arXiv subjects

H. E. Stanley

Publications and source records attributed to H. E. Stanley.

80 records · Page 5Linked to original sources

arXiv1997

Scaling behavior in economics: II. Modeling of company growth

In the preceding paper we presented empirical results describing the growth of publicly-traded United States manufacturing firms within the years 1974--1993. Our results suggest that the data can be described by a scaling approach. Here, we propose models that may lead to some insight into these phenomena. First, we study a model in which the growth rate of a company is affected by a tendency to retain an ``optimal'' size. That model leads to an exponential distribution of the logarithm of the growth rate in agreement with the empirical results. Then, we study a hierarchical tree-like model of a company that enables us to relate the two parameters of the model to the exponent $β$, which describes the dependence of the standard deviation of the distribution of growth rates on size. We find that $β= -\ln Π/ \ln z$, where $z$ defines the mean branching ratio of the hierarchical tree and $Π$ is the probability that the lower levels follow the policy of higher levels in the hierarchy. We also study the distribution of growth rates of this hierarchical model. We find that the distribution is consistent with the exponential form found empirically.

cond-mat.stat-mech↗

arXiv1996

Avalanches in the lung: A statistical mechanical model

We study a statistical mechanical model for the dynamics of lung inflation which incorporates recent experimental observations on the opening of individual airways by a cascade or avalanche mechanism. Using an exact mapping of the avalanche problem onto percolation on a Cayley tree, we analytically derive the exponents describing the size distribution of the first avalanches and test the analytical solution by numerical simulations. We find that the tree-like structure of the airways together with the simplest assumptions concerning opening threshold pressures of each airway, is sufficient to explain the existence of power-law distributions observed experimentally.

cond-mat↗

arXiv1995

Quantitative Characterization of Permeability Fluctuations in Sandstone

Sedimentary rocks have complicated permeability fluctuations arising from the geological processes that formed them. These permeability fluctuations significantly affect the flow of fluids through the rocks. We analyze data on two sandstone samples from different geological environments, and find that the permeability fluctuations display long-range power-law correlations characterized by an exponent $H$. For both samples, we find $H \approx 0.82-0.90$.

cond-mat↗

arXiv1995

Stochastic Model for Surface Erosion Via Ion-Sputtering: Dynamical Evolution from Ripple Morphology to Rough Morphology

Surfaces eroded by ion-sputtering are sometimes observed to develop morphologies which are either ripple (periodic), or rough (non-periodic). We introduce a discrete stochastic model that allows us to interpret these experimental observations within a unified framework. We find that a periodic ripple morphology characterizes the initial stages of the evolution, whereas the surface displays self-affine scaling in the later time regime. Further, we argue that the stochastic continuum equation describing the surface height is a noisy version of the Kuramoto-Sivashinsky equation.

cond-mat↗

arXiv1995

Scaling properties of driven interfaces in disordered media

We perform a systematic study of several models that have been proposed for the purpose of understanding the motion of driven interfaces in disordered media. We identify two distinct universality classes: (i) One of these, referred to as directed percolation depinning (DPD), can be described by a Langevin equation similar to the Kardar-Parisi-Zhang equation, but with quenched disorder. (ii) The other, referred to as quenched Edwards-Wilkinson (QEW), can be described by a Langevin equation similar to the Edwards-Wilkinson equation but with quenched disorder. We find that for the DPD universality class the coefficient $λ$ of the nonlinear term diverges at the depinning transition, while for the QEW universality class either $λ= 0$ or $λ\to 0$ as the depinning transition is approached. The identification of the two universality classes allows us to better understand many of the results previously obtained experimentally and numerically. However, we find that some results cannot be understood in terms of the exponents obtained for the two universality classes {\it at\/} the depinning transition. In order to understand these remaining disagreements, we investigate the scaling properties of models in each of the two universality classes {\it above\/} the depinning transition. For the DPD universality class, we find for the roughness exponent $α_P = 0.63 \pm 0.03$ for the pinned phase, and $α_M = 0.75 \pm 0.05$ for the moving phase. For the growth exponent, we find $β_P = 0.67 \pm 0.05$ for the pinned phase, and $β_M = 0.74 \pm 0.06$ for the moving phase. Furthermore, we find an anomalous scaling of the prefactor of the width on the driving force. A new exponent $φ_M = -0.12 \pm 0.06$, characterizing the scaling of this prefactor, is shown to relate the values of the roughness

