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Pré-Publication, Document De Travail Année : 2012

An approach to anomalous diffusion in the n-dimensional space generated by a self-similar Laplacian

Résumé

We analyze a quasi-continuous linear chain with self-similar distribution of harmonic interparticle springs as recently introduced for one dimension (Michelitsch et al., Phys. Rev. E 80, 011135 (2009)). We define a continuum limit for one dimension and generalize it to $n=1,2,3,..$ dimensions of the physical space. Application of Hamilton's (variational) principle defines then a self-similar and as consequence non-local Laplacian operator for the $n$-dimensional space where we proof its ellipticity and its accordance (up to a strictly positive prefactor) with the fractional Laplacian $-(-\Delta)^\frac{\alpha}{2}$. By employing this Laplacian we establish a Fokker Planck diffusion equation: We show that this Laplacian generates spatially isotropic Lévi stable distributions which correspond to Lévi flights in $n$-dimensions. In the limit of large scaled times $\sim t/r^{\alpha} >>1$ the obtained distributions exhibit an algebraic decay $\sim t^{-\frac{n}{\alpha}} \rightarrow 0$ independent from the initial distribution and spacepoint. This universal scaling depends only on the ratio $n/\alpha$ of the dimension $n$ of the physical space and the Lévi parameter $\alpha$.
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Dates et versions

hal-00721688 , version 1 (29-07-2012)
hal-00721688 , version 2 (22-11-2012)

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Thomas Michelitsch, Gérard Maugin, Andrzej F. Nowakowski, Franck C.G.A Nicolleau, Mujibur Rahman. An approach to anomalous diffusion in the n-dimensional space generated by a self-similar Laplacian. 2012. ⟨hal-00721688v2⟩
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