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

TL;DR

This paper introduces a self-similar, non-local Laplacian operator in n-dimensional space, leading to a generalized diffusion model that produces Lévi stable distributions with universal algebraic decay depending on space dimension and Lévi parameter.

Contribution

It generalizes a self-similar Laplacian to n-dimensions, linking it to fractional Laplacians and Lévi stable distributions, and analyzes the resulting anomalous diffusion behavior.

Findings

The self-similar Laplacian is elliptic and matches the fractional Laplacian up to a positive factor.

The derived diffusion equation generates isotropic Lévi stable distributions in n-dimensions.

Distributions decay algebraically with a universal scaling depending on n and α.

Abstract

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 $\alpha$.