Inverse Power Law Research Articles

Fluctuations of Power Versus Energy for Random Fields Near a Perfectly Conducting Boundary

The standard deviations of the energy and Poynting power densities for an isotropic random field near a perfectly conducting planar boundary are characterized based on quartic plane-wave expansions. For normal and transverse components, different rates of decay exist as a function of electrical distance from the boundary. At large distances, the envelopes for the power are more strongly damped than for the energy, both showing inverse power law decay. The decay for the standard deviation is generally one order faster than that for the corresponding mean. For the normally directed power flux, its standard deviation near the boundary increases linearly with distance. The relative uncertainty of the scalar power is much smaller than that for the Poynting power. Poynting’s theorem for standard deviations is obtained and demonstrates larger standard deviations of the energy imbalance and power flux than their mean values.

Particle escape problem in semi-open system

<sec>We investigated the tunneling properties of a particle in a semi-open system. Starting initially from the eigenstate of the particle in the one-dimensional infinite well, we quench the infinitely high barrier on the right into a series of <inline-formula><tex-math id="M2">\begin{document}$\delta$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220450_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220450_M2.png"/></alternatives></inline-formula> barriers to observe the survival probability which is defined as the fidelity to the initial state. This constitutes a semi-Dirac comb model consisting of an infinitely high wall and multiple equally spaced <inline-formula><tex-math id="M3">\begin{document}$\delta$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220450_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220450_M3.png"/></alternatives></inline-formula>-potential barriers. We first solve the exact analytical solution of this model and obtain the closed analytic form of the eigen function expressed by a recursive relation. For a single barrier, multiple potential barriers, the disordered potential barriers, the closed-form expression of the survival probability i.e., the initial state fidelity, is given for any evolution time and it is used to reveal the mechanism of the particle escape process. The dependence of survival probability on the strength of barrier, number of barriers, and disorder strength is calculated numerically based on fast Fourier transform method. The relevant parameters are used to control and suppress the particle escape problem. We found that for a single <inline-formula><tex-math id="M4">\begin{document}$\delta$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220450_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220450_M4.png"/></alternatives></inline-formula>-potential barrier, the survival probability of the particle follows different trends in different decay time ranges. The particle in the ground state or excited states decays exponentially in a short time. After some time, the decay of the excited state will proceed with the same decay constant as that of the ground state. Finally, the survival probability follows a long-time inverse power law. The curve changes abruptly at different decay time intervals and is accompanied by significant oscillations. These oscillations in the transition region are caused by the interference of the exponential rate and the inverse power-law term, while the long-time non-exponential decay is due to the fact that the system energy spectrum has a lower bound. Increasing the barrier strength will greatly increase the probability of particles remaining in the well.</sec><sec>For multiple potential barriers, the reflection and transmission of particles between the potential barriers interfere with each other. When the strength of the potential barrier is small, the particle still decays exponentially. For a larger potential barrier strength, the probability of particle reflection increases, and the particles that tunnel out may be bounced back. The survival probability oscillates sharply, reaching higher fidelity at certain moments. The oscillatory maximum of the survival probability decreases linearly with the number of barriers, while the moment corresponding to the oscillatory maximum shows a parabolic increase with the number of barriers. The introduction of a series disordered barriers can significantly improve the survival probability and greatly suppress its oscillations over time.</sec><sec>Our calculation is expected to find applications in quantum control of particle escape problem in the disordered system.</sec>

Crucial Events and Biology

In the last years of the 20th century, the attention of physicists working in statistical physics moved from equilibrium processes characterized by stationary correlation functions and Poisson dynamics to biological processes exhibiting ergodicity breaking. The discovery of these processes raised a debate on whether basic properties such as the Onsager principle had to be abandoned or properly revisited. 1 The discovery of Levy processes led many researchers to replace the conventional central limit theorem with the generalized central limit theorem, responsible for a striking departure from the ordinary Gaussian statistics. The discovery of the processes of self-organization made the study of avalanches become very popular. 2 The observation of turbulent processes led to the discovery of new waiting time distribution densities, characterized by inverse power laws 3 and a new stochastic central limit theorem was invented to explain the emergence of Mittag-Leffler function, which is now widely used for the foundation of fractional derivatives. 4 The traditional Linear Response Theory of Kubo was replaced by a new form of linear response, compatible with the ergodic breakdown of complex systems, and this new form of linear response was used for the foundation of Complexity Matching (CM). 5 I plan to prove that crucial events are responsible for ergodicity breaking and that the CM phenomenon is a manifestation of crucial events. One problem still open in this field of research is the origin of [Formula: see text] noise that is traditionally interpreted as a manifestation of the Mandelbrot Fractional Brownian Motion (FBM). 6 I plan to show that the [Formula: see text] noise proposed in 5 for the foundation of the CM phenomenon has a completely different nature, involving crucial events rather than the FBM infinite memory. A recent result of my research group 7 proved that the progress of autonomic neuropathy makes the heartbeats of healthy individuals, dominated by crucial events, turn into FBM. Quite surprisingly, the same phenomenon of transition from the crucial event to the FBM regime was observed in the germination process of lentils 8 in the absence of light. The transition from Levy to Gauss statistics is supposed to be generated by environmental fluctuations and I will illustrate the experimental and theoretical research works that will shed light into their nature.

