Evolution of a vibrational wave packet on a disordered chain (original) (raw)

Asymptotic energy profile of a wave packet in disordered chains

Physical Review E, 2010

We investigate the long-time behavior of a wave packet initially localized at a single site n 0 in translationally invariant harmonic and anharmonic chains with random interactions. In the harmonic case, the energy profile ͗e n ͑t͒͘ averaged on time and disorder decays for large ͉n − n 0 ͉ as a power law ͗e n ͑t͒͘ Ϸ C͉n − n 0 ͉ − , where =5/ 2 and 3/2 for initial displacement and momentum excitations, respectively. The prefactor C depends on the probability distribution of the harmonic coupling constants and diverges in the limit of weak disorder. As a consequence, the moments ͗m ͑t͒͘ of the energy distribution averaged with respect to disorder diverge in time as t ␤͑͒ for Ն 2, where ␤ = +1− for Ͼ − 1. Molecular-dynamics simulations yield good agreement with these theoretical predictions. Therefore, in this system, the second moment of the wave packet diverges as a function of time despite the wave packet is not spreading. Thus, this only criterion, often considered earlier as proving the spreading of a wave packet, cannot be considered as sufficient in any model. The anharmonic case is investigated numerically. It is found for intermediate disorder that the tail of the energy profile becomes very close to those of the harmonic case. For weak and strong disorders, our results suggest that the crossover to the harmonic behavior occurs at much larger ͉n − n 0 ͉ and larger time.

Delocalization of wave packets in disordered nonlinear chains

Physical Review E, 2009

We consider the spatiotemporal evolution of a wave packet in disordered nonlinear Schrödinger and anharmonic oscillator chains. In the absence of nonlinearity all eigenstates are spatially localized with an upper bound on the localization length (Anderson localization). Nonlinear terms in the equations of motion destroy Anderson localization due to nonintegrability and deterministic chaos. At least a finite part of an initially localized wave packet will subdiffusively spread without limits. We analyze the details of this spreading process. We compare the evolution of single site, single mode and general finite size excitations, and study the statistics of detrapping times. We investigate the properties of mode-mode resonances, which are responsible for the incoherent delocalization process.

Diffusion of vibrations in disordered systems

JETP Letters, 2013

We consider diffusion of vibrations in random lattices with translational invariance. Above the frequency ω IR corresponding to the Ioffe-Regel crossover (and depending on the strength of disorder), phonons cannot propagate through the lattice and transfer energy. At the same time, most of the vibrations in this range are not localized. We show that these delocalized excitations are similar to diffusons introduced by P. B. Allen, J. L. Feldman, J. Fabian, and F. Wooten (see, e.g., Phil. Mag. B 79, 1715 (1999)) to describe heat transport in glasses. In this range the energy in the lattice is transferred by means of diffusion of vibrational excitations. We have calculated the diffusivity of the modes D(ω) using both the direct numerical solution of Newton equations and the Edwards-Thouless formula. It is nearly constant above ω IR and goes to zero at the local ization threshold.

Wave localization in strongly nonlinear Hertzian chains with mass defect

Physical Review E, 2009

We investigate the dynamical response of a mass defect in a one-dimensional non-loaded horizontal chain of identical spheres which interact via the nonlinear Hertz potential. Our experiments show that the interaction of a solitary wave with a light intruder excites localized mode. In agreement with dimensional analysis, we find that the frequency of localized oscillations exceeds the incident wave frequency spectrum and nonlinearly depends on the size of the intruder and on the incident wave strength. The absence of tensile stress between grains allows some gaps to open, which in turn induce a significant enhancement of the oscillations amplitude. We performed numerical simulations that precisely describe our observations without any adjusting parameters.

Dynamics of Chain Molecules in Disordered Materials

Physical Review Letters, 2006

The dynamic behavior of hard chains in disordered materials composed of fixed hard spheres is studied using discontinuous molecular dynamics simulations. The matrix induces entanglements in the chain fluid, i.e., for high matrix densities the diffusion coefficient D scales with the chain length N as D N ÿ2. At high matrix densities the rotational relaxation time becomes very large but the translational diffusion is not affected significantly; i.e., the chains display a dynamic heterogeneity reminiscent of probe diffusion in supercooled liquids and glasses. We show that this is because some chains are trapped, and move via a hopping mechanism. There are no signatures of this dynamic heterogeneity in the matrix static structure, however, which is identical to that of a hard-sphere liquid.

