Pure Power Law Research Articles

Self-similarity and scaling in forest communities

Ecological communities exhibit pervasive patterns and interrelationships between size, abundance, and the availability of resources. We use scaling ideas to develop a unified, model-independent framework for understanding the distribution of tree sizes, their energy use, and spatial distribution in tropical forests. We demonstrate that the scaling of the tree crown at the individual level drives the forest structure when resources are fully used. Our predictions match perfectly with the scaling behavior of an exactly solvable self-similar model of a forest and are in good accord with empirical data. The range, over which pure power law behavior is observed, depends on the available amount of resources. The scaling framework can be used for assessing the effects of natural and anthropogenic disturbances on ecosystem structure and functionality.

Curvature in metabolic scaling

For more than three-quarters of a century it has been assumed that basal metabolic rate increases as body mass raised to some power p. However, there is no broad consensus regarding the value of p: whereas many studies have asserted that p is 3/4 (refs 1-4; 'Kleiber's law'), some have argued that it is 2/3 (refs 5-7), and others have found that it varies depending on factors like environment and taxonomy. Here we show that the relationship between mass and metabolic rate has convex curvature on a logarithmic scale, and is therefore not a pure power law, even after accounting for body temperature. This finding has several consequences. First, it provides an explanation for the puzzling variability in estimates of p, settling a long-standing debate. Second, it constitutes a stringent test for theories of metabolic scaling. A widely debated model based on vascular system architecture fails this test, and we suggest modifications that could bring it into compliance with the observed curvature. Third, it raises the intriguing question of whether the scaling relation limits body size.

Functional renormalization group study of the interacting resonant level model in and out of equilibrium

We investigate equilibrium and steady-state nonequilibrium transport properties of a spinless resonant level locally coupled to two conduction bands of width $\ensuremath{\sim}\ensuremath{\Gamma}$ via a Coulomb interaction $U$ and a hybridization ${t}^{\ensuremath{'}}$. In order to study the effects of finite bias voltages beyond linear response, a generalization of the functional renormalization group to Keldysh frequency space is employed. Being mostly unexplored in the context of quantum impurity systems out of equilibrium, we benchmark this method against recently published time-dependent density matrix renormalization group data. We thoroughly investigate the scaling limit $\ensuremath{\Gamma}\ensuremath{\rightarrow}\ensuremath{\infty}$ characterized by the appearance of power laws. Most importantly, at the particle-hole symmetric point the steady-state current decays like $J\ensuremath{\sim}{V}^{\ensuremath{-}{\ensuremath{\alpha}}_{J}}$ as a function of the bias voltage $V⪢{t}^{\ensuremath{'}}$, with an exponent ${\ensuremath{\alpha}}_{J}(U)$ that we calculate to leading order in the Coulomb interaction strength. In contrast, we do not observe a pure power-law (but more complex) current-voltage-relation if the energy $ϵ$ of the resonant level is pinned close to either one of the chemical potentials $\ifmmode\pm\else\textpm\fi{}V/2$.

Reply to “Comment on ‘Tests of scaling and universality of the distributions of trade size and share volume: Evidence from three distinct markets’ ”

Analyzing trade-by-trade data for three distinct markets, we showed that the cumulative distributions of trade size display power-law tails $P{q>x}\ensuremath{\sim}{x}^{\ensuremath{-}{\ensuremath{\zeta}}_{q}}$, with exponents ${\ensuremath{\zeta}}_{q}$ in the ``L\'evy stable domain'' $({\ensuremath{\zeta}}_{q}<{\ensuremath{\zeta}}_{q}^{\ensuremath{\ast}}=2)$. Moreover we reported that the exponent values are consistent for all stocks irrespective of stock-specific variables such as market capitalization, industry sector, or the specific market where the stock is traded. Our conclusions were based on using two distinct estimation methods. R\'acz et al. now propose that one of the estimators we used has slow convergence for a pure power law, particularly as tail exponents approach the boundary ${\ensuremath{\zeta}}_{q}^{\ensuremath{\ast}}=2$. We examine the robustness of our results to specific estimation method by additional analysis using five distinct techniques to estimate ${\ensuremath{\zeta}}_{q}$. We find results that are fully consistent with those we had reported, providing compelling evidence that our conclusions hold regardless of estimation procedure.

