CS14: Complex Systems II

Organizer: Marcin Magdziarz (Wrocław University of Science and Technology)

The role of the fractional material derivative in Lévy walks

Marek Teuerle

The Lévy walk model, a key framework for describing anomalous diffusion processes with finite-second moments, is distinguished by its intrinsic coupling of jump lengths and waiting times \([1]\). This model has proven invaluable for capturing the dynamics of diverse systems, from the collective movement of biological organisms like bacteria and marine predators to physical phenomena involving cold atoms and blinking quantum dots (see \([2,3]\) for examples).

Recent theoretical advances have clarified the macroscopic behavior of Lévy walks. It has been shown that their scaling limits, under Skorokhod’s \(\mathbb{J}_1\) convergence, are \(\alpha\)-stable processes governed by a strongly dependent inverse \(\alpha\)-stable subordinator or by some combinations of these kind of processes \([4]\). It is also a well-established fact that the governing dynamics of the Lévy walks’ scaling limit can be described by the fractional material derivative, an extension of the classical material derivative to fractional calculus \([4,5]\).

This presentation focuses on the numerical approximation of this fractional material derivative. We introduce a novel finite-volume upwind scheme that incorporates spatiotemporal coupling of the underlying process and builds upon known techniques for fractional derivatives \([6,7]\). We confirm the stability of our proposed method. Furthermore, we will demonstrate the scheme’s accuracy by applying it to a one-sided probability problem (jumps/displacements only to the left or only to the right) associated with Lévy walks and comparing the results with traditional Monte Carlo simulations. Finally, we will outline an extension of our numerical approach to accommodate Lévy walks with arbitrary step directions, so-called biased Lévy walks.

Bibliography

\([2]\) Jane Smith. "Title of the Article." Journal Name, vol. X, no. Y, Year, pp. Z.

\([1]\) J. Klafter, A. Blumen, M.F. Shlesinger. "Random walks with infinite spatial and temporal moments." J. Stat. Phys., vol.27, 1982, pp. 499–512.

\([2]\) R. Metzler, J. Klafter. "The random walk’s guide to anomalous diffusion: a fractional dynamics approach." Phys. Rep., vol. 339, 2000, pp. 1–77.

\([3]\) V. Zaburdaev, S. Denisov, J. Klafter. "Lévy walks." Rev. Mod. Phys., vol. 87, 2015, pp. 483–530.

\([4]\) M. Magdziarz, M. Teuerle. "Asymptotic properties and numerical simulation of multidimensional Lévy walks." Commun. Nonlinear Sci. Numer. Simul., vol. 20, 2015, pp. 489–505.

\([5]\) I.M. Sokolov, R. Metzler. "Towards deterministic equations for Lévy walks: The fractional material derivative." Phys. Rev. E., vol. 67, 2003, pp. 010101(R).

\([6]\) Ł. Płociniczak. "Linear Galerkin-Legendre spectral scheme for a degenerate nonlinear and nonlocal parabolic equation arising in climatology." Appl. Numer. Math., vol. 179, 2022, pp. 105–124.

\([7]\) M. Teuerle, Ł. Płociniczak. "From Lévy walks to fractional material derivative: Pointwise representation and a numerical scheme." Commun. Nonlinear Sci. Numer. Simul., vol. 139, 2024, pp. 108316.