Onset of anomalous diffusion from local motion rules

Anomalous diffusion processes, in particular superdiffusive ones, are known to be efficient strategies for searching and navigation in animals and also in human mobility. One way to create such regimes are Lévy flights, where the walkers are allowed to perform jumps, the “flights,” that can eventually be very long as their length distribution is asymptotically power-law distributed. In our work, we present a model in which walkers are allowed to perform, on a one-dimensional lattice, “cascades” of $n$ unitary steps instead of one jump of a randomly generated length, as in the Lévy case, where $n$ is drawn from a cascade distribution $p_n$. We show that this local mechanism may give rise to superdiffusion or normal diffusion when $p_n$ is distributed as a power law. We also introduce waiting times that are power-law distributed as well and therefore the probability distribution scaling is steered by the two local distributions power-law exponents. As a perspective, our approach may engender a possible generalization of anomalous diffusion in context where distances are difficult to define, as in the case of complex networks, and also provide an interesting model for diffusion in temporal networks.

Figure 1

The walker is allowed to perform at each step $n$ elementary moves, each one having equal probability $1/2$ to be in the positive or negative direction (a). In panel (b), we present a possible avalanche, where the walker performs $n=6$ hops, two in the positive direction and 4 in the negative one, for a total effective displacement of $m=-2$.

Figure 2

In the continuous time frame, at each time $t_i$ the walker can, as before, perform $n$ microscopic moves: in panel (a) these macromovements result in displacements of $m=2, m=5$, and $m=2$. If we introduce the waiting times, the displacements are interspersed by waiting intervals: in panel (b) the walker stays put from $t_1$ to $t_1^{\prime }$ and from $t_2$ to $t_2^{\prime }$.

Figure 3

Probability distribution $P_k\left(d\right)$ in the discrete case for $\gamma =1.9$ for $N={10}^5$ and a simulation time $k={10}^6$ with $\mu =\frac{1}{2(\gamma -1)}$ and $\xi =\frac{d}{k^{1/2(\gamma -1)}}$. In (a) the curves superposition illustrates the self-similarity of the PDF and we display in (b) the power-law tail of the Lévy distribution. The color code in (a) and symbols in (b) refer to different $k$ as specified in the legend.

Figure 4

Probability distribution $P(d,t)$ in the continuous time case for (a) $\alpha =0.5, \gamma =1.6$ and (b) $\alpha =0.8, \gamma =1.7$ for $N={10}^5$, where $\mu =\frac{\alpha }{2(\gamma -1)}$ and $\xi =\frac{d}{t^{\alpha /2(\gamma -1)}}$. Again, the self-similarity of the PDF emerges once the curves are rescaled with respect to the $\mu$ exponent.