Power-law Probability Distribution Function Research Articles

Entropy variation in a fractal phase space

In this paper we recall for physicists how it is possible, using the principle of maximization of the Boltzmann-Shannon entropy, to derive the Burr-Singh-Maddala (BurrXII) double power law probability distribution function and its approximations (Pareto, loglogistic ..) and extension (GB2…) first used in econometrics. This is possible using a deformation of the power function, as this has been done in complex systems for the exponential function. We give to that distribution a deep stochastic interpretation using the theory of Weron et al. Applied to thermodynamics the entropy nonextensivity can be accounted for by assuming that the asymptotic exponents are scale dependent. Therefore functions which describe phenomena presenting power-law asymptotic behaviour can be obtained without introducing exotic forms of the entropy.

Avalanches and mixing behavior of porous 316L stainless steel under tension

The investigation of acoustic emission (AE) reveals mixing of avalanche processes in porous 316L stainless steel. One avalanche mechanism relates to the movement of dislocations, the other to crack propagation. Both mechanisms occur under different external tensions: small tension dislodges dislocations while crack propagation occurs at much higher tension close to the mechanical failure point. In an intermediate overlap regime, both effects occur simultaneously. The avalanche related power laws show a mixing behavior where the higher avalanche exponents form an upper limit for the mixing curve and the lower exponent is approached asymptotically for strong AE signals. The power law probability distribution functions, show a characteristic upwards bend near the crossover between the two mechanisms. The three regimes, namely, the dislocation movement, mixing, and crack propagation, are confirmed by other avalanche characteristics: the sparseness of the AE spectra, the avalanche exponents of energy, amplitude, and duration. Only the crack propagation follows mean field predictions, while the dislocation movements (in confined spaces) deviates significantly from mean field behavior with much greater exponents than those predicted in this approximation. We demonstrate that apparent deviations from scale invariance are, in reality, the result of two superimposed avalanche processes whereby each of them remains scale invariant. The seeming deviations from scale invariance are actually superposition effects.

Automodel solutions for superdiffusive transport by Lévy walks

The method of approximate automodel solution for the Green’s function of time-dependent superdiffusive (nonlocal) transport equations (Kukushkin A B and Sdvizhenskii P A 2016 J. Phys. A: Math. Theor. 49 255002) is extended to the case of a finite velocity of carriers. This corresponds to an extension from Lévy flight based transport to transport of a type which belongs to the class of ‘Lévy walks + rests’, to allow for retardation effects in Lévy flights. This problem covers the cases of transport by resonant photons in astrophysical gases and plasmas, heat transport by electromagnetic waves in plasmas, migration of predators, and other applications. We treat a model case of 1D transport on a uniform background with a simple power-law step-length probability distribution function (PDF). A solution for the arbitrary superdiffusive PDF is suggested, and verification of the solution for a particular power-law PDF, which corresponds, e.g., to Lorentzian wings of atomic spectral line shapes for the emission of photons, is carried out using the computation of the exact solution.

Probability distribution functions of gas surface density in M 33

Aims. We examine the interstellar medium (ISM) of M 33 to unveil fingerprints of self-gravitating gas clouds throughout the star-forming disk. Methods. The probability distribution functions (PDFs) for atomic, molecular, and total gas surface densities are determined at a resolution of about 50 pc over regions that share coherent morphological properties and considering cloud samples at different evolutionary stages in the star formation cycle. Results. Most of the total gas PDFs are well fit by log-normal functions whose width decreases radially outward. Because the HI velocity dispersion is approximately constant throughout the disk, the decrease in PDF width is consistent with a lower Mach number for the turbulent ISM at large galactocentric radii where a higher fraction of HI is in the warm phase. The atomic gas is found mostly at face-on column densities below NHlim = 2.5 × 1021 cm−2, with small radial variations of NHlim. The molecular gas PDFs do not show strong deviations from log-normal functions in the central region where molecular fractions are high. Here the high pressure and rate of star formation shapes the PDF as a log-normal function, dispersing self-gravitating complexes with intense feedback at all column densities that are spatially resolved. Power-law PDFs for the molecules are found near and above NHlim, in the southern spiral arm and in a continuous dense filament extending at larger galactocentric radii. In the filament nearly half of the molecular gas departs from a log-normal PDF, and power laws are also observed in pre-star-forming molecular complexes. The slope of the power law is between −1 and −2. This slope, combined with maps showing where the different parts of the power law PDFs come from, suggests a power-law stratification of the density within molecular cloud complexes, in agreement with the dominance of self-gravity.

Complexity and Anisotropy of Plastic Flow of α-Ti Probed by Acoustic Emission and Local Extensometry.

