Huygens principle for diffusion and anomalous diffusion in spatially extended systems

We present a universal view on diffusive behavior in chaotic spatially extended systems for anisotropic and isotropic media. For anisotropic systems, strong chaos leads to diffusive behavior (Brownian motion with drift) and weak chaos leads to superdiffusive behavior (Lévy processes with drift). For...

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Bibliographic Details
Published in:Proceedings of the National Academy of Sciences - PNAS 2013-05, Vol.110 (21), p.8411-8416
Main Authors: Gottwald, Georg A., Melbourne, Ian
Format: Article
Language:English
Subjects:
Anisotropy
Brownian motion
Central limit theorem
Chaos theory
Dynamical systems
Ergodic theory
Huygens principle
Lattice theory
Mathematics
Physical Sciences
Symmetry
Traveling waves
Waves
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Summary:We present a universal view on diffusive behavior in chaotic spatially extended systems for anisotropic and isotropic media. For anisotropic systems, strong chaos leads to diffusive behavior (Brownian motion with drift) and weak chaos leads to superdiffusive behavior (Lévy processes with drift). For isotropic systems, the drift term vanishes and strong chaos again leads to Brownian motion. We establish the existence of a nonlinear Huygens principle for weakly chaotic systems in isotropic media whereby the dynamics behaves diffusively in even space dimension and exhibits superdiffusive behavior in odd space dimensions.
ISSN:0027-8424
1091-6490
1091-6490
DOI:10.1073/pnas.1217926110