New Probability Distributions in Astrophysics: VII. The Truncated Gamma-Pareto Distribution Applied to Cosmic Rays

Many astrophysical phenomena are modeled by an inverse power law distribution at high values of the random variable but often at low values of the random variable we have a departure from an inverse power law. In order to insert a continuous transition from low to high values of the random variable we analyse the truncated gamma-Pareto distribution in two versions by deriving the most important statistical parameters. The application of the results to the distribution in energy of cosmic rays allows deriving an analytical expression for the average energy, which is 2.6 GeV.

Observational constraints and predictions of the interacting dark sector with field-fluid mapping

We consider an interacting field theory model that describes the interaction between dark energy-dark matter interaction. Only for a specific interaction term, this interacting field theory description has an equivalent interacting fluid description. For inverse power law potentials and linear interaction function, we show that the interacting dark sector model with field-fluid mapping is consistent with four cosmological data sets — Hubble parameter measurements (Hz), Baryonic Acoustic Oscillation data (BAO), Supernova Type Ia data (SN), and High redshift HII galaxy measurements (HIIG). More specifically, these data sets prefer a negative value of interaction strength in the dark sector and lead to consistent best-fit values of Hubble constant and other cosmological parameters. Having established that this interacting field theory model is consistent with cosmological observations, we obtain quantifying tools to distinguish between the interacting and non-interacting dark sector scenarios. We focus on the variation of the scalar metric perturbed quantities as a function of redshift related to structure formation, weak gravitational lensing, and the integrated Sachs-Wolfe effect. We show that the difference in the evolution becomes significant for z < 20, for all length scales, and the difference peaks at smaller redshift values z < 5. We then discuss the implications of our results for the upcoming missions.

Accelerated tribological testing based life prediction of aluminium-graphite tribo composite

Electrical sliding contacts in electrical motors and generators are subjected to severe friction and wear conditions. Conventional materials such as carbon and electrographite do not have longer life due to the generation of carbon dust. This creates environmental pollution and also affects the operation of an electrical machine. Aluminium-graphite composites are well suited for such applications. However, the life of the components made out of aluminum-graphite composite is to be evaluated in order to optimize their performance at the design stage itself. Traditional life data analysis at normal usage conditions is time-consuming. In the present study, aluminum-graphite composites were manufactured through the energy-efficient microwave sintering, and the life of components was estimated using Accelerated Life Testing (ALT) procedure. The experimental wear data were analyzed using an inverse power law model. Reliability and failure rate models were developed using the time-to-failure data. Life at normal usage conditions was predicted using these models. The calculated reliability at normal usage conditions was more than five years at a 99% confidence limit.

Weierstrass function methodology for uncertainty analysis of random media criticality with spectrum range control

Randomized Weierstrass function (RWF) has been under development for evaluating the uncertainty of random media criticality due to the material mixture in disorder. In this work, the modelling capability of RWF is refined so that the spectrum range can be controlled by specifying its lower and upper ends of the frequency domain variable. As a result, it becomes possible to make fair criticality comparison among replicas of random media under inverse power law power spectra. Technically, the infinite sum of trigonometric terms in RWF is formally extended to cover the arbitrarily low frequency domain and then truncated to finite terms for the sole purpose of spectrum range control. This means that the refined RWF is free of the issue of convergence towards a reference fractal characteristic of Weierstrass function and thus termed Incomplete Randomized Weierstrass function (IRWF). As a demonstration, a three-dimensional version of IRWF is applied to the mixture of three fuels with different burnups in a sufficiently water-moderated environment. Monte Carlo criticality calculations are carried out to evaluate the uncertainty of neutron effective multiplication factor (keff) due to the indeterminacy of mixture formation.