Infrared gluon and ghost propagator exponents from lattice QCD

The compatibility of the pure power law infrared solution of QCD and lattice data for the gluon and ghost propagators in Landau gauge is discussed. For the gluon propagator, the lattice data are well described by a pure power law with an infrared exponent κ∼0.53, in the Dyson–Schwinger notation. κ is measured using a technique that suppresses finite volume effects. This value is consistent with a vanishing zero momentum gluon propagator, in agreement with the Gribov–Zwanziger confinement scenario. For the ghost propagator, the lattice data seem not to follow a pure power law, at least for the range of momenta accessed in our simulation.

Herman Yaggi Carr, PhD (1924–2008): A tribute

Herman Yaggi Carr, PhD (1924–2008): A tribute

Nonlinear preferential rewiring in fixed-size networks as a diffusion process

We present an evolving network model in which the total numbers of nodes and edges are conserved, but in which edges are continuously rewired according to nonlinear preferential detachment and reattachment. Assuming power-law kernels with exponents alpha and beta , the stationary states which the degree distributions evolve toward exhibit a second-order phase transition-from relatively homogeneous to highly heterogeneous (with the emergence of starlike structures) at alpha=beta . Temporal evolution of the distribution in this critical regime is shown to follow a nonlinear diffusion equation, arriving at either pure or mixed power laws of exponents -alpha and 1-alpha .

Does the lattice zero-momentum gluon propagator for pure-gauge SU(3) Yang-Mills theory vanish in the infinite-volume limit?

The Cucchieri-Mendes bounds for the gluon propagator are discussed for the four dimensional pure-gauge SU(3) theory. Assuming a pure power law dependence on the inverse of the lattice volume, the lattice data gives a vanishing zero-momentum gluon propagator in the infinite-volume limit in agreement with the Gribov-Zwanziger horizon condition but contradicting the SU(2) analysis. The results are robust against variations of the lattice volumes and corrections to the power law. Our analysis considers also more general ansatze that, although not conclusive, open the possibility of having $D(0)\ensuremath{\ne}0$ in the infinite-volume limit. A solution to this puzzle requires further investigations.

Improvement of FDM by extrapolation on multiple grids

The extrapolation of results obtained on a series of 3 succeeding grids with halved mesh size is tested as a variant of the multigrid approach for solving the Laplace and Poisson equations in 2D. Based on corresponding experience with BEM for electric [1] and magnetic [2] field problems a pure power law is applied instead of the famous Richardson extrapolation [3]. On those grid points, which are common to all 3 grids, the potential values are extrapolated to an arbitrary fine discretization. On the points of the finest grid in between those of the coarser ones the potentials then are obtained by only few iterations to perform the interpolation. Both, the common 5-point discretization and the famous 9-point discretization by E. Kasper [5] are investigated and compared with respect to the possible win of accuracy by extrapolation. As an interesting result of this kind of extrapolation, the accumulated local discretization errors of the 5-point discretization are partially cured and the high accuracy by the 9-point formula of Kasper makes extrapolation inefficient. Like for classical MG (multi grid) [6] the acceleration of potential

