Scaling forms of particle densities for Lévy walks and strong anomalous diffusion

Marco Dentz (1) , Tanguy Le Borgne (2) , Daniel R. Lester (3) , Felipe P.J. de Barros (4)

1 IDAEA - Institute of Environmental Assessment and Water Research
2 GR - Géosciences Rennes
3 School of Civil, Environmental and Chemical Engineering
4 Sonny Astani Department of Civil and Environmental Engineering

We study the scaling behavior of particle densities for Lévy walks whose transition length r is coupled with the transition time t as |r| ∝ t α with an exponent α > 0. The transition-time distribution behaves as ψ(t) ∝ t −1−β with β > 0. For 1 < β < 2α and α 1, particle displacements are characterized by a finite transition time and confinement to |r| < t α while the marginal distribution of transition lengths is heavy tailed. These characteristics give rise to the existence of two scaling forms for the particle density, one that is valid at particle displacements |r| t α and one at |r| t α. As a consequence, the Lévy walk displays strong anomalous diffusion in the sense that the average absolute moments |r| q scale as t qν(q) with ν(q) piecewise linear above and below a critical value q c. We derive explicit expressions for the scaling forms of the particle densities and determine the scaling of the average absolute moments. We find that |r| q ∝ t qα/β for q < q c = β/α and |r| q ∝ t 1+αq−β for q > q c. These results give insight into the possible origins of strong anomalous diffusion and anomalous behaviors in disordered systems in general.

Domains: Hydrology

Marco Dentz, Tanguy Le Borgne, Daniel R. Lester, Felipe P.J. de Barros. Scaling forms of particle densities for Lévy walks and strong anomalous diffusion. Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, 2015, 92 (3), pp.Art. n°032128. ⟨10.1103/PhysRevE.92.032128⟩. ⟨insu-01216539⟩