Self-reinforcing directionality generates Lévy walks without the power-law assumption

Abstract

We introduce a persistent random walk model with finite velocity and self-reinforcing directionality, which explains how exponentially distributed runs self-organize into truncated Lévy walks observed in active intracellular transport by Chen et. al. [\textit{Nat. mat.}, 2015]. We derive the non-homogeneous in space and time, hyperbolic PDE for the probability density function (PDF) of particle position. This PDF exhibits a bimodal density (aggregation phenomena) in the superdiffusive regime, which is not observed in classical linear hyperbolic and Lévy walk models. We find the exact solutions for the first and second moments and criteria for the transition to superdiffusion.

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• Statistical Mechanics of Interacting Run-and-Tumble Bacteria
• Lévy walks
• First-passage times in complex scale-invariant media
• First-passage time of run-and-tumble particles
• Asymptotic densities of ballistic Lévy walks
• Exact moments in a continuous time random walk with complete memory of its history
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• Anomalous stochastic transport of particles with self-reinforcement and Mittag-Leffler distributed rest times