Exploring the nature of non-Fickian transport in laboratory experiments

Observations of non-Fickian transport in sandbox experiments [Levy M, Berkowitz B. Measurement and analysis of non-Fickian dispersion in heterogeneous porous media. J Contam Hydrol 2003;64:203–26] were analyzed previously using a power law tail ψ( t) ∼ t −1− β with 0 < β < 2 for the spectrum of transition times comprising a tracer plume migration. For each sandbox medium a choice of β resulted in an excellent fit to the breakthrough curve (BTC) data, and the value of β decreased slowly with increasing flow velocity. Here, the data are reanalyzed with the full spectrum of ψ( t) gleaned from analytical calculations [Cortis A, Chen Y, Scher H, Berkowitz B. Quantitative characterization of pore-scale disorder effects on transport in “homogeneous” granular media. Phys Rev E 2004;10(70):041108. doi: 10.1103/PhysRevE.70.041108], numerical simulations [Bijeljic B, Blunt MJ. Pore-scale modeling and continuous time random walk analysis of dispersion in porous media. Water Resour Res 2006;42:W01202. doi: 10.1029/2005WR004578] and permeability fields [Di Donato G, Obi E-O, Blunt MJ. Anomalous transport in heterogeneous media demonstrated by streamline-based simulation. Geophys Res Lett 2003;30:1608–12s. doi: 10.1029/2003GL017196]. We represent the main features of the full spectrum of transition times with a truncated power law (TPL), ψ( t) ∼ ( t 1 + t) −1− β exp(− t/ t 2), where t 1 and t 2 are the limits of the power law spectrum. An excellent fit to the entire BTC data set, including the changes in flow velocity, for each sandbox medium is obtained with a single set of values of t 1, β, t 2. The influence of the cutoff time t 2 is apparent even in the regime t < t 2. Significantly, we demonstrate that the previous apparent velocity dependence of β is a result of choosing a pure power law tail for ψ( t). The key is the change in the log–log slope of the TPL form of ψ( t) with a shifting observational time window caused by the change in the mean velocity. Hence, the use of the full spectrum of ψ( t) is not only necessary for the transition to Fickian behavior, but also to account for the dynamics of these laboratory observations of non-Fickian transport.

The Growth of Luminous Red Galaxies by Merging

We study the role of major and minor mergers in the mass growth of luminous red galaxies. We present small-scale (0.01 h−1 Mpc < r < 8 h−1 Mpc ) projected cross-correlation functions of 23,043 luminous early-type galaxies from the Sloan Digital Sky Survey (SDSS) luminous red galaxy (LRG) sample (0.16 < z < 0.30, M0.3 i − 5log h ≈ − 22.75 mag) with all their companions in the SDSS imaging sample, split into color and luminosity subsamples with M0.3 i − 5log h < − 18 mag. We deproject the two-dimensional functions to obtain three-dimensional real-space LRG-galaxy cross-correlation functions for each companion subsample. We find that the cross-correlation functions are not pure power laws and that there is a clear one-halo to two-halo transition near 1 h−1 Mpc. We convert these results into close pair statistics and estimate the LRG accretion rate from each companion galaxy subsample using timescales from dynamical friction arguments for each subsample of the companions. We find that the accretion onto LRGs is dominated by dry mergers of galaxies that are more luminous than L*. We integrate the luminosity accretion rate from mergers over all companion galaxy subsamples and find that LRGs are growing by 1.7 ± 0.1 h percent per Gyr, on average, from merger activity at redshift z ~ 0.25. This rate is almost certainly an overestimate, because we have assumed that all close pairs are merging as quickly as dynamical friction allows; nonetheless, it is on the low side of the panoply of measurements in the literature, and it is lower than any rate predicted from theory.

Jet Breaks and Missing Breaks in the X‐Ray Afterglow of Gamma‐Ray Bursts

The X-ray afterglows (AGs) of gamma-ray bursts (GRBs) and X-ray flashes (XRFs) have, after the fast-decline phase of their prompt emission, a temporal behavior varying between two extremes. A large fraction of these AGs has a canonical light curve which, after an initial shallow-decay plateau phase, breaks smoothly into a fast power-law decline. Very energetic GRBs, contrariwise, appear to not have a break: their AGs decline like a power law from the start of the observations. Breaks and ‘‘missing breaks’’ are intimately related to the geometry and deceleration of the jets responsible for GRBs. In the frame of the cannonball (CB) model of GRBs and XRFs, we analyze the cited extreme behaviors (canonical and pure power law) and intermediate cases spanning the observed range of X-ray AG shapes.Weshow that the entire panoplyof X-ray light-curve shapes—measured withSwift and other satellites—are as anticipated in the CB model. We test the expected correlations between the AG’s shape and the peak and isotropic energies of the prompt radiation, strengthening a simple conclusion of the analysis of AG shapes: in energetic GRBs the break is not truly missing, it is hidden under the tail of the prompt emission, or it occurs too early to be recorded. Wealsoverifythatthespectralindexof theunabsorbedAGsandthetemporalindexesof theirlatepower-lawdecline differ by half a unit, as predicted. Subject headingg gamma rays: bursts — gamma rays: theory Online material: color figures

