Properties of stable ensembles of Euclidean random matrices
2024-01-17, Baumgärtel, Philipp, Vogel, Florian, Fuchs, Matthias
We study the spectrum of a system of coupled disordered harmonic oscillators in the thermodynamic limit. This Euclidean random matrix ensemble has been suggested as a model for the low temperature vibrational properties of glass. Exact numerical diagonalization is performed in three and two spatial dimensions, which is accompanied by a detailed finite size analysis. It reveals a low-frequency regime of sound waves that are damped by Rayleigh scattering. At large frequencies localized modes exist. In between, the central peak in the vibrational density of states is well described by Wigner's semicircle law for not too large disorder, as is expected for simple random matrix systems. We compare our results with predictions from two recent self-consistent field theories.
Vibrational Phenomena in Glasses at Low Temperatures Captured by Field Theory of Disordered Harmonic Oscillators
2023-06-07, Vogel, Florian, Fuchs, Matthias
We investigate the vibrational properties of topologically disordered materials by analytically studying particles that harmonically oscillate around random positions. Exploiting classical field theory in the thermodynamic limit at T=0, we build up a self-consistent model by analyzing the Hessian utilizing Euclidean random matrix theory. In accordance with earlier findings [T. S. Grigera et al.J. Stat. Mech. (2011) P02015.], we take nonplanar diagrams into account to correctly address multiple local scattering events. By doing so, we end up with a first principles theory that can predict the main anomalies of athermal disordered materials, including the boson peak, sound softening, and Rayleigh damping of sound. In the vibrational density of states, the sound modes lead to Debye’s law for small frequencies. Additionally, an excess appears in the density of states starting as ω4 in the low frequency limit, which is attributed to (quasi-) localized modes.
Stress correlation function and linear response of Brownian particles
2020, Vogel, Florian, Fuchs, Matthias
Abstract.We determine the non-local stress autocorrelation tensor in an homogeneous and isotropic systemof interacting Brownian particles starting from the Smoluchowski equation of the configurational probabil-ity density. In order to relate stresses to particle displacements as appropriate in viscoelastic states, we gobeyond the usual hydrodynamic description obtained in the Zwanzig-Mori projection-operator formalismby introducing the proper irreducible dynamics following Cichocki and Hess, andKawasaki. Differentlyfrom these authors, we include transverse contributions as well. This recovers theexpression for the stressautocorrelation including the elastic terms in solid states as found for Newtonian and Langevin systems, incase that those are evaluated in the overdamped limit. Finally, we arguethat the found memory functionreduces to the shear and bulk viscosity in the hydrodynamic limit of smooth and slow fluctuations andderive the corresponding hydrodynamic equations.