Coherent spatio-temporal coupling in fractional wanderings. Renewed approach to continuous-time Lévy flights

  • R. Kutner
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 519)


The one-dimensional continuous-time Lévy flights (CTLF) are reconsidered in the renewed framework of the nonseparable continuous-time random walk (CTRW) model in order to be able to treat the spatio-temporal relations in terms of the self-similar structure of the Lévy process. Hence, a novel spatio-temporal coupling is introduced by assuming that in each order of the structure the probability density for the flight and for waiting are joined. This (stochastic) structure is characterized by the spatial fractional dimension 1/β (representing the flights) and the temporal one 1/α (representing the waiting). Time was assumed here as the only independent truncation range. In the present work we study the asymptotic properties of our procedure. For example, by applying the method of steepest descents we obtained the particle propagator in the approximate scaling form, P(X, t) ∼ t -η(α,β)/2 F(ξ), where the scaling function \(\mathcal{F}(\xi ) = \xi \bar v(\alpha ,\beta )\) exp(−const(α,β) ξv(αβ)) and the scaling variable ξ =| X | /t η(α,β)/2 is large. The principal result of our analysis is that the exponents η, ν and \(\bar \upsilon\) depend on more fundamental, fractional dimensions α and β, what leads to a novel scaling. As a result of competition between exponents α and β an enhanced, dispersive or normal diffusion was recognized for a given topology of the structure in distinction from the prediction of the commonly used separable CTRW model where the enhanced diffusion is lost and the dispersive one is strongly limited. We compare here partially thermalized versions of both approaches where some initial fluctuations were also included in agreement with the spirit of the theory of the renewal processes. Having the propagator we calculated, for example, the mean-square displacement and found its novel scaling with time for enhanced particle diffusion, given by ∼ t 1+α(2/β−1), in distinction from its diverging for β<2 within the separable CTRW model. This renewed CTLF approach offers a possibility to properly model the time-dependence for any fractional (critical) wandering of jump type.


Anomalous Diffusion Normal Diffusion Laplace Domain Jump Model Single Jump 
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Copyright information

© Springer-Verlag 1999

Authors and Affiliations

  • R. Kutner
    • 1
  1. 1.Institute of Experimental PhysicsWarsaw UniversityWarsawPoland

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