Current progress in the prediction of mechanical behavior of solids requires understanding of spatiotemporal complexity of plastic flow caused by self-organization of crystal defects. It may be particularly important in hexagonal materials because of their strong anisotropy and combination of different mechanisms of plasticity, such as dislocation glide and twinning. These materials often display complex behavior even on the macroscopic scale of deformation curves, e.g., a peculiar three-stage elastoplastic transition, the origin of which is a matter of debates. The present work is devoted to a multiscale study of plastic flow in α-Ti, based on simultaneous recording of deformation curves, 1D local strain field, and acoustic emission (AE). It is found that the average AE activity also reveals three-stage behavior, but in a qualitatively different way depending on the crystallographic orientation of the sample axis. On the finer scale, the statistical analysis of AE events and local strain rates testifies to an avalanche-like character of dislocation processes, reflected in power-law probability distribution functions. The results are discussed from the viewpoint of collective dislocation dynamics and are confronted to predictions of a recent micromechanical model of Ti strain hardening.

A dual power-law distribution for the stellar initial mass function

We introduce a new dual power law (DPL) probability distribution function for the mass distribution of stellar and substellar objects at birth, otherwise known as the initial mass function (IMF). The model contains both deterministic and stochastic elements, and provides a unified framework within which to view the formation of brown dwarfs and stars resulting from an accretion process that starts from extremely low mass seeds. It does not depend upon a top down scenario of collapsing (Jeans) masses or an initial lognormal or otherwise IMF-like distribution of seed masses. Like the modified lognormal power law (MLP) distribution, the DPL distribution has a power law at the high mass end, as a result of exponential growth of mass coupled with equally likely stopping of accretion at any time interval. Unlike the MLP, a power law decay also appears at the low mass end of the IMF. This feature is closely connected to the accretion stopping probability rising from an initially low value up to a high value. This might be associated with physical effects of ejections sometimes (i.e., rarely) stopping accretion at early times followed by outflow driven accretion stopping at later times, with the transition happening at a critical time (therefore mass). Comparing the DPL to empirical data, the critical mass is close to the substellar mass limit, suggesting that the onset of nuclear fusion plays an important role in the subsequent accretion history of a young stellar object.

The effect of magnetic fields and ambipolar diffusion on the column density probability distribution function in molecular clouds

Simulations generally show that non-self-gravitating clouds have a lognormal column density ($\Sigma$) probability distribution function (PDF), while self-gravitating clouds with active star formation develop a distinct power-law tail at high column density. Although the growth of the power law can be attributed to gravitational contraction leading to the formation of condensed cores, it is often debated if an observed lognormal shape is a direct consequence of supersonic turbulence alone, or even if it is really observed in molecular clouds. In this paper we run three-dimensional magnetohydrodynamic simulations including ambipolar diffusion with different initial conditions to see the effect of strong magnetic fields and nonlinear initial velocity perturbations on the evolution of the column density PDFs. Our simulations show that column density PDFs of clouds with supercritical mass-to-flux ratio, with either linear perturbations or nonlinear turbulence, quickly develop a power-law tail such that $dN/d \log \Sigma \propto \Sigma^{-\alpha}$ with index $\alpha \simeq 2$. Interestingly, clouds with subcritical mass-to-flux ratio also proceed directly to a power-law PDF, but with a much steeper index $\alpha \simeq 4$. This is a result of gravitationally-driven ambipolar diffusion. However, for nonlinear perturbations with a turbulent spectrum ($v_{k}^{2} \propto k^{-4}$), the column density PDFs of subcritical clouds do retain a lognormal shape for a major part of the cloud evolution, and only develop a distinct power-law tail with index $\alpha \simeq 2$ at greater column density when supercritical pockets are formed.

Predicting mining collapse: Superjerks and the appearance of record-breaking events in coal as collapse precursors.

The quest for predictive indicators for the collapse of coal mines has led to a robust criterion from scale-model tests in the laboratory. Mechanical collapse under uniaxial stress forms avalanches with a power-law probability distribution function of radiated energy P∼E^{-}^{ɛ}, with exponent ɛ=1.5. Impending major collapse is preceded by a reduction of the energy exponent to the mean-field value ɛ=1.32. Concurrently, the crackling noise increases in intensity and the waiting time between avalanches is reduced when the major collapse is approaching. These latter criteria were so-far deemed too unreliable for safety assessments in coal mines. We report a reassessment of previously collected extensive collapse data sets using "record-breaking analysis," based on the statistical appearance of "superjerks" within a smaller spectrum of collapse events. Superjerks are defined as avalanche signals with energies that surpass those of all previous events. The final major collapse is one such superjerk but other "near collapse" events equally qualify. In this way a very large data set of events is reduced to a sparse sequence of superjerks (21 in our coal sample). The main collapse can be anticipated from the sequence of energies and waiting times of superjerks, ignoring all weaker events. Superjerks are excellent indicators for the temporal evolution, and reveal clear nonstationarity of the crackling noise at constant loading rate, as well as self-similarity in the energy distribution of superjerks as a function of the number of events so far in the sequence E_{sj}∼n^{δ} with δ=1.79. They are less robust in identifying the precise time of the final collapse, however, than the shift of the energy exponents in the whole data set which occurs only over a short time interval just before the major event. Nevertheless, they provide additional diagnostics that may increase the reliability of such forecasts.

