Power-law Distribution Function Research Articles

Effect of porosity on the nonlinear free vibrational behavior of two-directional functionally graded porous cone-shaped shells resting on elastic substrates

The present article describes an analytical approach to investigate the nonlinear free vibrational behavior of a two-directional functionally graded porous (TDFGP) cone-shaped shell resting on elastic substrates. A mixture model with the power-law distribution functions is utilized to obtain the TDFGP shell’s mechanical properties. In addition, it is assumed that the internal porosities in the matrix materials can be dispersed into two independent patterns, either even or uneven porosity distribution. The field displacement in the TDFGP shell is approximated using first-order shear deformation theory. Hamilton’s principle and von Karman’s large deformation assumptions give a dynamic model for the TDFGP shell. The Galerkin decomposition method is employed to discretize the governing partial differential equations into time-dependent ordinary differential equations. The harmonic balance method is used to analytically solve the final time-dependent equations and acquire the nonlinear frequency response of the TDFGP shell. Research has been conducted on porosity, gradient indexes, elastic substrates, boundary conditions, and geometric ratios through parametric studies to investigate the outcomes and their applicability to real-world issues.

Kinetic effects of dust size distribution on Alfvén waves in magnetized space plasmas

ABSTRACT Dust populations in space plasmas are often described by a size distribution function, generally a power law distribution. In view of that, we include this feature in the kinetic description of a homogeneous magnetized dusty plasma with electrically charged immobile dust grains, in order to study its effects in the propagation and damping of Alfvén waves. The dispersion relation is numerically solved using parameters typically found in the dust-driven stellar winds of carbon-rich stars and in Earth’s auroral acceleration region, two space systems with unalike plasma parameters and in which Alfvén waves are known to play important roles in the plasma acceleration and heating processes. We show that the characteristics of the normal modes, namely the ion cyclotron and whistler modes, will change when one considers a power law distribution of dust sizes in the theory, as compared to a mono-sized dust population; and that these differences will depend on the exponent p of the power law, which alters the plasma charge imbalance between electrons and ions. We also notice that power-law distribution functions will modify the waves’ damping rate values. In particular, we show that in a stellar wind environment the ion cyclotron mode at very small wavenumber decreases with the reduction of p, while for higher wavenumber the damping of this mode increases with the reduction of p. For the Earth’s magnetosphere, the results obtained show that the wave damping increases with the decrease of p for all wavenumbers, for the parameters considered in the analysis.

Consistent Modeling of Transport Processes and Travel Times—Coupling Soil Hydrologic Processes With StorAge Selection Functions

AbstractUnderstanding the transport processes and travel times of pollutants in the subsurface is crucial for an effective management of drinking water resources. Transport processes and soil hydrologic processes are inherently linked to each other. In order to account for this link, we couple the process‐based hydrologic model RoGeR with StorAge Selection (SAS) functions. We assign to each hydrological process a specific SAS function (e.g., power law distribution function). To represent different transport mechanisms, we combined a specific set of SAS functions into four transport model structures: complete‐mixing, piston flow, advection‐dispersion and advection‐dispersion with time‐variant parameters. In this study, we conduct modeling experiments at the Rietholzbach lysimeter, Switzerland. All modeling experiments are benchmarked with HYDRUS‐1D. We compare our simulations to the measured hydrologic variables (percolation and evapotranspiration fluxes and soil water storage dynamics) and the measured water stable isotope signal (18O) in the lysimeter seepage for a period of ten years (1997–2007). An additional virtual bromide tracer experiment was used to benchmark the models. Additionally, we carried out a sensitivity analysis and provided Sobol indices for hydrologic model parameters and SAS parameters. Our results indicate that the advection‐dispersion transport model produces the best results. And thus, advective‐dispersive transport processes play a dominant role at Rietholzbach lysimeter. Our modeling approach provides the capability to test hypotheses of different transport mechanisms and to improve process understanding and predictions of transport processes. Overall, the combined model allows a very effective simulation of combined flux and transport processes at various temporal and spatial scales.

