The fractional Laplacian as a limiting case of a self-similar spring model and applications to n-dimensional anomalous diffusion

Abstract

We analyze the “fractional continuum limit” and its generalization to n dimensions of a self-similar discrete spring model which we introduced recently [21]. Application of Hamilton’s (variational) principle determines in rigorous manner a self-similar and as consequence non-local Laplacian operator. In the fractional continuum limit the discrete self-similar Laplacian takes the form of the fractional Laplacian $- ( - \Delta )^{\tfrac{\alpha } {2}}$ with 0 < α < 2. 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: We find a characteristic scaling law for the density of normal modes ∼ $\omega ^{\tfrac{{2n}} {\alpha } - 1}$ with a positive exponent $\tfrac{{2n}} {\alpha } - 1 > n - 1$ being always greater than n−1 characterizing a regular lattice with local interparticle interactions. Furthermore, we study in this setting anomalous diffusion generated by this Laplacian which is the source of Lévi flights in n-dimensions. In the limit of “large scaled times” ∼ t/r ^α >> 1 we show that all distributions exhibit the same asymptotically algebraic decay ∼ t ^-n/α → 0 independent from the initial distribution and spatial position. This universal scaling depends only on the ratio n/α of the dimension n of the physical space and the Lévi parameter α.