Pure Power Law Research Articles

The breakdown of the direct relation between the density scaling exponent and the intermolecular interaction potential for molecular systems with purely repulsive intermolecular forces

In this work, we question the generally accepted statement that the character of intermolecular interactions can be directly determined from the scaling exponent. We show that the scaling exponent evidently differs on the molecular architecture for pure inverse power law systems. Since these systems are characterized by a well-defined dependency between the intermolecular virial and potential energy, the results contradict theoretical predictions and previous reports. Moreover, we suggest that the recently proposed “molecular force” method also returns the value that varies from the one scaling the dynamics. Finally, the obtained result allows us to conclude that the value of the density scaling exponent for real liquids depends on intramolecular interactions.

Under the weight of heavy tails: A power law perspective on the emergence of outliers in entrepreneurship

A fundamental discovery in entrepreneurship is that firm outcomes do not follow a symmetrical Gaussian curve. Instead, most are heavily right-skewed distributions in which a few extreme outliers (e.g., rock star firms like Airbnb, Tesla, and Uber) account for a disproportionate amount of the output. Although past research usually described outcome distributions as shaped following the power law, our study asks the following question: What other less extreme distributions of generalizable firm outcomes exist in entrepreneurship? Our investigation leverages four representative datasets from the U.S., Europe, and Australia, comprising 32 samples with about 22,000 ventures. We implemented a precise data-analytic approach that compares each sample (i.e., empirical distribution) against multiple theoretical distribution shapes to identify the best fit. Results showed that, across nearly all samples, the pure power law was not the dominant distribution. Instead, the annual revenue distribution is shaped as a power law with an exponential cutoff, and the number of employees distribution is shaped lognormally. Combined, these suggest the existence of top-down limitations on the highest performing firms. Accordingly, we offer an agenda for future research focused on (a) identifying and releasing systemic constraints, (b) examining and falsifying the underlying generative mechanisms that cause the emergence of heavy-tailed distributions and the outliers therein, and (c) conducting multi-level, mixed-method studies to investigate how micro-level interactions aggregate into macro-level heavy-tailed distributions. Our paper makes significant contributions to the power law perspective and future efforts to explain and predict the emergence of rock star firms in entrepreneurship.

A network model of social contacts with small-world and scale-free features, tunable connectivity, and geographic restrictions.

Small-world networks and scale-free networks are well-known theoretical models within the realm of complex graphs. These models exhibit "low" average shortest-path length; however, key distinctions are observed in their degree distributions and average clustering coefficients: in small-world networks, the degree distribution is bell-shaped and the clustering is "high"; in scale-free networks, the degree distribution follows a power law and the clustering is "low". Here, a model for generating scale-free graphs with "high" clustering is numerically explored, since these features are concurrently identified in networks representing social interactions. In this model, the values of average degree and exponent of the power-law degree distribution are both adjustable, and spatial limitations in the creation of links are taken into account. Several topological metrics are calculated and compared for computer-generated graphs. Unexpectedly, the numerical experiments show that, by varying the model parameters, a transition from a power-law to a bell-shaped degree distribution can occur. Also, in these graphs, the degree distribution is most accurately characterized by a pure power-law for values of the exponent typically found in real-world networks.

Algorithms for Fractional Dynamical Behaviors Modelling Using Non-Singular Rational Kernels

This paper proposes algorithms to model fractional (dynamical) behaviors using non-singular rational kernels whose interest is first demonstrated on a pure power law function. Two algorithms are then proposed to find a non-singular rational kernel that allows the input-output data to be fitted. The first one derives the impulse response of the modeled system from the data. The second one finds the interlaced poles and zeros of the rational function that fits the impulse response found using the first algorithm. Several applications show the efficiency of the proposed work.

Early stage of bubble spreading in a viscous ambient liquid

We investigate numerically the spreading dynamics of a bubble coming into contact with a smooth solid substrate in a viscous liquid. The substrate is partially wettable, and the singularity of the moving contact line is relieved by adopting the Navier-slip model. The Stokes equations are solved by employing a boundary element method coupled with adaptive mesh refinement. This allows us to realize sufficiently small slip lengths down to $O(10^{-5})$ in dimensionless form, which is crucial to resolve the local interface structures at the early stage of spreading. The results show that the early-stage spreading of the bubble is always characterized by the growth and propulsion of a dewetting liquid rim close to the contact line, while the macroscopic interface remains unchanged. The evolution of the contact line and the morphology of the rim depends on the wettability and the slip length, and a parametric investigation is performed. Based on mass conservation, a relation between the rim size and the spreading radius is established. We also propose an analytical prediction of the temporal variation in the contact line radius at the early stage of spreading, which is found to follow a logarithmically corrected linear relation rather than a pure power law. Moreover, the early stages of bubble spreading are qualitatively similar for two-dimensional and axisymmetric configurations.

