The physics of granular materials (e.g. sand, soil etc.) is one of the most fashionable current topics in the modern condensed matter physics [1]. The vibrational dynamics of granular materials (e.g. sound propagation) is of particular interest in the field. A rigorous analytical theory of the  dynamics of granular materials has not yet been developed and only some simple analytical models have been suggested.

The aim of this project is to develop a simple structural model of a granular material and investigate its dynamical properties. The model will be based on a regular lattice of spheres (or discs in 2D) with a random distribution of  radii. Such a model can be linked to the model of atomic vibrations in  force-constant disordered lattices, that we have studied [2].  The first aim of the project is to find connections between  the distribution of the radii of granules and the distribution of the effective spring constants. This can  be done analytically and/or numerically. Following this, there are two possible independent aspects of the project, either:

(i) To use a mean-field approach for calculation of the vibrational spectrum of the model of granular materials based on a known (calculated or simulated) probability distribution of the effective force constants. This should be able to  be done analytically in some limiting cases (narrow distributions, low-energy limit) and generally numerically  by solving self-consistent integral equations,

or:

(ii) Exact numerical diagonalization (thereby obtaining the vibrational spectrum)  of the dynamical matrix of second derivatives of the interaction potential describing the vibrational dynamics of  models of granular materials. In this variant of the project, it will be desirable to go beyond the simplified lattice model (which is treatable analytically) and  to create (numerically) topologically disordered structural models of granular materials.