Topic

# Power law

About: Power law is a research topic. Over the lifetime, 6207 publications have been published within this topic receiving 180083 citations.

##### Papers published on a yearly basis

##### Papers

More filters

••

TL;DR: Some of the empirical evidence for the existence of power-law forms and the theories proposed to explain them are reviewed.

Abstract: When the probability of measuring a particular value of some quantity varies inversely as a power of that value, the quantity is said to follow a power law, also known variously as Zipf's law or the Pareto distribution. Power laws appear widely in physics, biology, earth and planetary sciences, economics and finance, computer science, demography and the social sciences. For instance, the distributions of the sizes of cities, earthquakes, forest fires, solar flares, moon craters and people's personal fortunes all appear to follow power laws. The origin of power-law behaviour has been a topic of debate in the scientific community for more than a century. Here we review some of the empirical evidence for the existence of power-law forms and the theories proposed to explain them.

4,734 citations

••

TL;DR: The characteristic features of alpha and beta relaxation in the supercooled state of non-network forming liquids and polymers are reviewed in this paper, where a crossover temperature Tc, located above the calorimetric glass transition temperature Tg, where the transport properties change from those typical for a strongly coupled liquid to those characteristic for a glass.

Abstract: The characteristic features of alpha - and beta - relaxation in the supercooled state of nonnetwork forming liquids and polymers are reviewed. Particular emphasis is put on properties observed recently by neutron scattering and molecular dynamics studies within the mesoscopic time region. The data indicate the existence of a crossover temperature Tc, located above the calorimetric glass transition temperature Tg, where the transport properties change from those typical for a strongly coupled liquid to those characteristic for a glass. Near Tc the alpha -process is specified by Debye-Waller factor anomalies, power law divergences of the relaxation scale and a crossover from alpha -scale universality to decoupling of the various relaxation processes. The beta -process is characterized by a complete absence of correlations between spatial and temporal motion and unconventional scaling laws. There appear two fractal time decay processes and two divergent time scales if the cross over temperature is approached. The results of the mode coupling theory for the supercooled liquid dynamics are reviewed and shown to give a unified and partly quantitative description of the Tc anomalies.

2,296 citations

••

TL;DR: A rich and long history is found of how lognormal distributions have arisen as a possible alternative to power law distributions across many fields, focusing on underlying generative models that lead to these distributions.

Abstract: Recently, I became interested in a current debate over whether file size distributions are best modelled by a power law distribution or a lognormal distribution. In trying to learn enough about these distributions to settle the question, I found a rich and long history, spanning many fields. Indeed, several recently proposed models from the computer science community have antecedents in work from decades ago. Here, I briefly survey some of this history, focusing on underlying generative models that lead to these distributions. One finding is that lognormal and power law distributions connect quite naturally, and hence, it is not surprising that lognormal distributions have arisen as a possible alternative to power law distributions across many fields.

1,787 citations

••

University of Pennsylvania

^{1}, New York University^{2}, Princeton University^{3}, Drexel University^{4}, Ohio State University^{5}, University of Chicago^{6}, Johns Hopkins University^{7}, University of Pittsburgh^{8}, University of Arizona^{9}, Fermilab^{10}, University of Colorado Boulder^{11}, Carnegie Mellon University^{12}, University of Hawaii^{13}, Apache Corporation^{14}, Massachusetts Institute of Technology^{15}, Spanish National Research Council^{16}, University of Sussex^{17}, University of Tokyo^{18}, University of Michigan^{19}, Rochester Institute of Technology^{20}, Pennsylvania State University^{21}TL;DR: In this paper, the authors employed a matrix-based method using pseudo-Karhunen-Loeve eigenmodes, producing uncorrelated minimum-variance measurements in 22 k-bands of both the clustering power and its anisotropy due to redshift-space distortions.

Abstract: We measure the large-scale real-space power spectrum P(k) by using a sample of 205,443 galaxies from the Sloan Digital Sky Survey, covering 2417 effective square degrees with mean redshift z ≈ 0.1. We employ a matrix-based method using pseudo-Karhunen-Loeve eigenmodes, producing uncorrelated minimum-variance measurements in 22 k-bands of both the clustering power and its anisotropy due to redshift-space distortions, with narrow and well-behaved window functions in the range 0.02 h Mpc-1 < k < 0.3 h Mpc-1. We pay particular attention to modeling, quantifying, and correcting for potential systematic errors, nonlinear redshift distortions, and the artificial red-tilt caused by luminosity-dependent bias. Our results are robust to omitting angular and radial density fluctuations and are consistent between different parts of the sky. Our final result is a measurement of the real-space matter power spectrum P(k) up to an unknown overall multiplicative bias factor. Our calculations suggest that this bias factor is independent of scale to better than a few percent for k < 0.1 h Mpc-1, thereby making our results useful for precision measurements of cosmological parameters in conjunction with data from other experiments such as the Wilkinson Microwave Anisotropy Probe satellite. The power spectrum is not well-characterized by a single power law but unambiguously shows curvature. As a simple characterization of the data, our measurements are well fitted by a flat scale-invariant adiabatic cosmological model with h Ωm = 0.213 ± 0.023 and σ8 = 0.89 ± 0.02 for L* galaxies, when fixing the baryon fraction Ωb/Ωm = 0.17 and the Hubble parameter h = 0.72; cosmological interpretation is given in a companion paper.

1,734 citations

••

TL;DR: From the investigation of these correlation functions, it is concluded that hopping processes are not important on the time scale of the $\beta$-relaxation for this system and for the temperature range investigated.

Abstract: We report the results of a large scale computer simulation of a binary supercooled Lennard-Jones liquid. We find that at low temperatures the curves for the mean squared displacement of a tagged particle for different temperatures fall onto a master curve when they are plotted versus rescaled time tD(T), where D(T) is the diffusion constant. The time range for which these curves follow the master curve is identified with the \ensuremath{\alpha}-relaxation regime of mode-coupling theory (MCT). This master curve is fitted well by a functional form suggested by MCT. In accordance with idealized MCT, D(T) shows a power-law behavior at low temperatures. The critical temperature of this power law is the same for both types of particles, and also the critical exponents are very similar. However, contrary to a prediction of MCT, these exponents are not equal to the ones determined previously for the divergence of the relaxation times of the intermediate scattering function [Phys. Rev. Lett. 73, 1376 (1994)]. At low temperatures, the van Hove correlation function (self as well as distinct part) shows almost no sign of relaxation in a time interval that extends over about three decades in time. This time interval can be interpreted as the \ensuremath{\beta}-relaxation regime of MCT. From the investigation of these correlation functions, we conclude the hopping processes are not important on the time scale of the \ensuremath{\beta} relaxation for this system and for the temperature range investigated. We test whether the factorization property predicted by MCT holds and find that this is indeed the case for all correlation functions investigated. The distance dependences of the critical amplitudes are in qualitative agreement with the ones predicted by MCT for some other mixtures. The non-Gaussian parameter for the self part of the van Hove correlation function for different temperatures follows a master curve when plotted against time t.

1,129 citations