Diffusion of vibrations in disordered systems (original) (raw)

Ioffe-Regel criterion and diffusion of vibrations in random lattices

Physical Review B, 2013

We consider diffusion of vibrations in 3d harmonic lattices with strong force-constant disorder. Above some frequency ωIR, corresponding to the Ioffe-Regel crossover, notion of phonons becomes ill defined. They cannot propagate through the system and transfer energy. Nevertheless most of the vibrations in this range are not localized. We show that they are similar to diffusons introduced by Allen, Feldman et al., Phil. Mag. B 79, 1715 to describe heat transport in glasses. The crossover frequency ωIR is close to the position of the boson peak. Changing strength of disorder we can vary ωIR from zero value (when rigidity is zero and there are no phonons in the lattice) up to a typical frequency in the system. Above ωIR the energy in the lattice is transferred by means of diffusion of vibrational excitations. We calculated the diffusivity of the modes D(ω) using both the direct numerical solution of Newton equations and the formula of Edwards and Thouless. It is nearly a constant above ωIR and goes to zero at the localization threshold. We show that apart from the diffusion of energy, the diffusion of particle displacements in the lattice takes place as well. Above ωIR a displacement structure factor S(q, ω) coincides well with a structure factor of random walk on the lattice. As a result the vibrational line width Γ(q) = Duq 2 where Du is a diffusion coefficient of particle displacements. Our findings may have important consequence for the interpretation of experimental data on inelastic x-ray scattering and mechanisms of heat transfer in glasses.

Vibrations in amorphous solids beyond the Ioffe-Regel criterion

Journal of Physics: Conference Series, 2013

We consider vibrations in 3d random harmonic lattices with translational invariance as a model of amorphous solid. Above some frequency ωIR, corresponding to the Ioffe-Regel crossover, notion of phonons becomes ill defined. They cannot propagate through the lattice and transfer energy. Nevertheless most of the vibrations in this range are not localized. Changing strength of disorder we can vary ωIR from zero value (when rigidity is zero and there are no phonons in the lattice) up to a typical frequency in the system. Above ωIR a displacement structure factor S(q, ω) coincides well with a structure factor of random walk on the lattice. As a result the vibrational line width Γ(q) = Duq 2 where Du is a diffusion coefficient of particle displacements. Our findings may have important consequence for the interpretation of experimental data on inelastic x-ray scattering in glasses.

Non-Markoffian diffusion in a one-dimensional disordered lattice

Journal of Statistical Physics, 1982

Recent treatments of diffusion in a one-dimensional disordered lattice by Machta using a renormalization-group approach, and by Alexander and Orbach using an effective medium approach, lead to a frequency-dependent (or non-Markoffian) diffusion coefficient. Their resUlts are confirmed by a direct calculation of the diffusion coefficient.

Kinetic effects in diffusion on a disordered square lattice

In this work, the effect of fluctuations in a disordered square lattice on diffusion of a test particle is studied using kinetic Monte Carlo simulations. Diffusion is relevant to a wide variety of problems, both within physics and outside of physics. Kinetic effects in diffusion are often hidden in a thermodynamical description of the problem. In this work, no assumptions based on energy are made, and diffusion occurs purely based on the attempt rate of the test particle and the occupation and fluctuation rate of the lattice. Although the average transition rate of the particle is the same for a static or fluctuating lattice with specific occupation, the diffusion constant is kinetically affected in a fluctuating, disordered lattice. If the lattice fluctuates faster than the attempt rate of the particle, diffusion is controlled by the attempt rate of the particle. However, if the lattice fluctuates slower than the attempt rate of the particle, diffusion is affected by the fluctuations. The slower the lattice fluctuates, the lower the diffusion constant. Furthermore, it is found that for fast fluctuating lattices, diffusion is due to Brownian motion. If the lattice fluctuates slower than the particle, diffusion becomes anomalous depending on the occupation of the lattice.

Heat transport and phonon localization in mass-disordered harmonic crystals

Physical Review B, 2010

We investigate the steady state heat current in two and three dimensional disordered harmonic crystals in a slab geometry, connected at the boundaries to stochastic white noise heat baths at different temperatures.The disorder causes short wavelength phonon modes to be localized so the heat current in this system is carried by the extended phonon modes which can be either diffusive or ballistic. Using ideas both from localization theory and from kinetic theory we estimate the contribution of various modes to the heat current and from this we obtain the asymptotic system size dependence of the current. These estimates are compared with results obtained from a numerical evaluation of an exact formula for the current, given in terms of a frequency transmission function, as well as from direct nonequilibrium simulations. These yield a strong dependence of the heat flux on boundary conditions. Our analytical arguments show that for realistic boundary conditions the conductivity is finite in three dimensions but we are not able to verify this numerically, except in the case where the system is subjected to an external pinning potential. This case is closely related to the problem of localization of electrons in a random potential and here we numerically verify that the pinned three dimensional system satisfies Fourier's law while the two dimensional system is a heat insulator. We also investigate the inverse participation ratio of different normal modes.

Biased random walk in energetically disordered lattices

Physical Review E, 1998

We utilize our previously reported model of energetically disordered lattices to study diffusion properties, where we now add the effect of a directional bias in the motion. We show how this leads to ballistic motion at low temperatures, but crosses over to normal diffusion with increasing temperature. This effect is in addition to the previously observed subdiffusional motion at early times, which is also observed here, and also crosses over to normal diffusion at long times. The interplay between these factors of the two crossover points is examined here in detail. The pertinent scaling laws are given for the crossover times. Finally, we deal with the case of the frequency dependent bias, which alternates ͑switches͒ its direction with a given frequency, resulting in a different type of scaling. ͓S1063-651X͑98͒11008-5͔

Small-world disordered lattices: spectral gaps and diffusive transport

New Journal of Physics

We investigate the dynamic behavior of lattices with disorder introduced through non-local network connections. Inspired by the Watts–Strogatz small-world model, we employ a single parameter to determine the probability of local connections being re-wired, and to induce transitions between regular and disordered lattices. These connections are added as non-local springs to underlying periodic one-dimensional (1D) and two-dimensional (2D) square, triangular and hexagonal lattices. Eigenmode computations illustrate the emergence of spectral gaps in various representative lattices for increasing degrees of disorder. These gaps manifest themselves as frequency ranges where the modal density goes to zero, or that are populated only by localized modes. In both cases, we observe low transmission levels of vibrations across the lattice. Overall, we find that these gaps are more pronounced for lattice topologies with lower connectivity, such as the 1D lattice or the 2D hexagonal lattice. We ...

Theory of vibration propagation in disordered media

Wave Motion, 2007

Irregularity can have a significant impact on the vibrational behavior of elastic systems and can effect a broad range of physical properties, ranging from the acoustic scattering cross section of marine structures to the thermal conductivity of semi-conductors. In many instances, the spatial behavior of the modes of the system is fundamentally altered by the irregularity and decays exponentially with distance. This behavior is known as Anderson localization and dynamically generates a new length scale, the localization length n. The modeling of such systems can be quite challenging, with numerical simulations often being misleading owing to finite size effects, and analytical methods being highly specialized and inaccessible to the non-expert in many body theory. We present here an approach which has been highly successful in recent years, the self-consistent diagrammatic theory. Published by Elsevier B.V.