Relations de dispersion pour cha\^ines lin\'eaires comportant des interactions harmoniques auto-similaires

TL;DR

This paper develops a mathematical framework for describing the mechanics and wave propagation in self-similar, fractal-like structures using non-local interactions and self-similar operators, revealing fractal dispersion relations.

Contribution

It introduces self-similar linear operators and equations to model the dynamics of self-similar chains with nonlocal interactions, linking them to fractional calculus.

Findings

Dispersion relation exhibits fractal, self-similar features.

Self-similar Laplacian relates to fractional integrals.

Low-frequency behavior follows a power law.

Abstract

Many systems in nature have arborescent and bifurcated structures such as trees, fern, snails, lungs, the blood vessel system, etc. and look self-similar over a wide range of scales. Which are the mechanical and dynamic properties that evolution has optimized by choosing self-similarity? How can we describe the mechanics of self-similar structures in the static and dynamic framework? Physical systems with self-similarity as a symmetry property require the introduction of non-local particle-particle interactions and a (quasi-) continuous distribution of mass. We construct self-similar functions and linear operators such as a self-similar variant of the Laplacian and of the D'Alembertian wave operator. The obtained self-similar linear wave equation describes the dynamics of a quasi-continuous linear chain of infinite length with a spatially self-similar distribution of nonlocal inter-particle springs. The self-similarity of the nonlocal harmonic particle-particle interactions results in a dispersion relation of the form of a Weierstrass-Mandelbrot function which exhibits self-similar and fractal features. We deduce a continuum approximation that links the self-similar Laplacian to fractional integrals and which yields in the low-frequency regime a power law frequency dependence for the oscillator density. For details of the present model we refer to our recent paper (Michelitsch et al., Phys. Rev. E 80, 011135 (2009)).