Wave Propagation in Scale Free Media Thomas Michelitsch, Gérard Maugin, Franck Nicolleau*, Andrzej Nowakowski*, Shahram Derogar* Institut Jean le Rond d'Alembert (Paris 6), CNRS UMR 7190 Paris, France Email: michel@lmm.jussieu.fr *University of Sheffield Sheffield, United Kingdom Email: F.Nicolleau@sheffield.ac.uk 6th Workshop on Synthetic Turbulence Models 5th-7th July 2010, Lyon ERCOFTAC/SIG 42 Overview • Motivation • Previous works and approaches • Self-similar functions and linear operators • Self-similar Laplacian operator and relationship to fractional derivatives • Physical model of a linear chain with self-similar harmonic interactions • Self-similar (fractal) dispersion relations and oscillator density • Perspectives Motivation Fractals (self-similar objects) in nature Why nature choses so often self-similarity (scaling invariance) as intrinsic symmetry? Which (mechanical) dynamic properties are optimized in such arborescent self-similar structures? --- The answer requires deeper understanding on: how the self-similarity (fractality) affects the mechanical especially dynamic properties, such as wave propagation, dispersion relation, vibrational density. --- However there is so far no generally accepted approach, hence, simple models catching essential features of self-similar and fractal systems are desirable. Motivation Previous works and approaches • Kigami J., A harmonic calculus on the Sierpinski spaces, Japan J. Appl. Math. 8 (1989), 259-290. • Bondarenko A.N., Levin, V.A., Self Similar Spectrum of a Fractal Lattice, Science and Technology (2005) KORUS Proceedings. The 9th Russian- Korean Int. Symposium, 33-35, ISBN: 0-7803-8943-3 • Tarasov V.E. , Chains with fractal dispersion law, J Phys A: Math Theor. 41 (2008), doi: 10.1088/1751-8113/41/3/035101 • Ostoja-Starzewski, M., Towards Thermomechanics of Fractal Media, Journal of Applied Mathematics and Physics (ZAMP), 58:6, 1085-1096, 2007. • Epstein M., Sniatytcki J., Fractal Mechanics , Physica D, 2006, vol. 220, no1, pp. 54-68 • Parisi G., Frisch U., “Multifractal analysis”, 1985 (caracterization of “irregular functions”, developed in the context of turbulence Self-similar functions and operators What means « self-similarity »? We call a function selfsimilar with repect to variable if it fulfills the condition for a given pair of real numbers (>1) and ( > 0 ) for all h>0. A (non-unique) solution of the affine problem can be represented as band of convergence: exigence for Self-similar functions constants or (oscillating) functions limited by constants Self-similar functions Some properties of with non-integer rest , uniquely determined by its values in precribed by any function defined in oscillating part (infinitely rapid arround h=0) : Hölder exponent (at h=0) self-similar (analogue of) Laplacian Conventional 1D Laplacian explicit form of self-similar Laplacian self-similar (with respect to h) Self-similar Laplacian operator remains finite only if Continuum limit: fractional derivatives in the limit where Laplacian Self-similar Laplacian operator exists for Self-similar Laplacian operator Continuum limit with where with the Riemann-Liouville fractional integral exists for Later: identification of D with fractal (Hausdorff) dimension of dispersion relation A 1D self-similar symmetry chain requires: Infinite length (inifinite number of DOF) Quasi-continuous 1D spatially homogeneous distribution of mass Non-locality of inter-particle interactions (self-similar spatial function) Hamiltonian with the self-similar elastic energy density finite (convergent) for Physical linear chain model Linear chain with self-similar harmonic interactions Equation of motion Self-similar wave equation and d’Alembertian wave operator Self-similar dispersion relation Self-similar Weierstrass-Mandelbrot functions convergent for  dispersion relation Self-similar dispersion relation The Weierstrass-Mandelbrot function is a fractal curve for with fractal (Hausdorff) dimension Self-similar dispersion relation Self-similar dispersion relation Oscillator density Continuum limit where power law for « small » |kh| number of normal oscillators per frequency intervall and unit length power law with  is a (physically) observable quantity fractal dimension exists for since with universal « footprint » of self-similarity (?) Perspectives Towards --- • A Mechanics of self-similar (fractal) media, • Dynamic characteristics of self-similar (fractal) structures (arborescent, infinitely branched structures) such as Sierpinski fractals, • Construction of self-similar wave-packages e.g. Perspectives • Green’s functions ---  complete information of the dynamic (spectral) information,  Scaling-relations and critical exponents, such as the controversial (controversal Alexander-Orbach conjecture) about the low frequency behaviour of the spectral density. d “spectral” dimension , Paper: T. Michelitsch, G. Maugin, F. Nicolleau, S. Derogar Phys. Rev. E 80, 011135 (2009) Online : http://link.aps.org/doi/10.1103/PhysRevE.80.011135 E-print: michel@lmm.jussieu.fr http://www.dalembert.upmc.fr/home/michelitsch/ Merci !