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

Ralf Metzler·Igor M. Sokolov
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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.

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Ralf Metzler, Igor M. Sokolov. 2001-07-24. Fractional Klein-Kramers equation for superdiffusive transport: normal versus anomalous time evolution in a differential L{é}vy walk model. https://doi.org/10.1209/epl%2Fi2002-00421-1

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