The dynamical structure factor in topologically disordered systems

Abstract

A computation of the dynamical structure factor of topologically disordered systems, where the disorder can be described in terms of euclidean random matrices, is presented. Among others, structural glasses and supercooled liquids belong to that class of systems. The computation describes their relevant spectral features in the region of the high frequency sound. The analytical results are tested with numerical simulations and are found to be in very good agreement with them. Our results may explain the findings of inelastic X-ray scattering experiments in various glassy systems.Comment: Version to be published in J. Chem. Phy

Brillouin and boson peaks in glasses from vector Euclidean random matrix theory

“…For example, the repetition scheme (28) involved 16 diagrams in the previous version (see appendix of ref. 13), while it now corresponds to just one. The other two diagrams for Σ (2) , which will be neglected in the resummation used below (see Fig.…”

“…This three diagrams (see Fig.2), vanish independently for p = 0, and (in the scalar case) are equivalent to the 39 diagrams of the previuosly published expansion [13]. Given these rules, it is a well-known combinatorial result (Dyson equation) that the sum of all planar diagrams (cactus approximation) takes the form of a self-consistent integral equation for the self-energy:…”

“…For the moment, let us show the main steps of the computation in this simplified situation. The computation of (5) is the generalization of the scalar vibrations case [13,14], so we shall only outline it, stressing the differences. Proceeding as in the scalar case, one first expands the propagator in 1/z:…”

“…Organizing the calculation as explained in ref. 13, one easily finds A (r,r) µν (p) in terms off µν (p), the Fourier transform of f µν (r), that can be separated in longitudinal transverse parts:…”

“…There are basically two such approaches: modified mode-coupling theory [11] (which is not limited to harmonic excitations), and euclidean random matrix theory (ERMT) [12,13,14]. ERMT owes its name to the fact that it formulates the vibrational problem as random matrix problem [15].…”