Paper detail

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

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 $-(-Δ)^\fracα{2}$. 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 $\sim t/r^α >>1$ the obtained distributions exhibit an algebraic decay $\sim t^{-\frac{n}α} \rightarrow 0$ independent from the initial distribution and spacepoint. This universal scaling depends only on the ratio $n/α$ of the dimension $n$ of the physical space and the Lévi parameter $α$.