Continuous-time random-walk model for anomalous diffusion in expanding media

Expanding media are typical in many different fields, e.g., in biology and cosmology. In general, a medium expansion (contraction) brings about dramatic changes in the behavior of diffusive transport properties such as the set of positional moments and the Green's function. Here, we focus on the characterization of such effects when the diffusion process is described by the continuous-time random-walk (CTRW) model. As is well known, when the medium is static this model yields anomalous diffusion for a proper choice of the probability density function (pdf) for the jump length and the waiting time, but the behavior may change drastically if a medium expansion is superimposed on the intrinsic random motion of the diffusing particle. For the case where the jump length and the waiting time pdfs are long-tailed, we derive a general bifractional diffusion equation which reduces to a normal diffusion equation in the appropriate limit. We then study some particular cases of interest, including Lévy flights and subdiffusive CTRWs. In the former case, we find an analytical exact solution for the Green's function (propagator). When the expansion is sufficiently fast, the contribution of the diffusive transport becomes irrelevant at long times and the propagator tends to a stationary profile in the comoving reference frame. In contrast, for a contracting medium a competition between the spreading effect of diffusion and the concentrating effect of contraction arises. In the specific case of a subdiffusive CTRW in an exponentially contracting medium, the latter effect prevails for sufficiently long times, and all the particles are eventually localized at a single point in physical space. This “big crunch” effect, totally absent in the case of normal diffusion, stems from inefficient particle spreading due to subdiffusion. We also derive a hierarchy of differential equations for the moments of the transport process described by the subdiffusive CTRW model in an expanding medium. From this hierarchy, the full time evolution of the second-order moment is obtained for some specific types of expansion. In the case of an exponential expansion, exact recurrence relations for the Laplace-transformed moments are obtained, whence the long-time behavior of moments of arbitrary order is subsequently inferred. Our analytical and numerical results for both Lévy flights and subdiffusive CTRWs confirm the intuitive expectation that the medium expansion hinders the mixing of diffusive particles occupying separate regions. In the case of Lévy flights, we quantify this effect by means of the so-called “Lévy horizon.”

Figure 1

Time evolution of the physical propagator for $H=-{10}^{-2}$ and Lévy exponent $\mu =3/2$. Solid thin lines represent the analytical propagator $P(y,t)$ for $t=25,50,100,200,500$ (from top to bottom at $y=0$). The corresponding symbols (solid circle, solid square, open circle, down triangle, and up triangle) represent simulation results obtained with ${10}^6$ realizations. The thick line represents the asymptotic theoretical stationary propagator given by Eq. (57) and overlaps with the thin line corresponding to the analytical result for $t=500$. In the simulations we used $\lambda \left(y\right)={\mathsf{L}}_{3/2}\left(y;{\sigma }^{3/2}=1/2\right)$ and a waiting time pdf $\phi \left(t\right)$ given by the exponential pdf (7) with $\tau =1$. According to Eq. (28), this parameter choice implies ${\mathfrak{D}}_{1,3/2}=1/2$.

Figure 2

Pdf corresponding to two Lévy pulses with $\mu =1$ (Cauchy pulses) at $t=2^{17}$ in the case of a power-law expansion with $\gamma =2, t_0={10}^3$, and ${\mathfrak{D}}_{1,1}=1/2$. This implies $x_{\text{Lh}}\left(2^{17}\right)=496.2$ and $x_{\text{Leh}}=500$. The pulses depart from $x=\pm x_0$ with (a) $x_0={10}^2$ (merged pulses), (b) $x_0={10}^3$ (mutually influenced pulses), and (c) $x_0={10}^4$ (clearly separated pulses). Solid lines represent the analytical propagator $W_2\left(x,t=2^{17}\right)$, dashed lines depict the asymptotic stationary state and dots represent simulation results (${10}^6$ realizations) with $\lambda \left(y\right)$ given by Eq. (60) with $\sigma =1/2$ and $\phi \left(t\right)$ given by Eq. (7) with $\tau =1$. The solid and dashed lines are almost indistinguishable from one another because the chosen time $t=2^{17}$ is so large that the propagator $W\left(x,t=2^{17}\right)$ is already very close to the asymptotic one.

Figure 3

Double-logarithmic representation of the comoving second-order moment for a subdiffusive CTRW with $\alpha =1/2$ in a medium subjected to a power-law expansion with $t_0={10}^3$ and (a) $\gamma =1/10$ (diffusion-dominated regime), (b) $\gamma =\alpha /2=1/4$ (marginal case), and (c) $\gamma =1$ (expansion dominated regime). Solid lines represent analytical results obtained from Eq. (64), whereas dashed lines represent the long-time asymptotics described by Eq. (66a). The dots correspond to numerical simulations performed up to $t=2^{34}$ (${10}^6$ realizations) with $\lambda \left(y\right)$ given by Eq. (5) with ${\sigma }^2=1/2$ and $\phi \left(t\right)$ given by Eq. (67) with $\xi =1/\pi$.

Figure 4

Double-logarithmic representation of the physical second-order moment corresponding to a subdiffusive random walk in an exponentially expanding medium with (a) $H={10}^{-9}$ and (b) $H=-{10}^{-9}$ (contracting medium). In all cases $\alpha =1/2$ and ${\mathfrak{D}}_{\alpha }=1/2$. Solid lines represent analytical results obtained from Eq. (69). The dots correspond to numerical simulations performed up to $t=2^{34}$ with ${10}^6$ realizations where $\lambda \left(y\right)$ is given by Eq. (5) with ${\sigma }^2=1/2$ and $\phi \left(t\right)$ is given by Eq. (67) with $\xi =1/\pi$. For the contracting medium, panel (b), one sees that $\langle y^2\rangle$ goes to zero at sufficiently long times.