The Broadband Spectrum of Galaxy Clusters

We examine whether nonthermal protons energized during a cluster merger are simultaneously responsible for the Coma cluster's diffuse radio flux (via secondary decay) and the departure of its intra-cluster medium (ICM) from a thermal profile via Coulomb collisions between the quasithermal electrons and the hadrons. Rather than approximating the influence of nonthermal proton/thermal electron collisions as extremely rare events which cause an injection of nonthermal, power-law electrons (the `knock-on' approximation), we self-consistently solve (to our knowledge, for the first time) the covariant kinetic equations for the two populations. The electron population resulting from these collisions is out of equilibrium, yet not a power law, and importantly displays a higher bremsstrahlung radiative efficiency than a pure power law. Observations with GLAST will test this model directly.

Discovery of very high energy gamma-ray emission coincident with molecular clouds in the W 28 (G6.4-0.1) field

We observed the W28 field (for ~40 h) at Very High Energy (VHE) gamma-ray energies (E>0.1 TeV) with the H.E.S.S. Cherenkov telescopes. A reanalysis of EGRET E>100 MeV data was also undertaken. Results from the NANTEN 4m telescope Galactic plane survey and other CO observations have been used to study molecular clouds. We have discovered VHE gamma-ray emission (HESSJ1801-233) coincident with the northeastern boundary of W28, and a complex of sources (HESSJ1800-240A, B and C) ~0.5 deg south of W28, in the Galactic disc. The VHE differential photon spectra are well fit by pure power laws with indices Gamma~2.3 to 2.7. The NANTEN ^{12}CO(J=1-0) data reveal molecular clouds positionally associating with the VHE emission, spanning a ~15 km s^{-1} range in local standard of rest velocity. The VHE/molecular cloud association could indicate a hadronic origin for HESSJ1801-233 and HESSJ1800-240, and several cloud components in projection may contribute to the VHE emission. The clouds have components covering a broad velocity range encompassing the distance estimates for W28 (~2 kpc), and extending up to ~4 kpc. Assuming a hadronic origin, and distances of 2 and 4 kpc for cloud components, the required cosmic ray density enhancement factors (with respect to the solar value) are in the range ~10 to ~30. If situated at 2 kpc distance, such cosmic ray densities may be supplied by a SNR like W28. Additionally and/or alternatively, particle acceleration may come from several catalogued SNRs and SNR candidates, the energetic ultra compact HII region W28A2, and the HII regions M8 and M20 along with their associated open clusters. Further sub-mm observations would be recommended to probe in detail the dynamics of the molecular clouds at velocites >10 km s^{-1}, and their possible connection to W28.

Sandpile on scale-free networks with assortative mixing

We numerically investigate the Bak–Tang–Wiesenfeld sandpile model on scale-free networks with assortative mixing, where the threshold height of each node is equal to its degree. It is observed that a large fraction of multiple topplings are included in avalanches on assortative networks, which is absent on uncorrelated networks. We introduce a parameter to characterize the fraction of multiple topplings in avalanches of area a. The fraction of multiple topplings increases dramatically with the degree of assortativity and has a peak for small a whose height also increase with the assortativity of the networks. Unlike the case on uncorrelated networks, the distributions of avalanche size, area and duration do not follow pure power law, but deviate more obviously from pure power law with the growing degree of assortativity. The results show that the assortative mixing has a strong influence on the behavior of avalanche dynamics on complex networks.

