Asymptotic Power Law Research Articles

Connections between kinks with different asymptotics

In this letter, we show how to build bridges between field-theoretic models that have kink solutions with different asymptotic behavior. We study transformational properties of kinks in models with a real scalar field in two-dimensional space-time. We show how to obtain new models with solitons having different asymptotic behavior, using deformations with specific functions. In particular, starting from the well-known model, we obtain kinks with super-exponential, super-super-exponential, power-law, and logarithmic asymptotics. We also comment on how asymptotic behavior of stability potential and zero mode can be obtained for deformed kinks.

Multibody interactions between protein inclusions in the pointlike curvature model for tense and tensionless membranes.

The pointlike curvature constraint (PCC) model and the disk detachment angle (DDA) model for the deformation-mediated interaction of conical integral protein inclusions in biomembranes are compared in the small deformation regime. Given the radius of membrane proteins, which is comparable to the membrane thickness, it is not obvious which of the two models should be considered the most adequate. For two proteins in a tensionless membranes, the PCC and DDA models coincide at the leading-order in their separation but differ at the next order. Yet, for distances larger than twice the proteins diameter, the difference is less than . Like the DDA model, the PCC model includes all multibody interactions in a non-approximate way. The asymptotic many-body energy of triangular and square protein clusters is exactly the same in both models. Pentagonal clusters, however, behave differently; they have a vanishing energy in the PCC model, while they have a non-vanishing weaker asymptotic power law in the DDA model. We quantify the importance of multibody interactions in small polygonal clusters of three, four and five inclusions with identical or opposite curvatures in tensionless or tense membranes. We find that the pairwise approximation is almost always very poor. At short separation, the three-body interaction is not sufficient to account for the full many-body interaction. This is confirmed by equilibrium Monte Carlo simulations of up to ten inclusions.

Discovery of a Long Thermonuclear X-Ray Burst from the Ultracompact Binary 4U 1850–087

We report the detection of a long X-ray burst from the ultracompact binary 4U 1850–087. The type I X-ray burst was observed on MJD 60171.65 by the Monitor of All-sky X-ray Image and Neutron Star Interior Composition Explorer (NICER). We analyze the NICER data between MJD 60095.19 and 60177.43, which includes the observation covering parts of the long X-ray burst decay phase, i.e., 0.15–3.8 hr after the trigger. The persistent spectra are well described by a multicolor disk blackbody, with an inner temperature of ∼0.5 keV, a thermally Comptonized continuum, an asymptotic power-law photon index of Γ ∼ 2.2, and an electron temperature of kT e ∼ 20–30 keV. The mean persistent flux was around 3.8 × 10−10 erg cm−2 s−1, corresponding to a local mass accretion rate of ∼1%ṁEdd . Part of the time-resolved burst spectra show a clear deviation from the blackbody model, which can be improved by considering the enhanced persistent emission due to the Poynting–Robertson drag or the reflected disk emission illuminated by the burst. From the burst flux during the cooling phase, we estimate the burst duration, τ ≈ 0.78 hr, the burst fluence, E b ≈ 4.1 × 1041 erg, and the ignition column depth, y ign ≈ 3.5 × 1010 g cm−2. These long X-ray burst parameters from 4U 1850–087 suggest a regime of unstable burning of a thick, pure helium layer slowly accreted from a helium donor star. Moreover, we identify 7σ significant ∼1 keV emission lines in the burst spectra, which may originate from the surrounding accretion disk.

Phase-space entropy cascade and irreversibility of stochastic heating in nearly collisionless plasma turbulence.

We consider a nearly collisionless plasma consisting of a species of "test particles" in one spatial and one velocity dimension, stirred by an externally imposed stochastic electric field-a kinetic analog of the Kraichnan model of passive advection. The mean effect on the particle distribution function is turbulent diffusion in velocity space-known as stochastic heating. Accompanying this heating is the generation of fine-scale structure in the distribution function, which we characterize with the collisionless (Casimir) invariant C_{2}∝∫∫dxdv〈f^{2}〉-a quantity that here plays the role of (negative) entropy of the distribution function. We find that C_{2} is transferred from large scales to small scales in both position and velocity space via a phase-space cascade enabled by both particle streaming and nonlinear interactions between particles and the stochastic electric field. We compute the steady-state fluxes and spectrum of C_{2} in Fourier space, with k and s denoting spatial and velocity wave numbers, respectively. In our model, the nonlinearity in the evolution equationfor the spectrum turns into a fractional Laplacian operator in k space, leading to anomalous diffusion. Whereas even the linear phase mixing alone would lead to a constant flux of C_{2} to high s (towards the collisional dissipation range) at every k, the nonlinearity accelerates this cascade by intertwining velocity and position space so that the flux of C_{2} is to both high k and high s simultaneously. Integrating over velocity (spatial) wave numbers, the k-space (s-space) flux of C_{2} is constant down to a dissipation length (velocity) scale that tends to zero as the collision frequency does, even though the rate of collisional dissipation remains finite. The resulting spectrum in the inertial range is a self-similar function in the (k,s) plane, with power-law asymptotics at large k and s. Our model is fully analytically solvable, but the asymptotic scalings of the spectrum can also be found via a simple phenomenological theory whose key assumption is that the cascade is governed by a "critical balance" in phase space between the linear and nonlinear timescales. We argue that stochastic heating is made irreversible by this entropy cascade and that, while collisional dissipation accessed via phase mixing occurs only at small spatial scales rather than at every scale as it would in a linear system, the cascade makes phase mixing even more effective overall in the nonlinear regime than in the linear one.

Near-critical behavior of the Zhong-Zhang model.