Watermelons on the half-plane

We study the watermelon probabilities in the uniform spanning forests on the two-dimensional semi-infinite square lattice near either the open or closed boundary to which the forests can or cannot be rooted, respectively. We derive universal power laws describing the asymptotic decay of these probabilities with the distance between the reference points growing to infinity, as well as their non-universal constant prefactors. The obtained exponents match with the previous predictions made for the related dense polymer models using the Coulomb gas technique and conformal field theory, as well as with the lattice calculations made by other authors in different settings. We also discuss the logarithmic corrections some authors argued to appear in the watermelon correlation functions on the infinite lattice. We show that the full account for diverging terms of the lattice Green function, which ensures the correct probability normalization, provides the pure power law decay in the case of semi-infinite lattice with the closed boundary studied here, as well as in the case of infinite lattice discussed elsewhere. The solution is based on the all-minors generalization of the Kirchhoff matrix tree theorem, the image method and the developed asymptotic expansion of the Kirchhoff determinants.

Spatiotemporal analysis of earthquake occurrence in synthetic and worldwide data

The present work presents spatiotemporal analyses of distributions between successive earthquakes. This study comprehends data produced by a modified version of the Olami–Feder–Christensen model with a small-world topology and actual worldwide earthquakes from 2000 to 2019. The distributions were studied from the nonextensive statistical mechanic’s viewpoint, which was shown to be a suitable approach since q-exponential functions produced better fittings to data than pure power laws. Our results show that, by applying scaling relationships, the probability distributions have data collapses in all cases. It reinforces the conception of a critical behavior in the seismological phenomenon and that there is no differentiation between the spatiotemporal statistical features of earthquakes, whether small or large in size or magnitude. In addition, the presence of q-exponential distributions and the ability of a small-world-like OFC model to reproduce spatiotemporal features of real worldwide earthquakes indicate self-organized criticality and long-range spatiotemporal correlations between earthquakes.

Slow-roll inflation at N3LO

The next generation of cosmological observations will be sensitive to small deviations from a pure power law in the primordial power spectrum of the curvature perturbations. In the context of slow-roll inflation, these deviations are expected, they correspond to the so-called running of the spectral index. Their measurement would bring as much information as the discovery of deviations from scale invariance. However, robust parameter inference requires to marginalize over any possible higher-order uncertainties which have been, up to now, not fully determined. We tackle this issue by deriving the inflationary scalar and tensor slow-roll power spectra at next-to-next-to-next to leading order, fully expanded around an observable pivot wave number, for all single-field inflationary models having minimal and nonminimal kinetic terms. Our result therefore encompasses string-inspired inflationary models having a varying speed of sound.

The Panchromatic Hubble Andromeda Treasury: Triangulum Extended Region (PHATTER). III. The Mass Function of Young Stellar Clusters in M33

Abstract We measure the star cluster mass function (CMF) for the Local Group galaxy M33. We use the catalog of stellar clusters selected from the Panchromatic Hubble Andromeda Treasury: Triangulum Extended Region survey. We analyze 711 clusters in M33 with 7.0 < log ( Age / yr ) < 8.5 , and log(M/M ⊙) > 3.0 as determined from color–magnitude diagram fits to individual stars. The M33 CMF is best described by a Schechter function with power-law slope α = − 2.06 − 0.13 + 0.14 , and truncation mass log(M c /M ⊙) = 4.24 − 0.13 + 0.16 . The data show strong evidence for a high-mass truncation, thus strongly favoring a Schechter function fit over a pure power law. M33's truncation mass is consistent with the previously identified linear trend between M c , and star formation rate surface density, ΣSFR. We also explore the effect that individual cluster mass uncertainties have on derived mass function parameters, and find evidence to suggest that large cluster mass uncertainties have the potential to bias the truncation mass of fitted mass functions at the 1σ level.

