Power-law Size Distribution Research Articles

A fractal model for characterizing fluid flow in fractured rock masses based on randomly distributed rock fracture networks

A fractal model that represents the geometric characteristics of rock fracture networks is proposed to link the fractal characteristics with the equivalent permeability of the fracture networks. The fracture networks are generated using the Monte Carlo method and have a power law size distribution. The fractal dimension DT is utilized to represent the tortuosity of the fluid flow, and another fractal dimension Df is utilized to represent the geometric distribution of fractures in the networks. The results indicate that the equivalent permeability of a fracture network can be significantly influenced by the tortuosity of the fluid flow, the aperture of the fractures and a random number used to generate the fractal length distribution of the fractures in the network. The correlation of fracture number and fracture length agrees well with the results of previous studies, and the calculated fractal dimensions Df are consistent with their theoretical values, which confirms the reliability of the proposed fractal length distribution and the stochastically generated fracture network models. The optimal hydraulic path can be identified in the longer fractures along the fluid flow direction. Using the proposed fractal model, a mathematical expression between the equivalent permeability K and the fractal dimension Df is proposed for models with large values of Df. The differences in the calculated flow volumes between the models that consider and those that do not consider the influence of fluid flow tortuosity are as high as 17.64–19.51%, which emphasizes that the effects of tortuosity should not be neglected and should be included in the fractal model to accurately estimate the hydraulic behavior of fracture networks.

Avalanche shape and exponents beyond mean-field theory

Elastic systems, such as magnetic domain walls, density waves, contact lines, and cracks, are pinned by substrate disorder. When driven, they move via avalanches, with power law distributions of size, duration and velocity. Their exponents, and the shape of an avalanche, defined as its mean velocity as a function of time, were studied. They are known approximatively from experiments and simulations, and were predicted from mean-field models, such as the Brownian force model (BFM), where each point of the elastic interface sees a force field which itself is a random walk. As we showed in EPL, 97 (2012) 46004, the BFM is the starting point for an expansion around the upper critical dimension, with for short-ranged elasticity, and for long-ranged elasticity. Here we calculate analytically the , i.e. 1-loop, correction to the avalanche shape at fixed duration T, for both types of elasticity. The exact expression, though different from the phenomenological form presented by Laurson et al. in Nat. Commun., 4 (2013) 2927, is well approximated by , 0 < x < 1. The asymmetry is negative for d close to , skewing the avalanche towards its end, as observed in numerical simulations in d = 2 and 3. The exponent is given by the two independent exponents at depinning, the roughness ζ and the dynamical exponent z. We propose a general procedure to predict other avalanche exponents in terms of ζ and z. We finally introduce and calculate the shape at fixed avalanche size, not yet measured in experiments or simulations.

Search for sub-kilometre trans-Neptunian objects using CoRoT asteroseismology data

We present the search for sub-kilometre trans-Neptunian objects (TNOs) by re-examining the CoRoT (Convection Rotation and Planetary Transits) asteroseismology observations. The total observation time employed in this work is about 144 000 star hours with signal-to-noise ratio larger than 1000 computed on 30-s intervals. 13 Possible Occultation Events (POEs) were found from the deviation method. These detections gives a density in the ecliptic sky plane of TNOs larger than 400-m radius of |$N(R > 400\,\mathrm{m})=1.4^{+4.2} _{-0.7} \times 10^{7}\,\mathrm{deg}^{-2}$|⁠. The fit of the density of TNOs with the believed break rbreak = 45 km provides a power-law size distribution index q larger than 3.5. This value is consistent with the detection of two potential events from Hubble Space Telescope (HST)/Fine Guidance Sensor (FGS) observations that found q = 3.8 ± 0.2. However, fitting the 13 POEs with the HST/FGS result alone gives a power-law size distribution index of q = 4.5 ± 0.2 in the size range of 0.2–2.0 km. This value is then compared with evolution models of the Kuiper belt.

