2012/07/30 by Thomas M. Michelitsch, Gérard A. Maugin, Michelitsch, Thomas +7
Mathematics · Biochemistry, Genetics and Molecular Biology · Physics and Astronomy · #Fractional Differential Equations Solutions #Diffusion and Search Dynamics #stochastic dynamics and bifurcation
paper · pdf · doi:10.48550/arxiv.1207.7132
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)^\fracα2. 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 ∼ t/rα >>1 the obtained distributions exhibit an algebraic decay ∼ t^-\fracnα → 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.