Introducing a new multi-particle collision method for the evolution of dense stellar systems

Context.In a previous paper we introduced a new method for simulating collisional gravitationalN-body systems with linear time scaling onN, based on the multi-particle collision (MPC) approach. This allows us to easily simulate globular clusters with a realistic number of stellar particles (105 − 106) in a matter of hours on a typical workstation.Aims.We evolve star clusters containing up to 106stars to core collapse and beyond. We quantify several aspects of core collapse over multiple realizations and different parameters while always resolving the cluster core with a realistic number of particles.Methods.We run a large set ofN-body simulations with our new code MPCDSS. The cluster mass function is a pure power law with no stellar evolution, allowing us to clearly measure the effects of the mass spectrum on core collapse.Results.Leading up to core collapse, we find a power-law relation between the size of the core and the time left to core collapse. Our simulations thus confirm the theoretical self-similar contraction picture but with a dependence on the slope of the mass function. The time of core collapse has a non-monotonic dependence on the slope, which is well fitted by a parabola. This also holds for the depth of core collapse and for the dynamical friction timescale of heavy particles. Cluster density profiles at core collapse show a broken-power-law structure, suggesting that central cusps are a genuine feature of collapsed cores. The core bounces back after collapse, with visible fluctuations, and the inner density slope evolves to an asymptotic value. The presence of an intermediate-mass black hole inhibits core collapse, making it much shallower, irrespective of the mass-function slope.Conclusions.We confirm and expand on several predictions of star cluster evolution before, during, and after core collapse. Such predictions were based on theoretical calculations or small-size directN-body simulations. Here we put them to the test in MPC simulations with a much larger number of particles, allowing us to resolve the collapsing core.

Deriving Population Scaling Rules from Individual-Level Metabolism and Life History Traits.

AbstractIndividual metabolism generally scales with body mass with an exponent around 3/4. From dimensional arguments it follows that maximum population growth rate (rmax) scales with a -1/4 exponent. However, the dimensional argument implicitly assumes that offspring size is proportional to adult size. Here, we calculate rmax from metabolic scaling at the level of individuals within size-structured populations while explicitly accounting for offspring size. We identify four general patterns of how rmax scales with adult mass based on four empirical life history patterns employed by groups of species. These life history patterns are determined by how traits of somatic growth rate and/or offspring mass relate to adult mass. One life history pattern-constant adult-to-offspring mass ratio and somatic growth rate independent of adult mass-leads to the classic -1/4 scaling of rmax. The other three life history patterns either lead to nonmetabolic population growth scaling with adult mass or do not follow a power-law relationship at all. Using life history data on five marine taxa and terrestrial mammals, we identify species groups that belong to one of each case. We predict that elasmobranchs, copepods, and mammals follow standard -1/4 power-law scaling, whereas teleost fish and bivalves do not have a pure power-law scaling. Our work highlights how taxa may deviate from the classic -1/4 metabolic scaling pattern of maximum population growth. The approach is generic and can be applied to any taxa.

Stochastic approach to gravitational waves from inflation

We propose a coarse-graining procedure for describing the superhorizon dynamics of inflationary tensor modes. Our aim is to formulate a stochastic description for the statistics of spin-2 modes which seed the background of gravitational waves from inflation. Using basic principles of quantum mechanics, we determine a probability density for coarse-grained tensor fields, which satisfies a stochastic Fokker-Planck equation at superhorizon scales. The corresponding noise and drift are computable, and depend on the cosmological system under consideration. Our general formulas are applied to a variety of cosmological scenarios, also considering cases seldom considered in the context of stochastic inflation, and which are important for their observational consequences. We start obtaining the expected expressions for noise and drift in pure de Sitter and power-law inflation, also including a discussion of effects of non-attractor phases. We then apply our methods to describe scenarios with a transition from inflation to standard cosmological eras of radiation and matter domination. We show how the interference between modes flowing through the cosmological horizon, and modes spontaneously produced at superhorizon scales, can affect the stochastic evolution of coarse-grained tensor quantities. In appropriate limits, we find that the corresponding spectrum of tensor modes at horizon crossing matches with the results of quantum field theory calculations, but we also highlight where differences can arise.

A Semianalytic Afterglow with Thermal Electrons and Synchrotron Self-Compton Emission

Abstract We extend previous work on gamma-ray burst afterglows involving hot thermal electrons at the base of a shock-accelerated tail. Using a physically motivated electron distribution based on first-principles simulations, we compute the broadband emission from radio to TeV gamma rays. For the first time, we present the effects of a thermal distribution of electrons on synchrotron self-Compton emission. The presence of thermal electrons causes temporal and spectral structure across the entire observable afterglow, which is substantively different from models that assume a pure power-law distribution for the electrons. We show that early-time TeV emission is enhanced by more than an order of magnitude for our fiducial parameters, with a time-varying spectral index that does not occur for a pure power law of electrons. We further show that the X-ray closure relations take a very different, also time-dependent, form when thermal electrons are present; the shape traced out by the X-ray afterglows is a qualitative match to observations of the traditional decay phase.

