Power-law distributions in empirical data

Power-law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man-made phenomena. Unfortunately, the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution -- the part of the distribution representing large but rare events -- and by the difficulty of identifying the range over which power-law behavior holds. Commonly used methods for analyzing power-law data, such as least-squares fitting, can produce substantially inaccurate estimates of parameters for power-law distributions, and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all. Here we present a principled statistical framework for discerning and quantifying power-law behavior in empirical data. Our approach combines maximum-likelihood fitting methods with goodness-of-fit tests based on the Kolmogorov-Smirnov statistic and likelihood ratios. We evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches. We also apply the proposed methods to twenty-four real-world data sets from a range of different disciplines, each of which has been conjectured to follow a power-law distribution. In some cases we find these conjectures to be consistent with the data while in others the power law is ruled out.Comment: 43 pages, 11 figures, 7 tables, 4 appendices; code available at http://www.santafe.edu/~aaronc/powerlaws

Centrality dependence of charged hadron production in deuteron+gold and nucleon+gold collisions at sqrt(s_NN)=200 GeV

We present transverse momentum (p_T) spectra of charged hadrons measured in deuteron-gold and nucleon-gold collisions at \sqrts = 200 GeV for four centrality classes. Nucleon-gold collisions were selected by tagging events in which a spectator nucleon was observed in one of two forward rapidity detectors. The spectra and yields were investigated as a function of the number of binary nucleon-nucleon collisions, \nu, suffered by deuteron nucleons. A comparison of charged particle yields to those in p+p collisions show that the yield per nucleon-nucleon collision saturates with \nu for high momentum particles. We also present the charged hadron to neutral pion ratios as a function of p_T.Comment: 330 authors, 15 pages text, 16 figures, 3 tables. Submitted to Phys. Rev. Lett. v2 has minor changes to reflect revisions during review process. Plain text data tables for the points plotted in figures for this and previous PHENIX publications are (or will be) publicly available at http://www.phenix.bnl.gov/papers.htm

Azimuthal Angle Correlations for Rapidity Separated Hadron Pairs in d+Au Collisions at sqrt(s_NN) = 200 GeV

We report on two-particle azimuthal angle correlations between charged hadrons at forward/backward (deuteron/gold going direction) rapidity and charged hadrons at mid-rapidity in deuteron-gold (d+Au) and proton-proton (p+p) collisions at sqrt(s_NN) = 200 GeV. Jet structures are observed in the correlations which we quantify in terms of the conditional yield and angular width of away side partners. The kinematic region studied here samples partons in the gold nucleus carrying nucleon momentum fraction x~0.1 to x~0.01. Within this range, we find no x dependence of the jet structure in d+Au collisions.Comment: 330 authors, 6 pages text, 4 figures, no tables. Submitted to Phys. Rev. Lett. Plain text data tables for the points plotted in figures for this and previous PHENIX publications are (or will be) publicly available at http://www.phenix.bnl.gov/papers.htm

Centrality Dependence of Charged Particle Multiplicity in Au-Au Collisions at sqrt(s_NN)=130 GeV

We present results for the charged-particle multiplicity distribution at mid-rapidity in Au - Au collisions at sqrt(s_NN)=130 GeV measured with the PHENIX detector at RHIC. For the 5% most central collisions we find $dN_{ch}/d\eta_{|\eta=0} = 622 \pm 1 (stat) \pm 41 (syst)$. The results, analyzed as a function of centrality, show a steady rise of the particle density per participating nucleon with centrality.Comment: 307 authors, 43 institutions, 6 pages, 4 figures, 1 table Minor changes to figure labels and text to meet PRL requirements. One author added: M. Hibino of Waseda Universit

Jet Structure from Dihadron Correlations in d+Au collisions at sqrt(s_NN) = 200 GeV

Dihadron correlations at high transverse momentum in d+Au collisions at sqrt(s_NN) = 200 GeV at midrapidity are measured by the PHENIX experiment at the Relativistic Heavy Ion Collider (RHIC). From these correlations we extract several structural characteristics of jets; the root-mean-squared (RMS) transverse momentum of fragmenting hadrons with respect to the jet sqrt(), the mean sine-squared angle between the scattered partons , and the number of particles produced within the dijet that are associated with a high-p_T particle (dN/dx_E distributions). We observe that the fragmentation characteristics of jets in d+Au collisions are very similar to those in p+p collisions and that there is also little dependence on the centrality of the d+Au collision. This is consistent with the nuclear medium having little influence on the fragmentation process. Furthermore, there is no statistically significant increase in the value of from p+p to d+Au collisions. This constrains the amount of multiple scattering that partons undergo in the cold nuclear medium before and after a hard-collision.Comment: 330 authors, 30 pages text, RevTeX4, 42 figures, 20 tables. Submitted to Physical Review C. Plain text data tables for the points plotted in figures for this and previous PHENIX publications are (or will be) publicly available at http://www.phenix.bnl.gov/papers.htm

Flow Measurements via Two-particle Azimuthal Correlations in Au + Au Collisions at sqrt(s_NN) = 130 GeV

