Lévy walks in nonhomogeneous environments

The Lévy walk process with rests is discussed. The jumping time is governed by an $\alpha$-stable distribution with $\alpha >1$ while a waiting time distribution is Poissonian and involves a position-dependent rate which reflects a nonhomogeneous trap distribution. The master equation is derived and solved in the asymptotic limit for a power-law form of the jumping rate. The relative density of resting and flying particles appears time-dependent, and the asymptotic form of both distributions obeys a stretched-exponential shape at large time. The diffusion properties are discussed, and it is demonstrated that, due to the heterogeneous trap structure, the enhanced diffusion, observed for the homogeneous case, may turn to a subdiffusion. The density distributions and mean squared displacements are also evaluated from Monte Carlo simulations of individual trajectories.

Figure 1

Time evolution of the density distributions $p_r(x,t)$ and ${\mathrm{p}}_v(x,t)$ for $\alpha =1.5$ and two values of $\theta$: (a) $\theta =1$ and (b) $\theta =-0.5$. The plots were obtained from Monte Carlo simulations by averaging over an ensemble of ${10}^7$ trajectories.

Figure 2

${\varphi }_v\left(t\right)$, calculated from Monte Carlo trajectory simulations, for $\alpha =1.5$ and the following values of $\theta$: $-0.5$, 0, 0.5, 1, 1.5, and 2 (from top to bottom on the right side). For the positive $\theta$, the form of ${\varphi }_v\left(t\right)$ for large time is approximately the power law with the following exponents: 0.24, 0.4, 0.52, and 0.6.

Figure 3

Diffusion exponent $\mu$ as a function of $\theta$ estimated from $\langle x^2\rangle \left(t\right)$ at small times ($t<{10}^3$) for $\alpha =1.5$ (squares) and $\alpha =1.2$ (points). Those results were obtained from Monte Carlo simulations. Red lines mark the asymptotic values, Eq. (62) and (63), for $\alpha =1.5$ (solid line) and 1.2 (dashed line). Inset: $\langle x^2\rangle \left(t\right)$ for $\theta =0.5$, 1, 1.5, and 2 (from top to bottom), evaluated from the trajectory simulations; straight red lines for small $t$ mark power-law fits while those for $t>{10}^4$ correspond to Eq. (62).