An approach to anomalous diffusion in the n-dimensional space generated by a self-similar Laplacian
| Autor: | Michelitsch, Thomas, Maugin, Gérard, Nowakowski, Andrzej, Nicolleau, Franck, Rahman, Mujibur |
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| Přispěvatelé: | Institut Jean le Rond d'Alembert (DALEMBERT), Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS), Department of Mechanical Engineering [Sheffield], University of Sheffield [Sheffield], General Electric Energy, Greenville, SC 29615, USA, Chercheur indépendant |
| Jazyk: | angličtina |
| Rok vydání: | 2012 |
| Předmět: |
Levi flights
self-similarity Fokker Planck equation [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] Classical Physics (physics.class-ph) FOS: Physical sciences Mathematics - Statistics Theory Physics - Classical Physics Mathematical Physics (math-ph) Statistics Theory (math.ST) [SPI.MECA.MSMECA]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Materials and structures in mechanics [physics.class-ph] [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH] [PHYS.MECA.MSMECA]Physics [physics]/Mechanics [physics]/Materials and structures in mechanics [physics.class-ph] fractional operator Fractional Laplacian anomalous diffusion 02.30.Rz 02.30.Vv 05.40.Fb [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST] Levi (stable) distributions scaling laws FOS: Mathematics Mathematical Physics non-locality |
| Popis: | 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 L\'evi stable distributions which correspond to L\'evi 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 L\'evi parameter $\alpha$. Comment: Submitted manuscript |
| Databáze: | OpenAIRE |
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