Two particle azimuthal correlation functions are presented for charged hadrons produced in Au + Au collisions at RHIC sqrt(s_NN) = 130 GeV. The measurements permit determination of elliptic flow without event-by-event estimation of the reaction plane. The extracted elliptic flow values v_2 show significant sensitivity to both the collision centrality and the transverse momenta of emitted hadrons, suggesting rapid thermalization and relatively strong velocity fields. When scaled by the eccentricity of the collision zone, epsilon, the scaled elliptic flow shows little or no dependence on centrality for charged hadrons with relatively low p_T. A breakdown of this epsilon scaling is observed for charged hadrons with p_T > 1.0 GeV/c for the most central collisions.Comment: 6 pages, RevTeX 3, 4 figures, 307 authors, submitted to Phys. Rev. Lett. on 11 April 2002. Plain text data tables for the points plotted in figures for this and previous PHENIX publications are (will be made) publicly available at http://www.phenix.bnl.gov/phenix/WWW/run/phenix/papers.htm

Double Helicity Asymmetry in Inclusive Mid-Rapidity neutral pion Production for Polarized p+p Collisions at sqrt(s)=200 GeV

We present a measurement of the double longitudinal spin asymmetry in inclusive pi^0 production in polarized proton-proton collisions at sqrt(s)=200 GeV. The data were taken at the Relativistic Heavy Ion Collider with average beam polarizations of 26%. The measurements are the first of a program to study the longitudinal spin structure of the proton, using strongly interacting probes, at collider energies. The asymmetry is presented for transverse momenta 1-5 GeV/c at mid-rapidity, where next-to-leading order perturbative quantum chromodynamic (NLO pQCD) calculations describe the unpolarized cross section well. The observed asymmetry is small and is compared with a NLO pQCD calculation with a range of polarized gluon distributions.Comment: 326 authors, 6 pages text, RevTeX, 3 figures, 1 table. To be submitted to PRL. Plain text data tables for the points plotted in figures for this and previous PHENIX publications are (or will be) publicly available at http://www.phenix.bnl.gov/papers.htm

Mid-Rapidity Direct-Photon Production in p+p Collisions at sqrt(s) = 200 GeV

A measurement of direct photons in p+p collisions at sqrt(s)=200 GeV is presented. A photon excess above background from pi^0 --> gamma+gamma, eta --> gamma+gamma, and other decays is observed in the transverse momentum range 5.5 < p_T < 7 GeV/c. The result is compared to a next-to-leading-order perturbative QCD calculation. Within errors, good agreement is found between the QCD calculation and the measured result.Comment: 330 authors, 7 pages text, RevTeX, 2 figures, 2 tables. Submitted to Physical Review D. Plain text data tables for the points plotted in figures for this and previous PHENIX publications are (or will be) publicly available at http://www.phenix.bnl.gov/papers.htm

Nuclear Modification Factors for Hadrons At Forward and Backward Rapidities in Deuteron-Gold Collisions at sqrt(s_NN) = 200 GeV

We report on charged hadron production in deuteron-gold reactions at sqrt(s_NN) = 200 GeV. Our measurements in the deuteron-direction cover 1.4 < eta < 2.2, referred to as forward rapidity, and in the gold-direction -2.0 < eta < -1.4, referred to as backward rapidity, and a transverse momentum range p_T = 0.5-4.0 GeV/c. We compare the relative yields for different deuteron-gold collision centrality classes. We observe a suppression relative to binary collision scaling at forward rapidity, sensitive to low momentum fraction (x) partons in the gold nucleus, and an enhancement at backward rapidity, sensitive to high momentum fraction partons in the gold nucleus.Comment: 330 authors, 6 pages text, 4 figures, REVTeX4. Published in Physical Review Letters. Minor changes over previous version in response to referee and editor comments, plus updating of references. Plain text data tables for the points plotted in figures for this and previous PHENIX publications are publicly available at http://www.phenix.bnl.gov/papers.htm

Event-by-event fluctuations in Mean pTp_T and Mean eTe_T in sqrt(s_NN) = 130 GeV Au+Au Collisions

Distributions of event-by-event fluctuations of the mean transverse momentum and mean transverse energy near mid-rapidity have been measured in Au+Au collisions at sqrt(s_NN) = 130 GeV at RHIC. By comparing the distributions to what is expected for statistically independent particle emission, the magnitude of non-statistical fluctuations in mean transverse momentum is determined to be consistent with zero. Also, no significant non-random fluctuations in mean transverse energy are observed. By constructing a fluctuation model with two event classes that preserve the mean and variance of the semi-inclusive p_T or e_T spectra, we exclude a region of fluctuations in sqrt(s_NN) = 130 GeV Au+Au collisions.Comment: 10 pages, RevTeX 3, 7 figures, 4 tables, 307 authors, submitted to Phys. Rev. C on 22 March 2002. Plain text data tables for the points plotted in figures for this and previous PHENIX publications are (will be made) publicly available at http://www.phenix.bnl.gov/phenix/WWW/run/phenix/papers.htm