1207.7132 (Thomas Michelitsch et al.)

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

Thomas Michelitsch, Gérard Maugin, Andrzej F. Nowakowski, Franck C. G. A Nicolleau, Mujibur Rahman

We analyze a quasi-continuous linear chain with self-similar distribution of harmonic interparticle springs as introduced for one dimension in PRE 80, 011135 (2009). We define a continuum limit for one dimension and generalize it to $n=1,2,3,..$ dimensions of the physical space. In this way we obtain a Hamiltonian functional in the $n=1,2,3,..$ dimensional 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. 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 $\alpha$.

View original: http://arxiv.org/abs/1207.7132