Functionally graded doubly-curved shell with temperature dependent material properties and surface-mounted MEE layers

In the present work, the geometrically nonlinear behavior of a simply supported functionally graded doubly curved shell with a surface-mounted magneto-electro-elastic layer is investigated based on Donnell’s theory. Kirchoff’s assumption and von Kármán’s strain-displacement relationship are used to obtain the governing differential equations. The through-the-thickness shell temperature is computed using Fourier’s heat conduction equation. Elastic material properties are assumed to be temperature-dependent and follow the power-law distribution function of the constituent ceramic-metal volume fractions. The magneto-electro-elastic layer is polarized along the thickness direction, which is subjected to electric and magnetic potentials. Gauss’s laws for electrostatics and magnetostatics are satisfied for magneto-electric materials. Nonlinear partial differential equations of motion are reduced to ordinary differential equations by introducing a stress function and using a single-term time-dependent assumed displacement function. Nonlinear free vibration response is obtained by direct numerical integration. After comparing the derived model with published literature results, numerical studies on the effects of applied electric and magnetic coupled fields of surface-mounted magneto-electro-elastic layers on shell vibration are reported. The nonlinear frequency ratios with respect to specified displacement for both spherical and cylindrical shells with thin Piezoelectric BaTiO 3 and Piezomagnetic CoF e 2 O 4 panels across material compositions, the radius of curvature, and thermal conduction are studied.

κmonty: a Monte Carlo Compton scattering code including non-thermal electrons

ABSTRACT Low-luminosity active galactic nuclei are strong sources of X-ray emission produced by Compton scattering originating from the accretion flows surrounding their supermassive black holes. The shape and energy of the resulting spectrum depend on the shape of the underlying electron distribution function (DF). In this work, we present an extended version of the grmonty code, called κmonty. The grmonty code previously only included a thermal Maxwell–Jütner electron DF. We extend the grmonty code with non-thermal electron DFs, namely the κ and power-law DFs, implement Cartesian Kerr–Schild coordinates, accelerate the code with mpi, and couple the code to the non-uniform adaptive mesh refinement grid data from the general relativistic magnetohydrodynamics code bhac. For the Compton scattering process, we derive two sampling kernels for both DFs. Finally, we present a series of code tests to verify the accuracy of our schemes. The implementation of non-thermal DFs opens the possibility of studying the effect of non-thermal emission on previously developed black hole accretion models.

The cluster initial mass function of the M82 disc super star clusters

ABSTRACT The presence of a population of a large number (∼400) of almost coeval (100–300 Myr) super star clusters (SSCs) in the disc of M82 offers an opportunity to construct the Cluster Initial Mass Function (CIMF) from the observed present-day Cluster Mass Function (CMF). We carry out the dynamical and photometric evolution of the CMF assuming that the clusters move in circular orbits under the gravitational potential of the host galaxy using the semi-analytical simulation code Evolve Me a Cluster of StarS. We explore power-law and lognormal functions for the CIMFs, and populate the clusters in the disc assuming uniform, power-law, and exponential radial distribution functions. We find that the observed CMF is best produced by a CIMF that is power law in form with an index of 1.8, for a power-law radial distribution function. More importantly, we establish that the observed turn-over in the present-day CMF is the result of observational incompleteness rather than due to dynamically induced effects, or an intrinsically lognormal CIMF, as was proposed for the fossil starburst region B of this galaxy. Our simulations naturally reproduce the mass–radius relation observed for a sub-sample of M82 SSCs.

A Network Clustering Algorithm for Protein Complex Detection Fused with Power-Law Distribution Characteristic

Network clustering for mining protein complexes from protein–protein interaction (PPI) networks has emerged as a prominent research area in data mining and bioinformatics. Accurately identifying complexes plays a crucial role in comprehending cellular organization and functionality. Network characteristics are often useful in enhancing the performance of protein complex detection methods. Many protein complex detection algorithms have been proposed, primarily focusing on local micro-topological structure metrics while overlooking the potential power-law distribution characteristic of community sizes at the macro global level. The effective use of this distribution characteristic information may be beneficial for mining protein complexes. This paper proposes a network clustering algorithm for protein complex detection fused with power-law distribution characteristic. The clustering algorithm constructs a cluster generation model based on scale-free power-law distribution to generate a cluster with a dense center and relatively sparse periphery. Following the cluster generation model, a candidate cluster is obtained. From a global perspective, the number distribution of clusters of varying sizes is taken into account. If the candidate cluster aligns with the constraints defined by the power-law distribution function of community sizes, it is designated as the final cluster; otherwise, it is discarded. To assess the prediction performance of the proposed algorithm, the gold standard complex sets CYC2008 and MIPS are employed as benchmarks. The algorithm is compared to DPClus, IPCA, SEGC, Core, SR-MCL, and ELF-DPC in terms of F-measure and Accuracy on several widely used protein–protein interaction networks. The experimental results show that the algorithm can effectively detect protein complexes and is superior to other comparative algorithms. This study further enriches the connection between analyzing complex network topology features and mining network function modules, thereby significantly contributing to the improvement of protein complex detection performance.

