Density of states on fractals : « fractons »

Abstract

The density of states on a fractal is calculated taking into account the scaling properties of both the volume and the connectivity. We use a Green's function method developed elsewhere which utilizes a relationship to the diffusion problem. It is found that proper mode counting requires a reciprocal space with new intrinsic fracton dimensionality d = 2 d/(2 + δ). Here, d is the effective dimensionality, and δ the exponent giving the dependence of the diffusion constant on distance. For example, we find for percolation clusters d = 4/3 within the numerical accuracy available, independent of the Euclidean dimensionality d. Crossover to normal behaviour at low frequencies is discussed for finite fractals and for percolation above the percolation threshold pc. Relevance to experimental results on proteins is also discussed