Phase transition in liquid drop fragmentation

A liquid droplet is fragmented by a sudden pressurized-gas blow, and the resulting droplets, adhered to the window of a flatbed scanner, are counted and sized by computerized means. The use of a scanner plus image recognition software enables us to automatically count and size up to tens of thousands of tiny droplets with a smallest detectable volume of approximately 0.02 nl . Upon varying the gas pressure, a critical value is found where the size distribution becomes a pure power law, a fact that is indicative of a phase transition. Away from this transition, the resulting size distributions are well described by Fisher's model at coexistence. It is found that the sign of the surface correction term changes sign, and the apparent power-law exponent tau has a steep minimum, at criticality, as previously reported in nuclear multifragmentation studies. We argue that the observed transition is not percolative, and introduce the concept of dominance in order to characterize it. The dominance probability is found to go to zero sharply at the transition. Simple arguments suggest that the correlation length exponent is nu=1/2 . The sizes of the largest and average fragments, on the other hand, do not go to zero abruptly but behave in a way that appears to be consistent with recent predictions of Ashurst and Holian.

Torque scaling in turbulent Taylor–Couette flow between independently rotating cylinders

Turbulent Taylor–Couette flow with arbitrary rotation frequencies ω1, ω2 of the two coaxial cylinders with radii r1 &lt; r2 is analysed theoretically. The current Jω of the angular velocity ω(x,t) = uϕ(r,ϕ,z,t)/r across the cylinder gap and and the excess energy dissipation rate ϵw due to the turbulent, convective fluctuations (the ‘wind’) are derived and their dependence on the control parameters analysed. The very close correspondence of Taylor–Couette flow with thermal Rayleigh–Bénard convection is elaborated, using these basic quantities and the exact relations among them to calculate the torque as a function of the rotation frequencies and the radius ratio η = r1/r2 or the gap width d = r2 − r1 between the cylinders. A quantity σ corresponding to the Prandtl number in Rayleigh–Bénard flow can be introduced, $\sigma = ((1 + \eta)/2)/\sqrt{\etaacute;)^4$. In Taylor–Couette flow it characterizes the geometry, instead of material properties of the liquid as in Rayleigh–Bénard flow. The analogue of the Rayleigh number is the Taylor number, defined as Ta ∝ (ω1 − ω2)2 times a specific geometrical factor. The experimental data show no pure power law, but the exponent α of the torque versus the rotation frequency ω1 depends on the driving frequency ω1. An explanation for the physical origin of the ω1-dependence of the measured local power-law exponents α(ω1) is put forward. Also, the dependence of the torque on the gap width η is discussed and, in particular its strong increase for η → 1.

Infrared gluon and ghost propagators from lattice QCD

We report on the infrared limit of the quenched lattice Landau gauge gluon and ghost propagators as well as the strong-coupling constant computed from large asymmetric lattices. The infrared lattice propagators are compared with the pure power law Solutions from Dyson-Schwinger equations (DSE). For the gluon propagator, the lattice data is compatible with the DSE solution. The preferred measured gluon exponent being ∼ 0.52, favouring a vanishing propagator at zero momentum. The lattice ghost propagator shows finite-volume effects and, for the volumes considered, the propagator does not follow a pure power law. Furthermore, the strong-coupling constant is computed and its infrared behaviour investigated.

Exploring the infrared gluon and ghost propagators using large asymmetric lattices

We report on the infrared limit of the quenched lattice Landau gauge gluon propagator computed from large asymmetric lattices. In particular, the compatibility of the pure power law infrared solution (q²)²kappa of the Dyson-Schwinger equations is investigated and the exponent kappa is measured. The lattice data favours kappa ~ 0.52, which would imply a vanishing zero momentum gluon propagator as predicted by the Kugo-Ojima confinement mechanism and the Zwanziger horizon condition. Results for the ghost propagator and for the running coupling constant are shown.

Power laws and the cosmic ray energy spectrum

Two separate statistical tests are applied to the AGASA and preliminary Auger cosmic ray energy spectra in an attempt to find deviation from a pure power-law. The first test is constructed from the probability distribution for the maximum event of a sample drawn from a power-law. The second employs the TP-statistic, a function defined to deviate from zero when the sample deviates from the power-law form, regardless of the value of the power index. The AGASA data show no significant deviation from a power-law when subjected to both tests. Applying these tests to the Auger spectrum suggests deviation from a power-law. However, potentially large systematics on the relative energy scale prevent us from drawing definite conclusions at this time.