Fractional Klein-Kramers equation for superdiffusive transport: normal versus anomalous time evolution in a differential L{é}vy walk model

Abstract

We introduce a fractional Klein-Kramers equation which describes sub-ballistic superdiffusion in phase space in the presence of a space-dependent external force field. This equation defines the differential L{é}vy walk model whose solution is shown to be non-negative. In the velocity coordinate, the probability density relaxes in Mittag-Leffler fashion towards the Maxwell distribution whereas in the space coordinate, no stationary solution exists and the temporal evolution of moments exhibits a competition between Brownian and anomalous contributions.

4 pages, REVTeX

References in corpus (4)

• Fractional Langevin equation
• Thermodynamics and Fractional Fokker-Planck Equations
• Diffusion on random site percolation clusters. Theory and NMR microscopy experiments with model objects
• Anomalous diffusion and the first passage time problem

Cited by in corpus (8)

• Lévy walks
• Fractional Diffusion Equation for a Power-Law-Truncated Levy Process
• Towards deterministic equations for Levy walks: the fractional material derivative
• Origin of hyperdiffusion in generalized Brownian motion
• Thermodynamic Uncertainty Relation Bounds the Extent of Anomalous Diffusion
• Feynman-Kac equation for anomalous processes with space- and time-dependent forces
• Fluctuation relations for anomalous dynamics generated by time-fractional Fokker-Planck equations
• Continuous Markovian model for Levy random walks with superdiffusive and superballistic regimes