cond-mat↗

arXiv1995

Scaling in Late Stage Spinodal Decomposition with Quenched Disorder

We study the late stages of spinodal decomposition in a Ginzburg-Landau mean field model with quenched disorder. Random spatial dependence in the coupling constants is introduced to model the quenched disorder. The effect of the disorder on the scaling of the structure factor and on the domain growth is investigated in both the zero temperature limit and at finite temperature. In particular, we find that at zero temperature the domain size, $R(t)$, scales with the amplitude, $A$, of the quenched disorder as $R(t) = A^{-β} f(t/A^{-γ})$ with $β\simeq 1.0$ and $γ\simeq 3.0$ in two dimensions. We show that $β/γ= α$, where $α$ is the Lifshitz-Slyosov exponent. At finite temperature, this simple scaling is not observed and we suggest that the scaling also depends on temperature and $A$. We discuss these results in the context of Monte Carlo and cell dynamical models for phase separation in systems with quenched disorder, and propose that in a Monte Carlo simulation the concentration of impurities, $c$, is related to $A$ by $A \sim c^{1/d}$.

cond-mat↗

arXiv1994

Avalanches and the Directed Percolation Depinning Model: Experiments, Simulations and Theory

We study the recently-introduced directed percolation depinning (DPD) model for interface roughening with quenched disorder for which the interface becomes pinned by a directed percolation (DP) cluster for $d = 1$, or a directed surface (DS) for $d > 1$. The mapping to DP enables us to predict some of the critical exponents of the growth process. For the case of $(1+1)$ dimensions, the theory predicts that the roughness exponent $α$ is given by $α= ν_{\perp} / ν_{\parallel}$, where $ν_{\perp}$ and $ν_{\parallel}$ are the exponents governing the divergence of perpendicular and parallel correlation lengths of the DP incipient infinite cluster. The theory also predicts that the dynamical exponent $z$ equals the exponent $d_{\rm min}$ characterizing the scaling of the shortest path on a isotropic percolation cluster. The exponent $α$ decreases monotonically with $d$ but does not seem to approach zero for any dimension calculated ($d \le 6$), suggesting that the DPD model has no upper critical dimension for the static exponents. On the other hand, $z$ appears to approach $2$ as $d \rightarrow 6$, as expected by the result $z = d_{\rm min}$, suggesting that $d_c = 6$ for the dynamics. We also perform a set of imbibition experiments, in both $(1+1)$ and $(2+1)$ dimensions, that can be used to test the DPD model. We find good agreement between experimental, theoretical and numerical approaches. Further, we study the properties of avalanches in the context of the DPD model. We relate the scaling properties of the avalanches in the DPD model to the scaling properties for the self-organized depinning (SOD) model, a variant of the DPD model. We calculate the exponent characterizing the avalanches distribution $τ_{\rm aval}$ for $d = 1$ to $d = 6$, and compare our results with recent theoretical

cond-mat↗

arXiv1994

Dynamics of Surface Roughening with Quenched Disorder

We study the dynamical exponent $z$ for the directed percolation depinning (DPD) class of models for surface roughening in the presence of quenched disorder. We argue that $z$ for $(d+1)$ dimensions is equal to the exponent $d_{\rm min}$ characterizing the shortest path between two sites in an isotropic percolation cluster in $d$ dimensions. To test the argument, we perform simulations and calculate $z$ for DPD, and $d_{\rm min}$ for percolation, from $d = 1$ to $d = 6$.

cond-mat↗