MACROSCOPIC DUST IN PROTOPLANETARY DISKS—FROM GROWTH TO DESTRUCTION

The collision dynamics of dusty bodies are crucial for planetesimal formation. Especially decimeter agglomerates are important in the different formation models. Therefore, in continuation of our experiments on mutual decimeter collisions, we investigate collisions of centimeter onto decimeter dust agglomerates in a small drop tower under vacuum conditions (p<5*$10^{-1}$ mbar) at a mean collision velocity of 6.68 +- 0.67m/s. We use quartz dust with irregularly shaped micrometer grains. Centimeter projectiles with different diameters, masses and heights are used, their typical volume filling factor is $\Phi_{p,m}$=0.466+-0.02. The decimeter agglomerates have a mass of about 1.5kg, a diameter and height of 12cm and a mean filling factor of $\Phi_{t,m}$=0.44+-0.004. At lower collision energies only the projectile gets destroyed and mass is transferred to the target. The accretion efficiency decreases with increasing obliquity and increasing difference in filling factor, if the projectile is more compact than the target. The accretion efficiency increases with increasing collision energy for collision energies under a certain threshold. Beyond this threshold at 298+-25 mJ catastrophic disruption of the target can be observed. This corresponds to a critical fragmentation strength Q*=190+-16 mJ/kg, which is a factor of four larger than expected. Analyses of the projectile fragments show a power law size distribution with average exponent of -3.8+-0.3. The mass distributions suggest that the fraction of smallest fragments increases for higher collision energies. This is interesting for impacts of small particle on large target bodies within protoplanetary disks, as smaller fragments couple better to the surrounding gas and re-accretion by gas drag is more likely.

Markers of criticality in phase synchronization.

The concept of the brain as a critical dynamical system is very attractive because systems close to criticality are thought to maximize their dynamic range of information processing and communication. To date, there have been two key experimental observations in support of this hypothesis: (i) neuronal avalanches with power law distribution of size and (ii) long-range temporal correlations (LRTCs) in the amplitude of neural oscillations. The case for how these maximize dynamic range of information processing and communication is still being made and because a significant substrate for information coding and transmission is neural synchrony it is of interest to link synchronization measures with those of criticality. We propose a framework for characterizing criticality in synchronization based on an analysis of the moment-to-moment fluctuations of phase synchrony in terms of the presence of LRTCs. This framework relies on an estimation of the rate of change of phase difference and a set of methods we have developed to detect LRTCs. We test this framework against two classical models of criticality (Ising and Kuramoto) and recently described variants of these models aimed to more closely represent human brain dynamics. From these simulations we determine the parameters at which these systems show evidence of LRTCs in phase synchronization. We demonstrate proof of principle by analysing pairs of human simultaneous EEG and EMG time series, suggesting that LRTCs of corticomuscular phase synchronization can be detected in the resting state and experimentally manipulated. The existence of LRTCs in fluctuations of phase synchronization suggests that these fluctuations are governed by non-local behavior, with all scales contributing to system behavior. This has important implications regarding the conditions under which one should expect to see LRTCs in phase synchronization. Specifically, brain resting states may exhibit LRTCs reflecting a state of readiness facilitating rapid task-dependent shifts toward and away from synchronous states that abolish LRTCs.

A stochastic model for the temporal aspects of flow intermittency in micropillar compression

Systematic experimental investigations have demonstrated that the plastic deformation of micropillar proceeds through a sequence of intermittent bursts, the sizes of which follow power-law statistics. In this study, a stochastic model based on the power-law distribution of burst size is formulated in the framework of crystal plasticity in order to investigate the temporal aspects of flow intermittency in micropillar compression. A Monte Carlo simulation scheme is developed to determine the burst size when a burst activity is captured. This burst size is considered as the displacement boundary condition of burst deformation. Three-dimensional finite element analysis of the model is performed and its predictions are validated by comparison with results from both micro-compression experiments and simulation tests of bulk crystals using the classic crystal plasticity finite element method (CPFEM). The model provides a reasonable prediction of stress–strain responses both at the macroscopic and microscopic scales. Finally, the capability of this model is shown with applications to the intermittent plastic deformation in micropillar compressions, in particular for their burst time durations and burst velocities. The results from such stochastic finite element analysis are shown to be consistent with earlier experimental findings and results of mean-field theory.