Fractional calculus ties the microscopic and macroscopic scales of complex network dynamics

A two-state, master equation-based decision-making model has been shown to generate phase transitions, to be topologically complex, and to manifest temporal complexity through an inverse power-law probability distribution function in the switching times between the two critical states of consensus. These properties are entailed by the fundamental assumption that the network elements in the decision-making model imperfectly imitate one another. The process of subordination establishes that a single network element can be described by a fractional master equation whose analytic solution yields the observed inverse power-law probability distribution obtained by numerical integration of the two-state master equation to a high degree of accuracy.

Molecular clouds have power-law probability distribution functions

In this Letter we investigate the shape of the probability distribution of column densities (PDF) in molecular clouds. Through the use of low-noise, extinction-calibrated \textit{Herschel}/\textit{Planck} emission data for eight molecular clouds, we demonstrate that, contrary to common belief, the PDFs of molecular clouds are not described well by log-normal functions, but are instead power laws with exponents close to two and with breaks between $A_K \simeq 0.1$ and $0.2\,\mathrm{mag}$, so close to the CO self-shielding limit and not far from the transition between molecular and atomic gas. Additionally, we argue that the intrinsic functional form of the PDF cannot be securely determined below $A_K \simeq 0.1\,\mathrm{mag}$, limiting our ability to investigate more complex models for the shape of the cloud PDF.

The Burr XII Distribution Family and the Maximum Entropy Principle: Power-Law Phenomena Are Not Necessarily “Nonextensive”

In this paper we recall for physicists how it is possible, using the principle of maximization of the Boltzmann-Shannon entropy, to derive the Burr-Bingh-Maddala (burr12) double power law probability distribution function and its approximations (Pareto, loglogistic ..) and extension first used in econometrics. this is possible using a deformation of the power function, as this has been done in complex systems for the exponential function. We give to that distribution a deep stochastic interpretation using the theory of Weron et al. applied to thermodynamics the entropy nonextensivity can be accounted for by assuming that the asymptotic exponents are scale dependent. Therefore functions which describe phenomena presenting power-law asymptotic behaviour can be obtained without introducing exotic forms of the entropy.

Velocity Inversion In Cylindrical Couette Gas Flows

We investigate a power-law probability distribution function to describe the mean free path of rarefied gas molecules in non-planar geometries. A new curvature-dependent model is derived by taking into account the boundary-limiting effects on the molecular mean free path for surfaces with both convex and concave curvatures. In comparison to a planar wall, we find that the mean free path for a convex surface is higher at the wall and exhibits a sharper gradient within the Knudsen layer. In contrast, a concave wall exhibits a lower mean free path near the surface and the gradients in the Knudsen layer are shallower. The Navier-Stokes constitutive relations and velocity-slip boundary conditions are modified based on a power-law scaling to describe the mean free path, in accordance with the kinetic theory of gases, i.e. transport properties can be described in terms of the mean free path.Velocity profiles for isothermal cylindrical Couette flow are obtained using the power-law model. We demonstrate that our model is more accurate than the classical slip solution, especially in the transition regime, and we are able to capture important non-linear trends associated with the non-equilibrium physics of the Knudsen layer. In addition, we establish a new criterion for the critical accommodation coefficient that leads to the non-intuitive phenomena of velocity-inversion. Our results are compared with conventional hydrodynamic models and direct simulation Monte Carlo data. The power-law model predicts that the critical accommodation coefficient is significantly lower than that calculated using the classical slip solution and is in good agreement with available DSMC data. Our proposed constitutive scaling for non-planar surfaces is based on simple physical arguments and can be readily implemented in conventional fluid dynamics codes for arbitrary geometric configurations.

