Emergence of Lévy Walks in Systems of Interacting Individuals

Abstract

Recent experiments (G. Ariel, et al., Nature Comm. 6, 8396 (2015)) revealed an intriguing behavior of swarming bacteria: they fundamentally change their collective motion from simple diffusion into a superdiffusive Lévy walk dynamics. We introduce a nonlinear non-Markovian persistent random walk model that explains the emergence of superdiffusive Lévy walks. We show that the alignment interaction between individuals can lead to the superdiffusive growth of the mean squared displacement and the power law distribution of run length with infinite variance. The main result is that the superdiffusive behavior emerges as a nonlinear collective phenomenon, rather than due to the standard assumption of the power law distribution of run distances from the inception. At the same time, we find that the repulsion/collision effects lead to the density dependent exponential tempering of power law distributions. This qualitatively explains experimentally observed transition from superdiffusion to the diffusion of mussels as their density increases (M. de Jager et al., Proc. R. Soc. B 281, 20132605 (2014)).

References in corpus (23)

• Collective motion
• Flocks, herds, and schools: A quantitative theory of flocking
• Anomalous transport in the crowded world of biological cells
• Statistical Mechanics of Interacting Run-and-Tumble Bacteria
• Intermittent search strategies
• Lévy walks
• Collective motion of self-propelled particles interacting without cohesion
• Self-organization in systems of self-propelled particles
• Pattern formation in self-propelled particles with density-dependent motility
• Diffusion, subdiffusion, and trapping of active particles in heterogeneous media
• Non-ergodic Intensity Correlation Functions for Blinking Nano Crystals
• Observation of anomalous diffusion and fractional self-similarity in one dimension
• Stabilization of active matter by flow-vortex lattices and defect ordering
• Colloids in a bacterial bath: simulations and experiments
• Alternating steady state in one-dimensional flocking
• Non-normalizable densities in strong anomalous diffusion: beyond the central limit theorem
• Sub-diffusion in External Potential: Anomalous hiding behind Normal
• Single integro-differential wave equation for Lévy walk
• Persistent random walk of cells involving anomalous effects and random death
• A Fractional Diffusion Equation for an n-Dimensional Correlated Levy Walk
• Nonlinear subdiffusive fractional equations and aggregation phenomenon
• Following fluctuating signs: anomalous active superdiffusion of swimmers in anisotropic media
• Self-organized anomalous aggregation of particles performing nonlinear and non-Markovian random walk

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• Cellular automaton models for time-correlated random walks: derivation and analysis
• Interacting particles with Lévy strategies: limits of transport equations for swarm robotic systems
• Orientational Memory of Active Particles in Multistate Non-Markovian Processes
• Self-reinforcing directionality generates Lévy walks without the power-law assumption
• Age representation of Levy walks: partial density waves, relaxation and first passage time statistics
• Transport in exclusion processes with one-step memory: density dependence and optimal acceleration
• Superdiffusion in self-reinforcing run-and-tumble model with rests
• Non-stationary Markovian Replication Process causing Diverse Diffusions
• Hydrodynamics of Active Lévy Matter
• Correlations and transport in exclusion processes with general finite memory
• Active Lévy Matter: Anomalous Diffusion, Hydrodynamics and Linear Stability
• In the folds of the Central Limit Theorem: Lévy walks, large deviations and higher-order anomalous diffusion
• Spectral properties of stochastic processes possessing finite propagation velocity
• Semiclassical scattering by edge imperfections in topological insulators under magnetic field
• Non-monotonic displacement distribution of active random walkers
• Individual particle persistence antagonizes global ordering in populations of nematically-aligning self-propelled particles
• Anomalous cumulative inertia in human behaviour