SOLAR ENERGETIC PARTICLE EVENTS IN DIFFERENT TYPES OF SOLAR WIND

We examine statistically some properties of 96 20 MeV gradual solar energetic proton (SEP) events as a function of three different types of solar wind (SW) as classified by Richardson and Cane. Gradual SEP (E > 10 MeV) events are produced in shocks driven by fast (V 900 km s–1) and wide (W > 60°) coronal mass ejections (CMEs). We find no differences among the transient, fast, and slow SW streams for SEP 20 MeV proton event timescales. It has recently been found that the peak intensities Ip of these SEP events scale with the ~2 MeV proton background intensities, which may be a proxy for the near-Sun shock seed particles. Both the intensities Ip and their 2 MeV backgrounds are significantly enhanced in transient SW compared to those of fast and slow SW streams, and the values of Ip normalized to the 2 MeV backgrounds only weakly correlate with CME V for all SW types. This result implies that forecasts of SEP events could be improved by monitoring both the Sun and the local SW stream properties and that the well known power-law size distributions of Ip may differ between transient and long-lived SW streams. We interpret an observed correlation between CME V and the 2 MeV background for SEP events in transient SW as a manifestation of enhanced solar activity.

SIZE DISTRIBUTION OF POSSIBLE DUST CARRIERS FOR THE EXTENDED RED EMISSION

Power-law size distributions expected to be applicable to possible carriers of extended red emission (ERE) have been examined using Monte Carlo (MC) simulations. Si nanoparticles and some polycyclic aromatic hydrocarbon complexes such as oligoacene and oligorylenes with energy gaps close to 2 eV have been considered. In the simplest case of unit quantum efficiency, the MC-generated size distributions are used to obtain photoluminescence (PL) spectra that are then corrected for dust extinction and reddening effects for comparison with observed ERE spectra. It is shown that a power-law size distribution with a decay exponent of α = 7/2, which closely agrees with starlight extinction data, fails to produce an ERE-like spectrum. However, size distributions with decay exponents of α = 19/12 and 3/2 are found to lead to acceptable spectra. Results indicate that energetic photon-induced breakup and competing aggregation effects dominate collisional effects in producing the observed steady-state mass distribution. It is shown that the peak wavelength of emission critically depends on the band gap, rather than cluster mass, which for oligoacenes and oligorylenes is widely different. The peak wavelength is also shown to be insensitive to dust attenuation.

Effect of morphology on light scattering by agglomerates

Using the discrete dipole approximation (DDA), we compute light scattering from irregularly shaped particles with three different types of agglomerate morphologies. The packing density of materials in the model particles spans the range from 0.169 through 0.336. We investigate four different refractive indices m=1.313+0i, 1.6+0.0005i, 1.5+0.1i, and 1.855+0.45i, which are representative of various cosmic and terrestrial materials in the visible. In each case, we consider a wide range of size parameters that makes it possible to average the light-scattering response over particle size with a power-law size distribution. We find that despite noticeable differences in particle morphology, their light-scattering responses are remarkably similar; the difference in light scattering often does not exceed the error bars occurring in laboratory measurements of micron-sized particles. On the contrary, the impact of refractive index and size distribution on light scattering appears to be much stronger compared with the morphology of complex, agglomerate particles. This finding may simplify considerably the interpretation of photo-polarimetric observations of atmospheric aerosols, cosmic dust particles, etc., because the precise specification of target-particle shape is unnecessary for the analysis.

Collective philanthropy: describing and modeling the ecology of giving.