Spatial Intermittency of Particle Distribution in Relativistic Plasma Turbulence

Relativistic magnetically dominated turbulence is an efficient engine for particle acceleration in a collisionless plasma. Ultrarelativistic particles accelerated by interactions with turbulent fluctuations form nonthermal power-law distribution functions in the momentum (or energy) space, f(γ)d γ ∝ γ −α d γ, where γ is the Lorenz factor. We argue that in addition to exhibiting non-Gaussian distributions over energies, particles energized by relativistic turbulence also become highly intermittent in space. Based on particle-in-cell numerical simulations and phenomenological modeling, we propose that the bulk plasma density has lognormal statistics, while the density of the accelerated particles, n, has a power-law distribution function, . We argue that the scaling exponents are related as β ≈ α + 1, which is broadly consistent with numerical simulations. Non-space-filling, intermittent distributions of plasma density and energy fluctuations may have implications for plasma heating and for radiation produced by relativistic turbulence.

Constraints From Dwarf Galaxies on Black Hole Seeding and Growth Models With Current and Future Surveys

Dwarf galaxies are promising test beds for constraining models of supermassive and intermediate-mass black holes (MBHs) via their BH occupation fraction (BHOF). Disentangling seeding from the confounding effects of mass assembly over a Hubble time is a challenging problem that we tackle in this study with a suite of semianalytical models (SAMs). We show how the measured BHOF depends on the lowest BH mass or active galactic nucleus (AGN) luminosity achieved by a survey. To tell seeding models apart, we need to detect or model all AGNs brighter than 1037 erg s−1 in galaxies of M * ∼ 108−10 M ⊙. Shallower surveys, like eRASS, cannot distinguish between seed models even with the compensation of a much larger survey volume. We show that the AMUSE survey, with its inference of the MBH population underlying the observed AGNs, strongly favors heavy seed models, growing with either a power-law Eddington ratio distribution function or one in which BH accretion is tied to the star formation rate (i.e., the AGN-main sequence, AGN-MS, model). These two growth channels can then be distinguished by the AGN luminosity function at >1040 erg s−1, with the AGN-MS model requiring more accretion than observed at z ∼ 0. Thus, current X-ray observations favor heavy seeds whose Eddington ratios follow a power-law distribution. The different models also predict different radio scaling relations, which we quantify using the fundamental plane of BH activity. We close with recommendations for the design of upcoming multiwavelength campaigns that can optimally detect MBHs in dwarf galaxies.

Free Vibration Characteristics of Bidirectional Graded Porous Plates with Elastic Foundations Using 2D-DQM

This manuscript develops for the first time a mathematical formulation of the dynamical behavior of bi-directional functionally graded porous plates (BDFGPP) resting on a Winkler–Pasternak foundation using unified higher-order plate theories (UHOPT). The kinematic displacement fields are exploited to fulfill the null shear strain/stress at the bottom and top surfaces of the plate without needing a shear factor correction. The bi-directional gradation of materials is proposed in the axial (x-axis) and transverse (z-axis) directions according to the power-law distribution function. The cosine function is employed to define the distribution of porosity through the transverse z-direction. Equations of motion in terms of displacements and associated boundary conditions are derived in detail using Hamilton’s principle. The two-dimensional differential integral quadrature method (2D-DIQM) is employed to transform partial differential equations of motion into a system of algebraic equations. Parametric analysis is performed to illustrate the effect of kinematic shear relations, gradation indices, porosity type, elastic foundations, geometrical dimensions, and boundary conditions (BCs) on natural frequencies and mode shapes of BDFGPP. The effect of the porosity coefficient on the natural frequency is dependent on the porosity type. The natural frequency is dependent on the coupling of gradation indices, boundary conditions, and shear distribution functions. The proposed model can be used in designing BDFGPP used in nuclear, marine, aerospace, and civil structures based on their topology and natural frequency constraints.