Wind Velocity Vertical Extrapolation by Extended Power Law

Wind energy gains more attention day by day as one of the clean renewable energy resources. We predicted wind speed vertical extrapolation by using extended power law. In this study, an extended vertical wind velocity extrapolation formulation is derived on the basis of perturbation theory by considering power law and Weibull wind speed probability distribution function. In the proposed methodology not only the mean values of the wind speeds at different elevations but also their standard deviations and the cross-correlation coefficient between different elevations are taken into consideration. The application of the presented methodology is performed for wind speed measurements at Karaburun/Istanbul, Turkey. At this location, hourly wind speed measurements are available for three different heights above the earth surface.

Exploiting the power law distribution properties of satellite fire radiative power retrievals: A method to estimate fire radiative energy and biomass burned from sparse satellite observations

[1] Instantaneous estimates of the power released by fire (fire radiative power, FRP) are available with satellite active fire detection products. The temporal integral of FRP provides an estimate of the fire radiative energy (FRE) that is related linearly to the amount of biomass burned needed by the atmospheric emissions modeling community. The FRE, however, is sensitive to satellite temporal and spatial FRP undersampling due to infrequent satellite overpasses, cloud and smoke obscuration, and failure to detect cool and/or small fires. Satellite FRPs derived over individual burned areas and fires have been observed to exhibit power law distributions. This property is exploited to develop a new way to derive FRE, as the product of the fire duration and the expected FRP value derived from the FRP power law probability distribution function. The method is demonstrated and validated by the use of FRP data measured with a dual-band radiometer over prescribed fires in the United States and by the use of FRP data retrieved from moderate resolution imaging spectroradiometer (MODIS) active-fire detections over Brazilian deforestation and Australian savanna fires. The biomass burned derived using the conventional FRP temporal integration and power law FRE estimation methods is compared with biomass burned measurements (prescribed fires) and available fuel load information reported in the literature (Australian and Brazilian fires). The results indicate that the FRE power law derivation method may provide more reliable burned biomass estimates under sparse satellite FRP sampling conditions and correct for satellite active-fire detection omission errors if the FRP power law distribution parameters and the fire duration are known.

Dynamics of sleep stage transitions in healthy humans and patients with chronic fatigue syndrome.

Physiological and/or pathological implications of the dynamics of sleep stage transitions have not, to date, been investigated. We report detailed duration and transition statistics between sleep stages in healthy subjects and in others with chronic fatigue syndrome (CFS); in addition, we also compare our data with previously published results for rats. Twenty-two healthy females and 22 female patients with CFS, characterized by complaints of unrefreshing sleep, underwent one night of polysomnographic recording. We find that duration of deep sleep (stages III and IV) follows a power-law probability distribution function; in contrast, stage II sleep durations follow a stretched exponential and stage I, and REM sleep durations follow an exponential function. These stage duration distributions show a gradually increasing departure from the exponential form with increasing depth of sleep toward a power-law type distribution for deep sleep, suggesting increasing complexity of regulation of deeper sleep stages. We also find a substantial number of REM to non-REM sleep transitions in humans, while this transition is reported to be virtually nonexistent in rats. The relative frequency of this REM to non-REM sleep transition is significantly lower in CFS patients than in controls, resulting in a significantly greater relative transition frequency of moving from both REM and stage I sleep to awake. Such an alteration in the transition pattern suggests that the normal continuation of sleep in light or REM sleep is disrupted in CFS. We conclude that dynamic transition analysis of sleep stages is useful for elucidating yet-to-be-determined human sleep regulation mechanisms with pathophysiological implications.

Power-law random walks

In this paper, random walks with independent steps distributed according to a Q-power-law probability distribution function with Q=1/(1-q) are studied. In the case q>1, we show that (i) a stochastic representation of the location of the walk after n steps can be explicitly given (for both finite and infinite variance) and (ii) a clear connection with the superstatistics framework can be established (including the anomalous diffusion case). In the case q<1, we prove that this random walk can be considered as the projection of an isotropic random walk, i.e., a random walk with fixed length steps and uniformly distributed directions. These results provide a natural extension of (i) the usual Gaussian framework and (ii) the infinite-covariance case of the superstatistics treatments.

Two-Threshold Model for Scaling Laws of Noninteracting Snow Avalanches

The sizes of snow slab failure that trigger snow avalanches are power-law distributed. Such a power-law probability distribution function has also been proposed to characterize different landslide types. In order to understand this scaling for gravity-driven systems, we introduce a two-threshold 2D cellular automaton, in which failure occurs irreversibly. Taking snow slab avalanches as a model system, we find that the sizes of the largest avalanches just preceding the lattice system breakdown are power-law distributed. By tuning the maximum value of the ratio of the two failure thresholds our model reproduces the range of power-law exponents observed for land, rock, or snow avalanches. We suggest this control parameter represents the material cohesion anisotropy.