Twitter as an innovation process with damping effect

In the existing literature about innovation processes, the proposed models often satisfy the Heaps’ law, regarding the rate at which novelties appear, and the Zipf’s law, that states a power law behavior for the frequency distribution of the elements. However, there are empirical cases far from showing a pure power law behavior and such a deviation is mostly present for elements with high frequencies. We explain this phenomenon by means of a suitable “damping” effect in the probability of a repetition of an old element. We introduce an extremely general model, whose key element is the update function, that can be suitably chosen in order to reproduce the behaviour exhibited by the empirical data. In particular, we explicit the update function for some Twitter data sets and show great performances with respect to Heaps’ law and, above all, with respect to the fitting of the frequency-rank plots for low and high frequencies. Moreover, we also give other examples of update functions, that are able to reproduce the behaviors empirically observed in other contexts.

Evidence for a high-energy tail in the gamma-ray spectra of globular clusters

ABSTRACT Millisecond pulsars are very likely the main source of gamma-ray emission from globular clusters. However, the relative contributions of two separate emission processes – curvature radiation from millisecond pulsar magnetospheres versus inverse Compton emission from relativistic pairs launched into the globular cluster environment by millisecond pulsars – have long been unclear. To address this, we search for evidence of inverse Compton emission in 8-yr Fermi–LAT data from the directions of 157 Milky Way globular clusters. We find a mildly statistically significant (3.8σ) correlation between the measured globular cluster gamma-ray luminosities and their photon field energy densities. However, this may also be explained by a hidden correlation between the photon field densities and the stellar encounter rates of globular clusters. Analysed in toto, we demonstrate that the gamma-ray emission of globular clusters can be resolved spectrally into two components: (i) an exponentially cut-off power law and (ii) a pure power law. The latter component – which we uncover at a significance of 8.2σ – has a power index of 2.79 ± 0.25. It is most naturally interpreted as inverse Compton emission by cosmic-ray electrons and positrons injected by millisecond pulsars. We find the luminosity of this power-law component is comparable to, or slightly smaller than, the luminosity of the curved component, suggesting the fraction of millisecond pulsar spin-down luminosity into relativistic leptons is similar to the fraction of the spin-down luminosity into prompt magnetospheric radiation.

Acceleration of cosmic rays by double shock waves in galaxy clusters: application to radio relics

Context. Radio relics in galaxy clusters are known to be good laboratories for verification of the applicability of the diffusive shock acceleration (DSA) model in its canonical version. The need for such verification stems from the inconsistencies in the shock properties resulting from radio observations compared to X-ray observations. Aims. In this article we aim to explore how the presence of a second shock in the vicinity of a relic modifies the spectrum of accelerated electrons and decipher which of the involved parameters can have a significant impact on their shape. Methods. We analytically studied DSA of cosmic rays in two stationary shocks aiming to investigate the change of the distribution function. The latter eventually leads to spectrum slope deviations visible in different observations and simulations that do not appear to be explained by the case wherein cosmic rays interact with a single shock wave. Results. We obtain a complex distribution function f(x, p) depending on many parameters (distance between two shocks, compression ratios, spatial diffusion coefficients, injection value, etc.). This function reveals modifications that occur because of the coupled acceleration in both shocks. Apparently, deviations in the particle spectrum from the pure power law depend on a few parameters such as Q1/Q2, κ1/κ2, r1/r2, and L. Although we do not verify this idea by taking a particular cluster as an example, we demonstrate a potential cause of spectral disturbances in radio relics. In general terms, our findings appear to correlate with results from the literature when the distance between the shocks is of the order of the width of a radio relic and κ1/κ2 ∝ 3.

Thermally rounded depinning of an elastic interface on a washboard potential.

The thermal rounding of the depinning transition of an elastic interface sliding on a washboard potential is studied through analytic arguments and very accurate numerical simulations. We confirm the standard view that well below the depinning threshold the average velocity can be calculated considering thermally activated nucleation of defects. However, we find that the straightforward extension of this analysis to near or above the depinning threshold does not fully describe the physics of the thermally assisted motion. In particular, we find that exactly at the depinning point the average velocity does not follow a pure power law of the temperature as naively expected by the analogy with standard phase transitions but presents subtle logarithmic corrections. We explain the physical mechanisms behind these corrections and argue that they are nonpeculiar collective effects which may also apply to the case of interfaces sliding on uncorrelated disordered landscapes.