Contact process on weighted planar stochastic lattice

We study the absorbing state phase transition in the contact process on the weighted planar stochastic (WPS) lattice. The WPS lattice is multifractal. Its dual network has a power-law degree distribution function and is also embedded in a bidimensional space. Moreover, it represents a novel way to introduce coordination disorder in lattice models. We investigated the critical behavior of the disordered system using extensive simulations. Our results show the critical behavior is distinct from that on a regular lattice, suggesting it belongs to a different universality class. We evaluate the exponent governing the bond fluctuations and our results agree with the Harris–Barghathi–Vojta criterium for relevant fluctuations.

Effects of nonlinear hygro-thermo-mechanical loading on the bending response of nanobeams using nonlocal strain gradient theory

This article analyzes the bending response of functionally graded nanobeams under hygro-thermo-mechanical loadings. The nonlocal strain gradient theory (NSGT) is implemented to consider both hardening and softening impacts on the system response. The nanobeam is constructed of functionally graded materials (FGMs), whose characteristics change smoothly across the thickness direction according to a power-law distribution function. The system is subjected to hygro-thermal environments according to the sinusoidally distributed loading. By employing Hamilton's principle, the static equations of the system are acquired. Static responses of the nanobeam are obtained with the help of the Galerkin technique. The effects of material distribution and different boundary conditions on the system deflection are examined. The impacts of several system parameters, such as the nonlocal parameter, length scale parameter, temperature gradient, and moisture variation, on the static responses of the functionally graded nanoscale beams are determined.

Research on the statistical damage constitutive model for composite solid propellant

The study of constitutive model of composite solid propellant is an important basis for analyzing the structural integrity of propellant grains of the solid rocket motor. From the view of microscopic, composite solid propellant is composed of a large number of solid particles with various sizes embedded in the matrix. In this paper, based on the distribution of solid particles with different sizes and the failure strength of the particle-matrix interfaces, the damage evolution was assumed to obey the Weibull distribution function and power law distribution function, and the two statistical damage constitutive models were first established. Then, the model parameters were identified and validated through the uniaxial tensile tests data of HTPB propellant. Finally, the prediction abilities of the developed two models were compared. The results show that both of the two statistical damage models can describe the stress-strain curves of HTPB propellant well.

A New Function for Frequency Analysis in the Risk Assessment of Offshore Oil-Related Platforms

_ Risk assessment of offshore oil-related platforms such as the floating production storage and offloading system (FPSO) is extremely important. In earthquakes prone to target sea areas, earthquakes should be considered as one of the important external forces that cause severe damage. In such risk assessments, for estimating the appearance frequency of seismic energy, the complementary cumulative distribution function (CCDF) of seismic energy of a considered area is usually based on the Gutenberg–Richter law. Furthermore, it is important to estimate the CCDF of the leak hole diameter when estimating the damage caused by an oil spill accident in an offshore oil-related platform. Some ship classification societies recommend using a formula based on the power law distribution as used in those processes. However, the power law distribution has a drawback in that, i.e., it is essentially impossible to accurately estimate the CCDF over the entire domain of the random variables. The authors have developed an enhanced power law distribution function that eliminates such defects in the power law distribution function and some practical methods for estimating its parameters. In this study, the developed function is applied to the CCDFs of the seismic energy and leak hole diameter. The results demonstrate that the function fits the existing data with sufficient accuracy. Therefore, the new function introduced here can be well used in the frequency analysis of the risk assessment of offshore oil-related platforms. Introduction In the risk assessment of offshore oil-related platforms like floating production storage and offloading system (FPSO), it is necessary to estimate probabilistic characteristics of external forces against them. Earthquakes are a major external force. Furthermore, it is necessary to estimate the damage when an external force is applied. The frequency of external forces, such as earthquakes, is low, but the damage is extensive. However, damage to offshore oil-related platforms is caused by not only these external forces, but also oil leaks resulting from some factors, such as operation errors and corrosion, that may cause accidents such as fire and explosion. Therefore, it is also essential to estimate the probabilistic characteristics of such damage for the risk assessment of offshore oil-related platforms.

