Moment-preserving theory of vibrational dynamics of topologically disordered systems

We investigate a class of simple mass-spring models for the vibrational dynamics of topologically disordered solids. The dynamical matrix of these systems corresponds to the Euclidean-Random-Matrix (ERM) scheme. We show that the self-consistent ERM approximation introduced by Ganter and Schirmacher [1] preserves the first two nontrivial moments of the level density exactly. We further establish a link between these approximations and the fluctuating-elasticity approaches. Using this correspondence we derive and solve a new, simplified mean-field theory for calculating the vibrational spectrum of disordered mass-spring models with topological disorder. We calculate and discuss the level density and the spectral moments for a model in which the force constants obey a Gaussian site-separation dependence. We find fair agreement between the results of the new theory and a numerical simulation of the model. For systems with finite size we find that the moments strongly depend on the number of sites, which poses a caveat for extrapolating finite-system simulations to the infinite-size limit.