THE FRACTIONAL LAPLACIAN AS A LIMITING CASE OF A SELF-SIMILAR SPRING MODEL AND APPLICATIONS TO n-DIMENSIONAL ANOMALOUS DIFFUSION

Thomas Michelitsch (1) , Gérard A. Maugin (1) , Andrzej F. Nowakowski (2) , Franck C.G.A Nicolleau (2) , Mujibur Rahman (3) , Rahman Mujibur (3)

1 DALEMBERT - Institut Jean Le Rond d'Alembert
2 Sheffield Fluid Mechanics Group
3 General Electrics

We analyze the "fractional continuum limit" and its generalization to n dimensions of a self-similar discrete spring model which we introduced recently (PRE 80, 011135 (2009)). In the fractional continuum limit the discrete self-similar Laplacian takes the form of the fractional Laplacian. We analyze the fundamental link of fractal vibrational features of the discrete self-similar spring model and the smooth regular ones of the corresponding fractional continuum limit model in n dimensions.Furthermore, we study in this setting anomalous diffusion generated by this Laplacian which is the source of Levi flights in n-dimensions.

Mots clés: Fractional Laplacian; self-similarity; power laws; scale invariance; fractals; Weierstrass-Mandelbrot function; continuum limit; fractional calculus; Fokker Planck equation; anomalous diffusion; L evi flights; Levi (stable) distributions; non-locality e

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Théorie [stat.TH]; Probabilités [math.PR]

Thomas Michelitsch, Gérard A. Maugin, Andrzej F. Nowakowski, Franck C.G.A Nicolleau, Mujibur Rahman, et al.. THE FRACTIONAL LAPLACIAN AS A LIMITING CASE OF A SELF-SIMILAR SPRING MODEL AND APPLICATIONS TO n-DIMENSIONAL ANOMALOUS DIFFUSION. 2013. ⟨hal-00839196⟩