Caputo definition for finding fractional moments of power law distribution functions

In this paper, we propose a modern technique to derive fractional moment by using Caputo definition of fractional derivative. Such technique represents a modification of fractional moments on a certain type of function which is known by the power law distribution functions. The results have been obtained show that the obtained fractional moments within such functions have closed form.

Turbulence and Particle Acceleration in a Relativistic Plasma

Abstract In a collisionless plasma, the energy distribution function of plasma particles can be strongly affected by turbulence. In particular, it can develop a nonthermal power-law tail at high energies. We argue that turbulence with initially relativistically strong magnetic perturbations (magnetization parameter σ ≫ 1) quickly evolves into a state with ultrarelativistic plasma temperature but mildly relativistic turbulent fluctuations. We present a phenomenological and numerical study suggesting that in this case, the exponent α in the power-law particle-energy distribution function, f(γ)d γ ∝ γ −α d γ, depends on magnetic compressibility of turbulence. Our analytic prediction for the scaling exponent α is in good agreement with the numerical results.

Характеристическая функция самоподобного случайного процесса

A stochastic differential equation is proposed for a characteristic function whose inverse function describes a self-similar random process with a power-law behavior of power spectra in a wide frequency range and a power-law amplitude distribution function. Gaussian “tails” for the characteristic distribution make it possible to evaluate its stability according to the formulas of classical statistics using the maximum of the Gibbs-Shannon entropy and, therefore, the stability of a random process given by an inverse function.

Characteristic function of a self-similar random process

A stochastic differential equation is proposed for a characteristic function whose inverse function describes a self-similar random process with a power-law behavior of power spectra in a wide frequency range and a power-law amplitude distribution function. Gaussian "tails" for the characteristic distribution make it possible to evaluate its stability according to the formulas of classical statistics using the maximum of the Gibbs-Shannon entropy and, therefore, the stability of a random process given by an inverse function. Keywords: self-similar random processes, stochastic equations, power spectrum, 1/f-noise, maximum entropy.

Finite Element Modeling for Static Bending Behaviors of Rotating FGM Porous Beams with Geometrical Imperfections Resting on Elastic Foundation and Subjected to Axial Compression

The static bending analysis of the FG porous beam resting on the two-parameter elastic foundation is initially carried out using a combination of Reddy’s high-order shear deformation theory and the finite element technique, where the initial geometrical imperfection and rotation movement in one fixed axis are calculated. Through the power-law distribution function with porosities, material characteristics vary constantly from one surface to the next in the direction of thickness, and the beam is concurrently impacted by an acting force perpendicular to the beam axis and an axial compressive force. The stiffness matrix of the beam element changes as a result, and the static bending response of this beam is significantly different from that of ordinary beams. Comparison cases with published findings are used to verify the computational theory. The calculations clearly reveal many innovations for rotating beams that are influenced by many different kinds of loads, which may be used to the designing, manufacturing, and usage of these structures in reality.

Secondary Energization in Compressing Plasmoids during Magnetic Reconnection

Abstract Plasmoids—magnetized quasi-circular structures formed self-consistently in reconnecting current sheets—were previously considered to be the graveyards of energetic particles. In this paper, we demonstrate the important role of plasmoids in shaping the particle energy spectrum in relativistic reconnection (i.e., with upstream magnetization σ up ≫ 1). Using 2D particle-in-cell simulations in pair plasmas with σ up = 10 and 100, we study a secondary particle energization process that takes place inside compressing plasmoids. We demonstrate that plasmoids grow in time, while their interiors compress, amplifying the internal magnetic field. The magnetic field felt by particles injected in an isolated plasmoid increases linearly with time, which leads to particle energization as a result of magnetic moment conservation. For particles injected with a power-law distribution function, this energization process acts in such a way that the shape of the injected power law is conserved, while producing an additional nonthermal tail f(E) ∝ E −3 at higher energies, followed by an exponential cutoff. The cutoff energy, which increases with time as , can greatly exceed σ up m e c 2. We analytically predict the secondary acceleration timescale and the shape of the emerging particle energy spectrum, which can be of major importance in certain astrophysical systems, such as blazar jets.