arXiv subjects

Gérard Maugin

Publications and source records attributed to Gérard Maugin.

3 recordsLinked to original sources

arXiv2015

A regularized representation of the fractional Laplacian in n dimensions and its relation to Weierstrass-Mandelbrot type fractal functions

We demonstrate that the fractional Laplacian (FL) is the principal characteristic operator of harmonic systems with {\it self-similar} interparticle interactions. We show that the FL represents the "{\it fractional continuum limit}" of a discrete "self-similar Laplacian" which is obtained by Hamilton's variational principle from a discrete spring model. We deduce from generalized self-similar elastic potentials regular representations for the FL which involve convolutions of symmetric finite difference operators of even orders extending the standard representation of the FL. Further we deduce a regularized representation for the FL $-(-Δ)^{\fracα{2}}$ holding for $α\in \R \geq 0$. We give an explicit proof that the regularized representation of the FL gives for integer powers $\fracα{2} \in \N\_0$ a distributional representation of the standard Laplacian operator $Δ$ including the trivial unity operator for $α\rightarrow 0$. We demonstrate that self-similar {\it harmonic} systems are {\it all} governed in a distributional sense by this {\it regularized representation of the FL} which therefore can be conceived as characteristic footprint of self-similarity.

math-ph↗

arXiv2012

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 $α$.

physics.class-ph↗

arXiv2011

Sur une généralisation de l'opérateur fractionnaire

The goal of this communication is to propose a generalized notion of the "traditional derivative". This generalization includes the fractional derivatives such as the Riemann-Liouville, Gruenwald-Letnikov, Weyl, Riesz, Caputo, Marchaud derivatives and other variants as special cases. The approach is useful to describe mechanical problems in material systems with microstructures without characteristic scale such as self-similar and fractal materials.

physics.class-ph↗