Hypothetical Control of Fatal Quarrel Variability.

Wars, terrorist attacks, as well as natural catastrophes typically result in a large number of casualties, whose distributions have been shown to belong to the class of Pareto’s inverse power laws (IPLs). The number of deaths resulting from terrorist attacks are herein fit by a double Pareto probability density function (PDF). We use the fractional probability calculus to frame our arguments and to parameterize a hypothetical control process to temper a Lévy process through a collective-induced potential. Thus, the PDF is shown to be a consequence of the complexity of the underlying social network. The analytic steady-state solution to the fractional Fokker-Planck equation (FFPE) is fit to a forty-year fatal quarrel (FQ) dataset.

Small-sample limit of the Bennett acceptance ratio method and the variationally derived intermediates.

Free energy calculations based on atomistic Hamiltonians provide microscopic insight into the thermodynamic driving forces of biophysical or condensed matter systems. Many approaches use intermediate Hamiltonians interpolating between the two states for which the free energy difference is calculated. The Bennett acceptance ratio (BAR) and variationally derived intermediates (VI) methods are optimal estimator and intermediate states in that the mean-squared error of free energy calculations based on independent sampling is minimized. However, BAR and VI have been derived based on several approximations that do not hold for very few sample points. Analyzing one-dimensional test systems, we show that in such cases BAR and VI are suboptimal and that established uncertainty estimates are inaccurate. Whereas for VI to become optimal, less than seven samples per state suffice in all cases; for BAR the required number increases unboundedly with decreasing configuration space densities overlap of the end states. We show that for BAR, the required number of samples is related to the overlap through an inverse power law. Because this relation seems to hold universally and almost independent of other system properties, these findings can guide the proper choice of estimators for free energy calculations.

Long Wavelength Thermal Density Fluctuations in Molecular and Polymer Glass-Forming Liquids: Experimental and Theoretical Analysis under Isobaric Conditions.

We establish via an in-depth analysis of experimental data that the dimensionless compressibility (proportional to the dimensionless amplitude of long wavelength thermal density fluctuations) of one-component normal and supercooled liquids of chemically complex nonpolar and weakly polar molecules and polymers follows extremely well a surprisingly simple and general temperature dependence over an exceptionally wide range of pressures and temperatures. A theoretical basis for this behavior is shown to exist in the venerable van der Waals model and its more modern interpretations. Although associated hydrogen-bonding (and to a lesser degree strongly polar) liquids display modestly more complex behavior, rather simple temperature and pressure dependences are also discovered. A new approach to collapse the temperature- and pressure-dependent dimensionless compressibility data onto a master curve is formulated that differs from the empirical thermodynamic scaling approach. As a practical matter, we also find that the dimensionless compressibility scales well as an inverse power law with temperature with an exponent that is system dependent and decreases with pressure. At very high pressures and low temperatures, the thermal liquid behavior appears to approach (but not reach) a repulsion-dominated random close packing limit. All these findings are relevant to our recent theoretical work on the problem of activated relaxation and vitrification of supercooled molecular and polymeric liquids.

Nonlinear dynamics of a self-mixing thin-slice solid-state laser subjected to Doppler-shifted optical feedback.

Chaotic oscillations of a linearly polarized single longitudinal-mode thin-slice Nd:GdVO_{4} laser placed in a self-mixing laser Doppler velocity scheme were dynamically characterized in terms of the intensity probability distribution, joint time-frequency analysis, and short-term Fourier transformation of temporal evolutions, and the degree of disorder in the amplitude and phase of the long-term temporal evolutions. The transition from chaotic relaxation oscillations (ROs) to chaotic spiking oscillations (SOs) was explored via the chaotic itinerancy (CI) regime by increasing the feedback ratio toward the laser from a rotating scattering object. The intensity probability distribution was found to change from an exponential decay in the RO regime to an inverse power law in the SO regime, which manifests itself in self-organized critical behavior, while stochastic subharmonic frequency locking among the two periodicities of RO and SO takes place in the CI regime featuring quantum-noise (spontaneous-emission)-induced order in the amplitude and phase of the spiking oscillations. All of the experimental results were reproduced by numerical simulations of a model equation of a single-mode self-mixing solid-state laser subjected to Doppler-shifted optical feedback from a rotating scattering object.