Empirical analysis of Zipf’s law, power law, and lognormal distributions in medical discharge reports

BackgroundBayesian modelling and statistical text analysis rely on informed probability priors to encourage good solutions. ObjectiveThis paper empirically analyses whether text in medical discharge reports follow Zipf’s law, a commonly assumed statistical property of language where word frequency follows a discrete power-law distribution. MethodWe examined 20,000 medical discharge reports from the MIMIC-III dataset. Methods included splitting the discharge reports into tokens, counting token frequency, fitting power-law distributions to the data, and testing whether alternative distributions—lognormal, exponential, stretched exponential, and truncated power-law—provided superior fits to the data. ResultDischarge reports are best fit by the truncated power-law and lognormal distributions. Discharge reports appear to be near-Zipfian by having the truncated power-law provide superior fits over a pure power-law. ConclusionOur findings suggest that Bayesian modelling and statistical text analysis of discharge report text would benefit from using truncated power-law and lognormal probability priors and non-parametric models that capture power-law behavior.

Stress‐Dependent Magnitudes of Induced Earthquakes in the Groningen Gas Field

AbstractGeological faults may produce earthquakes under induced stresses associated with hydrocarbon extraction, geothermal extraction, or CO2 storage. The associated risks depend on the frequency and magnitude of these earthquakes. Within seismic risk analysis, the exceedance probability of seismic moments, , is treated as a pure power law distribution, , where the power law exponent, β, may vary in time or space or with stress. Insights from statistical mechanics theories of brittle failure, statistical seismology, and acoustic emissions experiments all indicate this pure power law may contain an exponential taper, , where the taper strength, ζ, decreases with increasing stress. The role of this taper is to significantly reduce the probability of earthquakes larger than ζ −1 relative to the pure power law. We review the existing theoretical and observational evidence for a stress‐dependent exponential taper to motivate a range of magnitude models suitable for induced seismicity risk analysis. These include stress‐invariant models with and without a taper, stress‐dependent β models without a taper, and stress‐dependent ζ models. For each of these models, we evaluated their forecast performance within the Groningen gas field in the Netherlands using a combination of Bayesian inference, and simulations. Our results show that the stress‐dependent ζ model with constant β likely offers (75–85%) higher performance forecasts than the stress‐dependent β models with . This model also lowers the magnitudes with a 10% and 1% chance of exceedance over the next 5 years of gas production from 4.3 to 3.7 and from 5.5 to 4.3, respectively.

Planck2018 results

We report on the implications for cosmic inflation of the 2018 release of thePlanckcosmic microwave background (CMB) anisotropy measurements. The results are fully consistent with those reported using the data from the two previousPlanckcosmological releases, but have smaller uncertainties thanks to improvements in the characterization of polarization at low and high multipoles.Plancktemperature, polarization, and lensing data determine the spectral index of scalar perturbations to bens = 0.9649 ± 0.0042 at 68% CL. We find no evidence for a scale dependence ofns, either as a running or as a running of the running. The Universe is found to be consistent with spatial flatness with a precision of 0.4% at 95% CL by combiningPlanckwith a compilation of baryon acoustic oscillation data. ThePlanck95% CL upper limit on the tensor-to-scalar ratio,r0.002 < 0.10, is further tightened by combining with the BICEP2/Keck Array BK15 data to obtainr0.002 < 0.056. In the framework of standard single-field inflationary models with Einstein gravity, these results imply that: (a) the predictions of slow-roll models with a concave potential,V″(ϕ) < 0, are increasingly favoured by the data; and (b) based on two different methods for reconstructing the inflaton potential, we find no evidence for dynamics beyond slow roll. Three different methods for the non-parametric reconstruction of the primordial power spectrum consistently confirm a pure power law in the range of comoving scales 0.005 Mpc−1 ≲ k ≲ 0.2 Mpc−1. A complementary analysis also finds no evidence for theoretically motivated parameterized features in thePlanckpower spectra. For the case of oscillatory features that are logarithmic or linear ink, this result is further strengthened by a new combined analysis including thePlanckbispectrum data. The newPlanckpolarization data provide a stringent test of the adiabaticity of the initial conditions for the cosmological fluctuations. In correlated, mixed adiabatic and isocurvature models, the non-adiabatic contribution to the observed CMB temperature variance is constrained to 1.3%, 1.7%, and 1.7% at 95% CL for cold dark matter, neutrino density, and neutrino velocity, respectively.Planckpower spectra plus lensing set constraints on the amplitude of compensated cold dark matter-baryon isocurvature perturbations that are consistent with current complementary measurements. The polarization data also provide improved constraints on inflationary models that predict a small statistically anisotropic quadupolar modulation of the primordial fluctuations. However, the polarization data do not support physical models for a scale-dependent dipolar modulation. All these findings support the key predictions of the standard single-field inflationary models, which will be further tested by future cosmological observations.