Reflective of income and wealth distributions, philanthropic gifting appears to follow an approximate power-law size distribution as measured by the size of gifts received by individual institutions. We explore the ecology of gifting by analysing data sets of individual gifts for a diverse group of institutions dedicated to education, medicine, art, public support, and religion. We find that the detailed forms of gift-size distributions differ across but are relatively constant within charity categories. We construct a model for how a donor's income affects their giving preferences in different charity categories, offering a mechanistic explanation for variations in institutional gift-size distributions. We discuss how knowledge of gift-sized distributions may be used to assess an institution's gift-giving profile, to help set fundraising goals, and to design an institution-specific giving pyramid.

Fragmentation and limits to dynamical scaling in viscous coarsening: an interrupted in situ x-ray tomographic study.

X-ray microtomography was used to follow the coarsening of the structure of a ternary silicate glass experiencing phase separation in the liquid state. The volumes, surfaces, mean, and Gaussian curvatures of the domains of minority phase were measured after reconstruction of the 3D images and segmentation. A linear growth law of the characteristic length scale ℓ∼t was observed. A detailed morphological study was performed. While dynamical scaling holds for most of the geometrical observables under study, a progressive departure from scaling invariance of the distributions of local curvatures was evidenced. The latter results from a gradual fragmentation of the structure in the less viscous phase that also leads to a power-law size distribution of isolated domains.

A universal power-law scaling exponent for fracture apertures in sandstones

A high-resolution data set of kinematic aperture (opening displacement) of opening-mode fractures, from large (up to 2 m long) quartz-cemented sandstone samples, shows that microfractures are ubiquitous and that most natural-fracture sets are better fit by power-law size distributions than by exponential, normal, or log-normal distributions. The data set includes 3822 fractures within 68 scanlines from eight formations on three continents. Kinematic apertures were measured along scanlines using scanning electron microscope–based cathodoluminescence (SEM-CL) and, for field data, using a hand lens. Microtextural evidence from SEM-CL shows that power law–distributed fractures typically have crack-seal texture and are composed of opening increments having a narrow (characteristic) aperture size range. In contrast, rare non-power-law–distributed fracture populations lack crack-seal texture. Power-law exponents, as measured in one dimension, have values of −0.8 ± 0.1. Most variation among fracture sets results from power-law coefficients, which constitute a scale-invariant measure of fracture intensity. We show how observed scaling patterns can be used to improve estimations of large-fracture spacing in cases where fracture sampling is limited, as by the width of cores. The low (

Power law size-distributed heterogeneity explains colloid retention on soda lime glass in the presence of energy barriers.

This article concerns reading the nanoscale heterogeneity thought responsible for colloid retention on surfaces in the presence of energy barriers (unfavorable attachment conditions). We back out this heterogeneity on glass surfaces by comparing mechanistic simulations incorporating discrete heterogeneity with colloid deposition experiments performed across a comprehensive set of experimental conditions. Original data is presented for attachment to soda lime glass for three colloid sizes (0.25, 1.1, and 1.95 μm microspheres) under a variety of ionic strengths and fluid velocities in an impinging jet system. A comparison of mechanistic particle trajectory simulations incorporating discrete surface heterogeneity represented by nanoscale zones of positive charge (heterodomains) indicates that a power law size distribution of heterodomains ranging in size from 120 to 60 nm in radius was able to explain the observed retention for all conditions examined. In contrast, uniform and random placement of single-sized heterodomains failed to capture experimentally observed colloid retention across the range of conditions examined.

Undersampled critical branching processes on small-world and random networks fail to reproduce the statistics of spike avalanches.

