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 $g_L$ 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

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

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

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 $W$ and the number $N$ of generations. We first consider the Landauer transmission $T_N$. In the localized phase, its logarithm follows the traveling wave form $\ln T_N \simeq \bar{\ln T_N} + \ln t^*$ where (i) the disorder-averaged value moves linearly $\bar{\ln (T_N)} \simeq - \frac{N}{\xi_{loc}}$ and the localization length diverges as $\xi_{loc} \sim (W-W_c)^{-\nu_{loc}}$ with $\nu_{loc}=1$ (ii) the variable $t^*$ is a fixed random variable with a power-law tail $P^*(t^*) \sim 1/(t^*)^{1+\beta(W)}$ for large $t^*$ with $0<\beta(W) \leq 1/2$, so that all integer moments of $T_N$ are governed by rare events. In the delocalized phase, the transmission $T_N$ remains a finite random variable as $N \to \infty$, and we measure near criticality the essential singularity $\bar{\ln (T)} \sim - | W_c-W |^{-\kappa_T}$ with $\kappa_T \sim 0.25$. 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 $(W-W_c)^{- \nu_S}$ with $\nu_S \sim 1.5$, whereas it grows linearly in $N$ 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 $\nu$ at criticality.Comment: 28 pages, 21 figures, comments welcom

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 $d$ with random on-site energies $\epsilon_{\vec r}$ and critical long-ranged hoppings decaying typically as $V^{typ}(r) \sim V/r^d$, we show that the strong multifractality regime corresponding to small $V$ can be studied via the standard perturbation theory for eigenvectors in quantum mechanics. The Inverse Participation Ratios $Y_q(L)$, 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 $\tau_{typ}(q)$ and $\tau_{av}(q)$ as a function of $q$. For $q<1/2$, we obtain the non-vanishing limiting spectrum $\tau_{typ}(q)=\tau_{av}(q)=d(2q-1)$ as $V \to 0^+$. For $q>1/2$, this method yields the same disorder-averaged spectrum $\tau_{av}(q)$ of order $O(V)$ 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 $O(V)$, but with a different $q$-dependence $\tau_{typ}(q) \ne \tau_{av}(q)$ for all $q>q_c=1/2$. As a consequence, we find that the corresponding singularity spectra $f_{typ}(\alpha)$ and $f_{av}(\alpha)$ differ even in the positive region $f>0$, and vanish at different values $\alpha_+^{typ} > \alpha_+^{av}$, in contrast to the standard picture. We also obtain that the saddle value $\alpha_{typ}(q)$ of the Legendre transform reaches the termination point $\alpha_+^{typ}$ where $f_{typ}(\alpha_+^{typ})=0$ only in the limit $q \to +\infty$.Comment: 13 pages, 2 figures, v2=final versio

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

Comparison of two recombinant erythropoietin formulations in patients with anemia due to end-stage renal disease on hemodialysis: A parallel, randomized, double blind study

BACKGROUND: Recombinant human erythropoietin (EPO) is used for the treatment of last stage renal anemia. A new EPO preparation was obtained in Cuba in order to make this treatment fully nationally available. The aim of this study was to compare the pharmacokinetic, pharmacodynamic and safety properties of two recombinant EPO formulations in patients with anemia due to end-stage renal disease on hemodialysis. METHODS: A parallel, randomized, double blind study was performed. A single 100 IU/Kg EPO dose was administered subcutaneously. Heberitro (Heber Biotec, Havana, formulation A), a newly developed product and Eprex (CILAG AG, Switzerland, formulation B), as reference treatment were compared. Thirty-four patients with anemia due to end-stage renal disease on hemodialysis were included. Patients had not received EPO previously. Serum EPO level was measured by enzyme immunoassay (EIA) during 120 hours after administration. Clinical and laboratory variables were determined as pharmacodynamic and safety criteria until 216 hours. RESULTS: Both groups of patients were similar regarding all demographic and baseline characteristics. EPO kinetics profiles were similar for both formulations; the pharmacokinetic parameters were very close (i.e., AUC: 4667 vs. 4918 mIU.h/mL; Cmax: 119.1 vs. 119.7 mIU/mL; Tmax: 13.9 vs. 18.1 h; half-life, 20.0 vs. 22.5 h for formulations A and B, respectively). The 90% confidence intervals for the ratio between both products regarding these metrics were close to the 0.8 – 1.25 range, considered necessary for bioequivalence. Differences did not reach 20% in any case and were not determined by a formulation effect, but probably by a patients' variability effect. Concerning pharmacodynamic features, a high similitude in reticulocyte counts increments until 216 hours and the percentage decrease in serum iron until 120 hours was observed. There were no differences between formulations regarding the adverse events and their intensity. The more frequent events were pain at injection site (35.3%) and hypertension (29%). Additionally, further treatment of the patients with the study product yielded satisfactory increases in hemoglobin and hematocrit values. CONCLUSION: The formulations are comparable. The newly developed product should be acceptable for long-term application