Dynamics of fractal structures

Abstract

Dilation symmetry, as opposed to the degenerate limit of translation symmetry, requires (at least) three dimensionalities to contain a physical description:d, the Euclidean (or embedding) dimension;$\bar d$, the Haussdorf (or fractal) dimension;$\overline{\overline d}$, the fracton (or spectral) dimension. The dynamical properties of percolating networks are examined in this context. The vibrational density of states,N(ω), is calculated, and shown to be propertional toω ^d −1 in the phonon, or long-length-scale regime. A crossover is found at a frequency ω_c, propotional top−p _c , wherep is the bond occupancy probability, andp _c the critical percolation concentration. At short length scales,N(ω) is proportional to$\omega ^{\overline{\overline d} - 1}$ and the excitations are termedfractons. An effective mdium approximation (EMA) calculation of the vibration density of states exhibits a rapid rise inNω in the vicinity ofω _c . We suggest that this overall behavior has relevance to the vibrational properties of amorphous materials. The far infrared absorption spctra of a number of glasses and amorphous Ge exhibit structures which appear similar to the calculated EMANω. This lends credence to our previous analysis based solely on the thermal properties. We use the EMA to compute 〈r ²(t)〉 for a percolating network, and thence calculate the diffusion constantD(t). For short times, we obtain the Webman EMA result,D(t)∞t ^−1/2, with a smooth crossover to a constant value for long times. The vibrational dispersion curves are calculated within EMA. The velocity of soundv _s is found to vary as(p −p _c) ^1/2 in the phonon (small wave vectorq) regime. Whenq ≈ qc, (q _{cv s=ω} forp nearp _c , the dispersion curves flatten and bend over, then rise again withω ∞q ², looking somewhat “roton”-like. Forq >q _c , the “damping” becomes very large, so that the plane wave character of the solution fails. This peculiar double-valued structure ina>(ω) is responsible for the rapid rise inN(ω) nearω _c, and not the behavior of the diffusion constant. Our results suggest the following EMA values atd=3 in the fracton regime:$\overline{\overline d}$=1,$\bar d$=2, orθ=2 whereD(r) α r^∞θ.