The power-law size distributions obtained experimentally for neuronal avalanches are an important evidence of criticality in the brain. This evidence is supported by the fact that a critical branching process exhibits the same exponent . Models at criticality have been employed to mimic avalanche propagation and explain the statistics observed experimentally. However, a crucial aspect of neuronal recordings has been almost completely neglected in the models: undersampling. While in a typical multielectrode array hundreds of neurons are recorded, in the same area of neuronal tissue tens of thousands of neurons can be found. Here we investigate the consequences of undersampling in models with three different topologies (two-dimensional, small-world and random network) and three different dynamical regimes (subcritical, critical and supercritical). We found that undersampling modifies avalanche size distributions, extinguishing the power laws observed in critical systems. Distributions from subcritical systems are also modified, but the shape of the undersampled distributions is more similar to that of a fully sampled system. Undersampled supercritical systems can recover the general characteristics of the fully sampled version, provided that enough neurons are measured. Undersampling in two-dimensional and small-world networks leads to similar effects, while the random network is insensitive to sampling density due to the lack of a well-defined neighborhood. We conjecture that neuronal avalanches recorded from local field potentials avoid undersampling effects due to the nature of this signal, but the same does not hold for spike avalanches. We conclude that undersampled branching-process-like models in these topologies fail to reproduce the statistics of spike avalanches.

FOMALHAUT b AS A CLOUD OF DUST: TESTING ASPECTS OF PLANET FORMATION THEORY

We consider the ability of three models - impacts, captures, and collisional cascades - to account for a bright cloud of dust in Fomalhaut b. Our analysis is based on a novel approach to the power-law size distribution of solid particles central to each model. When impacts produce debris with (i) little material in the largest remnant and (ii) a steep size distribution, the debris has enough cross-sectional area to match observations of Fomalhaut b. However, published numerical experiments of impacts between 100 km objects suggest this outcome is unlikely. If collisional processes maintain a steep size distribution over a broad range of particle sizes (300 microns to 10 km), Earth-mass planets can capture enough material over 1-100 Myr to produce a detectable cloud of dust. Otherwise, capture fails. When young planets are surrounded by massive clouds or disks of satellites, a collisional cascade is the simplest mechanism for dust production in Fomalhaut b. Several tests using HST or JWST data - including measuring the expansion/elongation of Fomalhaut b, looking for trails of small particles along Fomalhaut b's orbit, and obtaining low resolution spectroscopy - can discriminate among these models.

HUBBLE SPACE TELESCOPEINVESTIGATION OF MAIN-BELT COMET 133P/ELST-PIZARRO

We report new observations of the prototype main-belt comet (active asteroid) 133P/Elst-Pizarro taken at high angular resolution using the Hubble Space Telescope. The object has three main components; a) a point-like nucleus, b) a long, narrow antisolar dust tail and c) a short, sunward anti-tail. There is no resolved coma. The nucleus has a mean absolute magnitude H_V = 15.70+/-0.10 and a lightcurve range 0.42 mag., the latter corresponding to projected dimensions 3.6 x 5.4 km (axis ratio 1.5:1), at the previously measured geometric albedo of 0.05+/-0.02. We explored a range of continuous and impulsive emission models to simultaneously fit the measured surface brightness profile, width and position angle of the antisolar tail. Preferred fits invoke protracted emission, over a period of 150 days or less, of dust grains following a differential power-law size distribution with index 3.25 < q < 3.5 and having a wide range of sizes. Ultra-low surface brightness dust projected in the sunward direction is a remnant from emission activity occurring in previous orbits, and consists of the largest (>cm-sized) particles. Ejection velocities of one micron-sized particles are comparable to the ~1.8 m/s gravitational escape speed of the nucleus, while larger particles are released at speeds less than the gravitational escape velocity. The observations are consistent with, but do not prove, a hybrid hypothesis in which mass loss is driven by gas drag from the sublimation of near-surface water ice, but escape is aided by centripetal acceleration from the rotation of the elongated nucleus. No plausible alternative hypothesis has been identified.

