ones found in a static medium. Our results are supported by numerical simulations.

Received 20 July 2018

DOI:https://doi.org/10.1103/PhysRevE.98.042117

Physics Subject Headings (PhySH)

Anomalous diffusion Brownian motion Continuous-time random walk Diffusion Growth processes Random walks

Statistical Physics & Thermodynamics

Authors & Affiliations

F. Le Vot and S. B. Yuste

Departamento de Física and Instituto de Computación Científica Avanzada (ICCAEx), Universidad de Extremadura, E-06071 Badajoz, Spain

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Vol. 98, Iss. 4 — October 2018

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Images

Figure 1
Propagator $W (x, t)$ for normal diffusive particles under a constant force in a power-law expanding medium with $γ = 2$ and $t_{0} = 10^{3}$ . The jump length distribution $λ^{*} (y)$ and waiting time pdf $φ (t)$ are given by Eqs. (3.5) and (3.8), respectively, with $σ^{2} = 1 / 2$ and $τ = 1$ . The probability of jumping to the right due to the force is $A = 3 / 4$ and, therefore, $v = 1 / \sqrt{2 π}$ . The symbols are simulation results for $t = 2^{8}$ (filled squares) $t = 2^{10}$ (open squares), $t = 2^{12}$ (filled circles), and $t = 2^{16}$ (open circles). The solid lines are the corresponding theoretical results given by Eq. (4.4). The broken line is the limit stationary propagator $W^{\infty} (x)$ .
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Figure 2
Propagator $W (x, t)$ for subdiffusive random walkers ( $α = 1 / 2$ and $D_{α} = 1 / 2$ ) in an exponentially expanding medium ( $t_{H} = 10^{4}$ ) subjected to an external force field ( $A - B = 1 / 2, v_{α} = 1 / \sqrt{2 π}$ ) at times $t = 2^{10}$ (squares) and $t = 2^{14}$ (circles). The lines represent the numerical solution of Eq. (3.16) obtained by means of the fractional Crank-Nicolson method with $Δ x = 0.1$ and $Δ t = 0.1$ for $t = 2^{10}$ , and with $Δ x = 0.1$ and $Δ t = 1$ for $t = 2^{14}$ . The symbols are simulation results for $10^{6}$ realizations where the jump length pdf of the walkers is the same as in Fig. 1, and their waiting time pdf is the Pareto distribution (3.27).
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Figure 3
Comoving variance for a subdiffusion-advection process ( $α = 1 / 2$ and $D_{α} = 1 / 2$ ) in the presence of an external force field ( $v_{α} = 1 / \sqrt{2 π}$ ) in a power-law expanding medium with $t_{0} = 10^{3}$ and, from top to bottom, $γ = 0, 1 / 4, 1 / 2, 2$ . Solid lines represent theoretical values obtained from Eqs. (4.22) and (4.30) whereas broken lines correspond to the force free case ( $v_{α} = 0$ ). The symbols are simulation results. The random walks were simulated as in Fig. 2.
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Figure 4
Physical variance for a subdiffusion-advection process ( $α = 1 / 2$ and $D_{α} = 1 / 2$ ) under an external force field ( $v_{α} = 1 / \sqrt{2 π}$ ) in an exponential contracting medium with $H = - 10^{- 4}$ . The symbols are simulation results obtained as described in Fig. 2. The thick solid line represents the theoretical results. The dashed line corresponds to the case with no external force. For comparison, we also provide the results for the case with external force but for a static medium (thin solid line). The short dashed line corresponds to the long-time asymptotic expression obtained from Eqs. (4.35) and (4.39).
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Figure 5
Propagator $W^{*} (y, t)$ for subdiffusive particles ( $α = 1 / 2$ and $D_{α} = 1 / 2$ ) subjected to an external force field ( $v_{α} = 1 / \sqrt{2 π}$ ) in an exponential contracting medium with $H = - 10^{- 4}$ . The symbols are simulation results obtained as described in Fig. 2 for $t = 2^{8}$ (filled circles), $t = 2^{10}$ (open circles), and $t = 2^{15}$ (squares). The lines correspond to numerical solutions of Eq. (3.16) by means of the fractional Crank-Nicolson method of Ref. [49] with $Δ x = Δ t = 0.1$ for $t = 2^{8}$ and $t = 2^{10}$ , and $Δ x = Δ t = 1$ for $t = 2^{15}$ .
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