Math. Model. Nat. Phenom.
Volume 11, Number 3, 2016Anomalous diffusion
|Page(s)||51 - 62|
|Published online||21 June 2016|
Lévy Transport in Slab Geometry of Inhomogeneous Media
1 Department of Physics, Technion, Haifa 32000, Israel
2 Max Planck Institute for the Physics of Complex Systems Nöthnitzer Strasse 38, 01187 Dresden, Germany
3 Radiation Safety Directorate, Partizanski odredi 143, P.O. Box 22, 1020 Skopje, Macedonia
⋆ Corresponding author. E-mail: email@example.com
We present a physical example, where a fractional (both in space and time) Schrödinger equation appears only as a formal effective description of diffusive wave transport in complex inhomogeneous media. This description is a result of the parabolic equation approximation that corresponds to the paraxial small angle approximation of the fractional Helmholtz equation. The obtained effective quantum dynamics is fractional in both space and time. As an example, Lévy flights in an infinite potential well are considered numerically. An analytical expression for the effective wave function of the quantum dynamics is obtained as well.
Mathematics Subject Classification: 60J60 / 33E12 / 26A33 / 34A08 / 35R11 / 60G50
Key words: fractional integration / parabolic equation approximation / fractional Schrödinger equation / Lévy flights
© EDP Sciences, 2016
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