62 research outputs found
A Model for the Propagation of Sound in Granular Materials
This paper presents a simple ball-and-spring model for the propagation of
small amplitude vibrations in a granular material. In this model, the
positional disorder in the sample is ignored and the particles are placed on
the vertices of a square lattice. The inter-particle forces are modeled as
linear springs, with the only disorder in the system coming from a random
distribution of spring constants. Despite its apparent simplicity, this model
is able to reproduce the complex frequency response seen in measurements of
sound propagation in a granular system. In order to understand this behavior,
the role of the resonance modes of the system is investigated. Finally, this
simple model is generalized to include relaxation behavior in the force network
-- a behavior which is also seen in real granular materials. This model gives
quantitative agreement with experimental observations of relaxation.Comment: 21 pages, requires Harvard macros (9/91), 12 postscript figures not
included, HLRZ preprint 6/93, (replacement has proper references included
Anderson transitions : multifractal or non-multifractal statistics of the transmission as a function of the scattering geometry
The scaling theory of Anderson localization is based on a global conductance
that remains a random variable of order O(1) at criticality. One
realization of such a conductance is the Landauer transmission for many
transverse channels. On the other hand, the statistics of the one-channel
Landauer transmission between two local probes is described by a multifractal
spectrum that can be related to the singularity spectrum of individual
eigenstates. To better understand the relations between these two types of
results, we consider various scattering geometries that interpolate between
these two cases and analyse the statistics of the corresponding transmissions.
We present detailed numerical results for the power-law random banded matrices
(PRBM model). Our conclusions are : (i) in the presence of one isolated
incoming wire and many outgoing wires, the transmission has the same
multifractal statistics as the local density of states of the site where the
incoming wire arrives; (ii) in the presence of backward scattering channels
with respect to the case (i), the statistics of the transmission is not
multifractal anymore, but becomes monofractal. Finally, we also describe how
these scattering geometries influence the statistics of the transmission off
criticality.Comment: 12 pages, 9 figure
Anderson transition on the Cayley tree as a traveling wave critical point for various probability distributions
For Anderson localization on the Cayley tree, we study the statistics of
various observables as a function of the disorder strength and the number
of generations. We first consider the Landauer transmission . In the
localized phase, its logarithm follows the traveling wave form where (i) the disorder-averaged value moves linearly
and the localization length
diverges as with (ii) the
variable is a fixed random variable with a power-law tail for large with , so that all
integer moments of are governed by rare events. In the delocalized phase,
the transmission remains a finite random variable as , and
we measure near criticality the essential singularity with . We then consider the
statistical properties of normalized eigenstates, in particular the entropy and
the Inverse Participation Ratios (I.P.R.). In the localized phase, the typical
entropy diverges as with , whereas it grows
linearly in in the delocalized phase. Finally for the I.P.R., we explain
how closely related variables propagate as traveling waves in the delocalized
phase. In conclusion, both the localized phase and the delocalized phase are
characterized by the traveling wave propagation of some probability
distributions, and the Anderson localization/delocalization transition then
corresponds to a traveling/non-traveling critical point. Moreover, our results
point towards the existence of several exponents at criticality.Comment: 28 pages, 21 figures, comments welcom
Non-hermitean delocalization in an array of wells with variable-range widths
Nonhermitean hamiltonians of convection-diffusion type occur in the
description of vortex motion in the presence of a tilted magnetic field as well
as in models of driven population dynamics. We study such hamiltonians in the
case of rectangular barriers of variable size. We determine Lyapunov exponent
and wavenumber of the eigenfunctions within an adiabatic approach, allowing to
reduce the original d=2 phase space to a d=1 attractor. PACS
numbers:05.70.Ln,72.15Rn,74.60.GeComment: 20 pages,10 figure
Collision and symmetry-breaking in the transition to strange nonchaotic attractors
Strange nonchaotic attractors (SNAs) can be created due to the collision of
an invariant curve with itself. This novel ``homoclinic'' transition to SNAs
occurs in quasiperiodically driven maps which derive from the discrete
Schr\"odinger equation for a particle in a quasiperiodic potential. In the
classical dynamics, there is a transition from torus attractors to SNAs, which,
in the quantum system is manifest as the localization transition. This
equivalence provides new insights into a variety of properties of SNAs,
including its fractal measure. Further, there is a {\it symmetry breaking}