Millimetre spectral indices of transition disks and their relation to the cavity radius

Transition disks are protoplanetary disks with inner depleted dust cavities and excellent candidates to investigate the dust evolution under the existence of a pressure bump. A pressure bump at the outer edge of the cavity allows dust grains from the outer regions to stop their rapid inward migration towards the star and efficiently grow to millimetre sizes. Dynamical interactions with planet(s) have been one of the most exciting theories to explain the clearing of the inner disk. We look for evidence of the presence of millimetre dust particles in transition disks by measuring their spectral index with new and available photometric data. We investigate the influence of the size of the dust depleted cavity on the disk integrated millimetre spectral index. We present the 3mm photometric observations carried out with PdBI of four transition disks: LkHa330, UXTauA, LRLL31, and LRLL67. We use available values of their fluxes at 345GHz to calculate their spectral index, as well as the spectral index for a sample of twenty transition disks. We compare the observations with two kind of models. In the first set of models, we consider coagulation and fragmentation of dust in a disk in which a cavity is formed by a massive planet located at different positions. The second set of models assumes disks with truncated inner parts at different radius and with power-law dust size distributions, where the maximum size of grains is calculated considering turbulence as the source of destructive collisions. We show that the integrated spectral index is higher for transition disks than for regular protoplanetary disks. For transition disks, the probability that the measured spectral index is positively correlated with the cavity radius is 95%. High angular resolution imaging of transition disks is needed to distinguish between the dust trapping scenario and the truncated disk case.

Optically significant particle sizes in seawater

Small particles (<10 μm) are often considered to play the dominant role in controlling scattering and absorption due to their relatively large numbers, which are typically found in the ocean. Here we present an approach for quantifying the size range of particles that contribute significantly to bulk inherent optical properties. We present a numerical assessment of the variability in optically significant particle sizes for simplistic populations that conform to the assumptions of homogeneous, spherical particles, and power-law size distributions. We use numerical predictions from Mie theory to suggest minimum and maximum particle sizes required for accurate predictions and observations of ocean optics for different particle size distributions (PSDs). When considering observed ranges of PSDs, our predictions suggest the need for measurements of optical properties and particles to capture information from particle sizes between diameters of 0.05-2000 μm in order to properly constrain relationships between particles and their associated optical properties. Natural particle populations in the ocean may present more complex PSDs that could be analyzed using the method presented here to establish optically significant size classes.

Gutenberg-Richter relation originates from Coulomb stress fluctuations caused by elastic rock heterogeneity

Based on measurements along boreholes, a characterization of the Earth's crust elastic heterogeneity is presented. We investigate its impact on Coulomb stress distribution and earthquake magnitude scaling. The analysis of elastic modulus distribution at various borehole locations in different regions reveals universal fractal nature of elastic heterogeneity. By applying a homogeneous far-field stress to a representative model of elastic rock heterogeneity, we show that it causes strong Coulomb stress fluctuations. In situ fluctuations of Coulomb stress are mainly controlled by in situ elastic moduli. Fluctuations caused by surrounding heterogeneities are only of minor importance. Hence, the fractal nature of elastic heterogeneity results in Coulomb stress fluctuations with power law size distribution. As a consequence, fault sizes and magnitudes of earthquakes scale according to the Gutenberg-Richter relation. Due to the universal fractal nature of elastic heterogeneity, the b value should be universal. Deviation from its universal value of b ≈1 occurs due to characteristic scales of seismogenic processes, which cause limitations or changes of fractal scaling. Scale limitations are also the reason for observed stress dependency of the b value. Our analysis suggests that the Gutenberg-Richter relation originates from Coulomb stress fluctuations caused by elastic rock heterogeneity.

The Burst Time Duration in Micropillar Deformation

Plastic flow of single crystal micropillars proceeds through a sequence of intermittent burst slips. The burst time durations are investigated based on an extended theoretical model which incorporates the observed power-law distribution of burst sizes in compression experiments of micropillars. The results show that the burst time durations exhibit a powerlaw behavior with an exponential cutoff, suggesting the same scaling behaviors as the burst sizes. In addition, the predicted scaling exponent is found to converge to a value of ∼1.6. It is demonstrated that our results are consistent with the experimental data.