Saturday, September 15, 2007

Fractional Dynamics

The fractional dynamics framework has been to descripte of anomalous transport in complex systems which, for slow diffusion, is close to equilibrium and satisfies linear response. On the basis of the generalised Chapman-Kolmogorov equation, fractional equations of the Klein-Kramers type in phase space and of the Fokker-Planck-Smoluchowski and diffusion types in position space have been derived in which the local time derivative is replaced by a non-local integrodifferential operator. We also proved that the fractional dynamics evolves from a multiple trapping process, during which the free motion events are governed by the standard Langevin equation. We derived similar fractional equations for Lévy flight type processes. Exact solutions can be obtained in the form of Fox "H-functions", and through separation of variables. An important feature is the Mittag-Leffler relaxation pattern that replaces the conventional exponential relaxation of modes and moments, turning from stretched exponential to inverse power-law. The case of spatiotemporally coupled Lévy walks is more subtle. Our recent findings indicate that systems governed by Lévy walk statistics equilibrate in the velocity coordinate, but may not possess a stationary state in the position coordinate. (This feature is connected to the understanding of generalised equilibrium principles for stochastic systems which do not relax to Gibbs-Boltzmann equilibrium.) These properties now have to be further investigated in an exact formulation obtained recently, in which we showed that the spatiotemporal coupling of Lévy walks translates into a fractional material derivative. Conversely, we could show that Lévy flights in a superharmonic external potential exhibit a bifurcation to a multimodal state at some critical time, and that the variance for this process is finite. An important issue for both Lévy flights and Lévy walks is the correct formulation of boundary value problems. We have established a correct way to formulate such first passage problems for Lévy flights, finding that the method of images (known from normal and subdiffusion) becomes inconsistent for processes with long jump lengths, and that the first passage time density differs from the probability density of first arrival.


Anonymous said...

Introductory Notes on Fractional Calculus:

richie007 said...

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james vick
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Anonymous said...

Why there are so many comments in this blog?