Selective renormalisation: a new approach to universal properties of granular structures

Abstract

The authors present a new theoretical approach to the problem of describing the dynamical properties of granular structures. For the first time a combination of the Lanczos method and a new selective renormalisation technique is used to determine the intermediate-wavelength properties of a microscopic model of a granular solid. The results show that mean-field theory correctly describes the ultra-long- and short-wavelength properties, but at intermediate wavelengths a solution of the full dynamical problem is required. A key parameter that determines the dynamical properties of a granular structure is the ratio C=Cg/Cb of the ultra-long-wavelength velocity of sound Cg to the short-wavelength velocity Cb. For the first time, they present results for the sound velocity ratio C, obtained from a microscopic model of a disordered granular structure. When Cg (corrected for porosity) decreases to less than approximately 90% of its bulk value Cb, the density of phonon states rho ( omega ) exhibits a low-frequency enhancement. This enhancement is expected to be a universal feature of weakly coupled granular structures. Detailed results are presented for rho ( omega ), both in the presence and in the absence of disorder.