Burstiness and Stochasticity in the Malleability of Physical Activity.

This study examined whether patterns of self-organization in physical activity (PA) predicted long-term success in a yearlong PA intervention. Increased moderate to vigorous PA (MVPA) was targeted in insufficiently active adults (N = 512) via goal setting and financial reinforcement. The degree to which inverse power law distributions, which are reflective of self-organization, summarized (a)daily MVPA and (b)time elapsed between meeting daily goals (goal attainment interresponse times) was calculated. Goal attainment interresponse times were also used to calculate burstiness, the degree to which meeting daily goals clustered in time. Inverse power laws accurately summarized interresponse times, but not daily MVPA. For participants with higher levels of MVPA early in the study, burstiness in reaching goals was associated with long-term resistance to intervention, while stochasticity in meeting goals predicted receptiveness to intervention. These results suggest that burstiness may measure self-organizing resistance to change, while PA stochasticity could be a precondition for behavioral malleability.

The inverse power law.

October’s BJGP theme is the young and the old. Much like general practice itself then. Three of Shakespeare’s Seven Ages of Man 1 make up the bulk of medical practice. For inpatient costs the figures for older care are even more marked — while the annual mean secondary care spend for a 50 year old is just over £300, this rises to £2000 by the age of 90.2 And we are well aware of the social determinants of health, influencing which of these young and old will need most care.3 So given that we all sign up to motherhood, apple pie, and patient-centred medicine, presumably parents, older children, and older people themselves have the greatest input into NHS health policy? Especially those on low incomes? But no, most …

감마과정 기반 양방향 접합 트랜지스터 잔여수명 예측

Purpose: The gamma process was used to predict the remaining useful life of a bipolar junction transistor, which is a widely used electronic component.<BR>Methods: The proposed method consists of six major steps: 1) data collection via accelerated degradation test, 2) data conversion for the gamma process, 3) parameter estimation of the gamma process, 4) shape function estimation using a fitting procedure, 5) degradation model generation using the inverse power law and Arrhenius relation, and 6) prediction of the remaining useful life by estimating the failure criterion.<BR>Results: The degradation characteristic of a bipolar junction transistor is the beta (β) value (current gain), and the transistor is considered a failure when the β value deteriorates by 33.3%.<BR>Conclusion: This paper proposes a new method for predicting the remaining life of a bipolar junction transistor.

The impact of training sample size on deep learning-based organ auto-segmentation for head-and-neck patients

To investigate the impact of training sample size on the performance of deep learning-based organ auto-segmentation for head-and-neck cancer patients, a total of 1160 patients with head-and-neck cancer who received radiotherapy were enrolled in this study. Patient planning CT images and regions of interest (ROIs) delineation, including the brainstem, spinal cord, eyes, lenses, optic nerves, temporal lobes, parotids, larynx and body, were collected. An evaluation dataset with 200 patients were randomly selected and combined with Dice similarity index to evaluate the model performances. Eleven training datasets with different sample sizes were randomly selected from the remaining 960 patients to form auto-segmentation models. All models used the same data augmentation methods, network structures and training hyperparameters. A performance estimation model of the training sample size based on the inverse power law function was established. Different performance change patterns were found for different organs. Six organs had the best performance with 800 training samples and others achieved their best performance with 600 training samples or 400 samples. The benefit of increasing the size of the training dataset gradually decreased. Compared to the best performance, optic nerves and lenses reached 95% of their best effect at 200, and the other organs reached 95% of their best effect at 40. For the fitting effect of the inverse power law function, the fitted root mean square errors of all ROIs were less than 0.03 (left eye: 0.024, others: <0.01), and the R square of all ROIs except for the body was greater than 0.5. The sample size has a significant impact on the performance of deep learning-based auto-segmentation. The relationship between sample size and performance depends on the inherent characteristics of the organ. In some cases, relatively small samples can achieve satisfactory performance.

