Dispersion Relations and Wave Operators in Self-Similar Quasi-Continuous Linear Chains

Abstract

We construct self-similar functions and linear operators to deduce a self-similar variant of the Laplacian operator and of the D'Alembertian wave operator. The exigence of self-similarity as a symmetry property requires the introduction of non-local particle-particle interactions. We derive a self-similar linear wave operator describing 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 also derive a continuum approximation which relates the self-similar Laplacian to fractional integrals and yields in the low-frequency regime a power law frequency-dependence of the oscillator density.

References in corpus (3)

• Chains with Fractal Dispersion Law
• Sierpinski signal generates $1/f^{\alpha }$ spectra
• Analysis of the Vibrational Mode Spectrum of a Linear Chain with Spatially Exponential Properties

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