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.

Propagation of mechanical waves in a one-dimensional nonlinear disordered lattice

Physical Review E, 2006

The propagation of transverse waves along a string loaded by N masses, each of them being fixed to a spring with a quadratic nonlinearity, is studied. After presenting the nonlinear model and stating the equation of propagation into a lattice with discrete nonlinearities and disorder, we propose a perturbation approach to wave propagation in a nonlinear lattice using the Green's function formalism. We show how the nonlinearity acts on the propagation into a disordered lattice. In the low-frequency approximation, an analytical expression of the boundary between the propagative regime and the evanescent one is found. Numerical results are compared to the analytical results and phase diagrams are proposed in the ordered and disordered cases. A behavior of the transmission coefficient is found, on an empirical basis, as a function of the length of the lattice and the localization length in the nonlinear case. Finally, a dynamic approach is developed and the ordered and disordered cases are addressed. This method is based on a finite difference equation and allows the construction of the Poincaré section describing the propagation of the wave into the lattice. This approach distinguishes between the properties of propagation in the lattice in a propagative regime and in an evanescent one.

Diffusive Propagation of Energy in a Non-acoustic Chain

Archive for Rational Mechanics and Analysis, 2016

We consider a non acoustic chain of harmonic oscillators with the dynamics perturbed by a random local exchange of momentum, such that energy and momentum are conserved. The macroscopic limits of the energy density, momentum and the curvature (or bending) of the chain satisfy a system of evolution equations. We prove that, in a diffusive space-time scaling, the curvature and momentum evolve following a linear system that corresponds to a damped Euler-Bernoulli beam equation. The macroscopic energy density evolves following a non linear diffusive equation. In particular the energy transfer is diffusive in this dynamics. This provides a first rigorous example of a normal diffusion of energy in a one dimensional dynamics that conserves the momentum.

Nonlinear waves in disordered diatomic granular chains

Physical Review E, 2010

We investigate the propagation and scattering of highly nonlinear waves in disordered granular chains composed of diatomic (two-mass) units of spheres that interact via Hertzian contact. Using ideas from statistical mechanics, we consider each diatomic unit to be a "spin", so that a granular chain can be viewed as a spin chain composed of units that are each oriented in one of two possible ways. Experiments and numerical simulations both reveal the existence of two different mechanisms of wave propagation: In low-disorder chains, we observe the propagation of a solitary pulse with exponentially decaying amplitude. Beyond a critical level of disorder, the wave amplitude instead decays as a power law, and the wave transmission becomes insensitive to the level of disorder. We characterize the spatio-temporal structure of the wave in both propagation regimes and propose a simple theoretical interpretation for such a transition. Our investigation suggests that an elastic spin chain can be used as a model system to investigate the role of heterogeneities in the propagation of highly nonlinear waves.

Vibrations in a Growing Nonlinear Chain

Discontinuity, Nonlinearity and Complexity, 2021

A one-dimensional chain describing the linear statistical increment of growing homogeneous atoms was arbitrarily built and investigated using an energy potential function. The analytic form of the considered potential has two exponential terms which describes chaotic behavior when the chain was excited. In order to investigate the dynamics of statistical attachment of individual atoms in the slender gold chain, the total energy of the entire system was changed by increasing the kinetic energy upon increment of homogeneous atoms in the chain. This resulted in a corresponding increase of the total energy in the system. On the other hand, the potential energy of the system on increment of homogeneous atoms equals zero, because the distance between corresponding atoms equals to the molecular distance (minimum potential distance). We considered the dynamical system with linear damping and without linear damping. Different initial points were investigated to obtain trends of vibration that includes chaotic and regular oscillations. At some initial point(s), the attached atom experiences an infinite jump which means it falls off the nonlinear slender chain and the chain was broken. The interpretation of this phenomenon means the gold chain will result into an unstable nanostructure. We compared the numerical simulation of the system with different built-in ordinary differential equation solvers of various computer algebra software. Numerical simulation were carried out by plotting the system of growing atoms' displacement against time. The system of linearly attached atoms were numerically simulated and inferences were stated from the study. In all cases considered, we inferred that amplitude of oscillation significantly increased at the end of the chain (terminal point) as compared to the initial point the oscillation started.