associated with the creation of SNAs which rigorously shows that the Lyapunov
exponent is nonpositive. By considering other related driven iterative
mappings, we show that these characteristics associated with the the appearance
of SNA are robust and occur in a large class of systems.Comment: To be appear in Physical Review Letter
Anderson localization transition with long-ranged hoppings : analysis of the strong multifractality regime in terms of weighted Levy sums
For Anderson tight-binding models in dimension with random on-site
energies and critical long-ranged hoppings decaying
typically as , we show that the strong multifractality
regime corresponding to small can be studied via the standard perturbation
theory for eigenvectors in quantum mechanics. The Inverse Participation Ratios
, which are the order parameters of Anderson transitions, can be
written in terms of weighted L\'evy sums of broadly distributed variables (as a
consequence of the presence of on-site random energies in the denominators of
the perturbation theory). We compute at leading order the typical and
disorder-averaged multifractal spectra and as a
function of . For , we obtain the non-vanishing limiting spectrum
as . For , this method
yields the same disorder-averaged spectrum of order as
obtained previously via the Levitov renormalization method by Mirlin and Evers
[Phys. Rev. B 62, 7920 (2000)]. In addition, it allows to compute explicitly
the typical spectrum, also of order , but with a different -dependence
for all . As a consequence, we find
that the corresponding singularity spectra and
differ even in the positive region , and vanish at
different values , in contrast to the standard
picture. We also obtain that the saddle value of the Legendre
transform reaches the termination point where
only in the limit .Comment: 13 pages, 2 figures, v2=final versio
Anderson localization on the Cayley tree : multifractal statistics of the transmission at criticality and off criticality
In contrast to finite dimensions where disordered systems display
multifractal statistics only at criticality, the tree geometry induces
multifractal statistics for disordered systems also off criticality. For the
Anderson tight-binding localization model defined on a tree of branching ratio
K=2 with generations, we consider the Miller-Derrida scattering geometry
[J. Stat. Phys. 75, 357 (1994)], where an incoming wire is attached to the root
of the tree, and where outcoming wires are attached to the leaves of
the tree. In terms of the transmission amplitudes , the total
Landauer transmission is , so that each channel
is characterized by the weight . We numerically measure the
typical multifractal singularity spectrum of these weights as a
function of the disorder strength and we obtain the following conclusions
for its left-termination point . In the delocalized phase ,
is strictly positive and is associated with a
moment index . At criticality, it vanishes and is
associated with the moment index . In the localized phase ,
is associated with some moment index . We discuss the
similarities with the exact results concerning the multifractal properties of
the Directed Polymer on the Cayley tree.Comment: v2=final version (16 pages
Trapping of Projectiles in Fixed Scatterer Calculations
We study multiple scattering off nuclei in the closure approximation. Instead
of reducing the dynamics to one particle potential scattering, the scattering
amplitude for fixed target configurations is averaged over the target
groundstate density via stochastic integration. At low energies a strong
coupling limit is found which can not be obtained in a first order optical
potential approximation. As its physical explanation, we propose it to be
caused by trapping of the projectile. We analyse this phenomenon in mean field
and random potential approximations.
(PACS: 24.10.-i)Comment: 15 page
Localization of a polymer in random media: Relation to the localization of a quantum particle
In this paper we consider in detail the connection between the problem of a
polymer in a random medium and that of a quantum particle in a random
potential. We are interested in a system of finite volume where the polymer is
known to be {\it localized} inside a low minimum of the potential. We show how
the end-to-end distance of a polymer which is free to move can be obtained from
the density of states of the quantum particle using extreme value statistics.
We give a physical interpretation to the recently discovered one-step
replica-symmetry-breaking solution for the polymer (Phys. Rev. E{\bf 61}, 1729
(2000)) in terms of the statistics of localized tail states. Numerical
solutions of the variational equations for chains of different length are
performed and compared with quenched averages computed directly by using the
eigenfunctions and eigenenergies of the Schr\"odinger equation for a particle
in a one-dimensional random potential. The quantities investigated are the
radius of gyration of a free gaussian chain, its mean square distance from the
origin and the end-to-end distance of a tethered chain. The probability
distribution for the position of the chain is also investigated. The glassiness
of the system is explained and is estimated from the variance of the measured
quantities.Comment: RevTex, 44 pages, 13 figure
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