Onset of anomalous diffusion from local motion rules

Abstract

Anomalous diffusion processes, in particular superdiffusive ones, are known to be efficient strategies for searching and navigation by 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 1D 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 PDF's 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.

10 pages,4 figures

References in corpus (6)

• Understanding individual human mobility patterns
• The scaling laws of human travel
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• Human Mobility in a Continuum Approach
• Evolution of optimal Lévy-flight strategies in human mental searches
• Input-output relationship in social communications characterized by spike train analysis

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• Fractional dynamics on circulant multiplex networks: optimal coupling and long-range navigation for continuous-time random walks
• Mean first-encounter times of simultaneous random walkers with resetting on networks
• Long-range connections and mixed diffusion in fractional networks
• Fractional networks, the new structure
• Long-range connections, real-world networks and rates of diffusion
• Non-monotonic displacement distribution of active random walkers
• Markov chain approach to anomalous diffusion on Newman-Watts networks