The investigation of stability of brane induced gravity model in the ‎presence of a scalar field with various kinds of potential‎

In this manuscript we consider the normal branch of DGP brane cosmological model in the presence of a ‎scalar field on the brane as the dark energy component. Using dynamical system approach we investigate ‎the stability properties of the system. We will see that one of the defined new variables lambda, that ‎depends on the potential function of the scalar field has a significant role in the evolution of the universe. ‎Therefore, we divide our discussion into two parts, the constant and the varying lambda. With a good ‎approximation we consider all the obtained critical points in the case of the constant lambda, as the ‎instantaneous critical points for the case of a varying lambda. We investigate two special scalar field ‎potential function: a double exponential potential and an inverse power law potential. In both of the cases we ‎study the evolution of the system and discuss the fate of the universe. We find that the inverse power law ‎potential function is in a good agreement with the current universe, but the double exponential potential ‎function can predict different future for the universe.‎

Modified Damage Equalization Method for Lifetime Estimation of Dielectrics

This paper presents a modified Damage Equalization Method with statistical features for life estimation of dielectrics. The paper postulates damage as a random variable, which resulted in establishing a relationship between the Weibull scale, shape parameters and the endurance coefficient `n' of inverse power law. By the modified DEM (m-DEM), for the first time, a range of values of n for each material is proposed, which is useful for design engineers. Until now, the endurance coefficient of inverse power law and strength constant were treated as constants. Here, it is shown that these coefficients do have statistical features or variations. Experiments are conducted on oil impregnated paper and LDPE samples using accelerated step-stresses for life estimation. The results of the estimation of endurance coefficient are compared with existing methods and reasonable conclusions are drawn in favor of the proposed method.

The Euler disk and its dynamic finite time singularity: Investigating a fascinating acoustical and mechanical phenomenon with simple means

Many have spun a coin or other disk on a table, observing — and listening! — that — while its motion comes to rest, the angular velocity of the contact point and the frequency of the emitted sound raise in a striking way. This dynamical system is known as the "Euler disk," and its physics is theoretically understood as a "finite time singularity" in form of an inverse power law. Various dissipation mechanisms lead to different exponents in the power law. The present investigation is about different configurations of the Euler disk using acoustical and mechanical data. We present results for the well-known Euler disk toy, well in accord with a model of viscous air dissipation. This is the first measurement using the acoustical signal for the dynamics of the Euler disk. Moreover, we present new cases where other dissipation mechanism must be dominant. Experiments are carried out with simple means and using a smartphone for recording and analysing data. This and interesting physics of the Euler disc make it a good candidate e.g. for undergraduate lab-work courses or research projects, which can spark students interest in acoustics, and more generally in applying physics to phenomena of everyday life.

Predicting the Relaxation Modulus for the Study of the Delayed Behaviour of Kenaf Fibres in Stress Relaxation

Plant fibres (PFs) are preferred reinforcements of bio-composites. Knowledge of their lifespan requires a study of their viscoelastic behaviour. In this paper, a stress relaxation analysis of kenaf fibres was performed at a constant rate of deformation at room temperature. A method for extracting the relaxation modulus in the deferred zone was proposed. This method was compared, using simulation, with the Zapas-Phillips method and experimental data via three predictive models: the stretched exponential function or KWW, the inverse power law of Nutting and the prony series. The results indicate that the relaxation modulus obtained by the method proposed is in good agreement with the experimental modulus. In addition, the estimated error is of the same order of magnitude as in the case of the Zapas-Phillips method. The parameters estimated from the KWW function (β = 0.4) and prony series model showed an important contribution in the study of the delayed response of kenaf fibres. These results can have a significant impact on the use of kenaf fibres in midterm and long-term loading applications.

Does FLASH deplete oxygen? Experimental evaluation for photons, protons, and carbon ions.

To investigate experimentally, if FLASH irradiation depletes oxygen within water for different radiation types such as photons, protons, and carbon ions. This study presents measurements of the oxygen consumption in sealed, 3D-printed water phantoms during irradiation with x-rays, protons, and carbon ions at varying dose rates up to 340Gy/s. The oxygen measurement was performed using an optical sensor allowing for noninvasive measurements. Oxygen consumption in water only depends on dose, dose rate, and linear energy transfer (LET) of the irradiation. The total amount of oxygen depleted per 10Gy was found to be 0.04% atm - 0.18% atm for 225kV photons, 0.04% atm - 0.25% atm for 224MeV protons, and 0.09% atm - 0.17% atm for carbon ions. Consumption depends on dose rate by an inverse power law and saturates for higher dose rates because of self-interactions of radicals. Higher dose rates yield lower oxygen consumption. No total depletion of oxygen was found for clinical doses. FLASH irradiation does consume oxygen, but not enough to deplete all the oxygen present. For higher dose rates, less oxygen was consumed than at standard radiotherapy dose rates. No total depletion was found for any of the analyzed radiation types for 10Gy dose delivery using FLASH.