The Zhong-Zhang (ZZ) model is a one-degree-of-freedom dynamical system describing the motion of an insulating plate of length d floating on the upper surface of a convecting fluid, with locking at the boundaries. In the absence of noise, the system away from the boundaries is described by linear differential equationswith a delay time τ. The d,τ plane consists of two domains separated by a critical curve. For asymptotically long times, subcritical orbits approach a nontrivial periodic attractor, while the supercritical ones tend to a stationary state at the origin. We investigate near-critical behavior using a modified fourth-order Runge-Kutta integration scheme. We then construct a piecewise analytic decomposition of the periodic attractor, which makes possible a far higher level of accuracy. Our results provide solid evidence for an asymptotic power-law approach to criticality of several observables. The power laws are fed back to determine the piecewise-analytic structure deep into the near-critical regime. In an Appendix, we explore the effect of introducing noise using modified order-3/2 Kloeden-Platen-Schurz stochastic integration, following several observable quantities through the near-critical parameter domain.

Generalized Pareto distribution and income inequality: an extension of Gibrat's law

<abstract> <p>Motivated by empirical observations, we proposed a possible extension of Gibrat's law. By applying it into the random growth theory of income distribution, we found that the income distribution is described by a generalized Pareto distribution (GPD) with three parameters. We observed that there is a parameter $ \eta $ in the GPD that plays a key role in determining the shape of income distribution. By using the Kolmogorov-Smirnov test, we empirically showed that, for typical market-economy countries, $ \eta $ is significantly close to 0, indicating that the income distribution is characterized by a two-class pattern: The bottom 90% of the population is approximated by an exponential distribution, while the richest 1%~3% is approximated by an asymptotic power law. However, we empirically found that in China during the period of the planned economy and the early stages of market reform (from 1978 to 1990), $ \eta $ deviated significantly from 0, indicating that the bottom of the population no longer conformed to an exponential distribution.</p> </abstract>

Optimized Self-Similar Borel Summation

The method of Fractional Borel Summation is suggested in conjunction with self-similar factor approximants. The method used for extrapolating asymptotic expansions at small variables to large variables, including the variables tending to infinity, is described. The method is based on the combination of optimized perturbation theory, self-similar approximation theory, and Borel-type transformations. General Borel Fractional transformation of the original series is employed. The transformed series is resummed in order to adhere to the asymptotic power laws. The starting point is the formulation of dynamics in the approximations space by employing the notion of self-similarity. The flow in the approximation space is controlled, and “deep” control is incorporated into the definitions of the self-similar approximants. The class of self-similar approximations, satisfying, by design, the power law behavior, such as the use of self-similar factor approximants, is chosen for the reasons of transparency, explicitness, and convenience. A detailed comparison of different methods is performed on a rather large set of examples, employing self-similar factor approximants, self-similar iterated root approximants, as well as the approximation technique of self-similarly modified Padé–Borel approximations.

Signature of functional enzyme dynamics in quasielastic neutron scattering spectra: The case of phosphoglycerate kinase.

We present an analysis of high-resolution quasi-elastic neutron scattering spectra of phosphoglycerate kinase which elucidates the influence of the enzymatic activity on the dynamics of the protein. We show that in the active state the inter-domain motions are amplified and the intra-domain asymptotic power-law relaxation ∝t-α is accelerated, with a reduced coefficient α. Employing an energy landscape picture of protein dynamics, this observation can be translated into a widening of the distribution of energy barriers separating conformational substates of the protein.

The asymptotic tails of limit distributions of continuous-time Markov chains

AbstractThis paper investigates tail asymptotics of stationary distributions and quasi-stationary distributions (QSDs) of continuous-time Markov chains on subsets of the non-negative integers. Based on the so-called flux-balance equation, we establish identities for stationary measures and QSDs, which we use to derive tail asymptotics. In particular, for continuous-time Markov chains with asymptotic power law transition rates, tail asymptotics for stationary distributions and QSDs are classified into three types using three easily computable parameters: (i) super-exponential distributions, (ii) exponential-tailed distributions, and (iii) sub-exponential distributions. Our approach to establish tail asymptotics of stationary distributions is different from the classical semimartingale approach, and we do not impose ergodicity or moment bound conditions. In particular, the results also hold for explosive Markov chains, for which multiple stationary distributions may exist. Furthermore, our results on tail asymptotics of QSDs seem new. We apply our results to biochemical reaction networks, a general single-cell stochastic gene expression model, an extended class of branching processes, and stochastic population processes with bursty reproduction, none of which are birth–death processes. Our approach, together with the identities, easily extends to discrete-time Markov chains.

Informational assessment of large scale self-similarity in nonlinear random field models

Large-scale behavior of a wide class of spatial and spatiotemporal processes is characterized in terms of informational measures. Specifically, subordinated random fields defined by nonlinear transformations on the family of homogeneous and isotropic Lancaster–Sarmanov random fields are studied under long-range dependence (LRD) assumptions. In the spatial case, it is shown that Shannon mutual information between random field components for infinitely increasing distance, which can be properly interpreted as a measure of large scale structural complexity and diversity, has an asymptotic power law decay that depends on the underlying LRD parameter scaled by the subordinating function rank. Sensitivity with respect to distortion induced by the deformation parameter under the generalized form given by divergence-based Rényi mutual information is also analyzed. In the spatiotemporal framework, a spatial infinite-dimensional random field approach is adopted. The study of the large-scale asymptotic behavior is then extended under the proposal of a functional formulation of the Lancaster–Sarmanov random field class, as well as of divergence-based mutual information. Results are illustrated, in the context of geometrical analysis of sample paths, considering some scenarios based on Gaussian and Chi-Square subordinated spatial and spatiotemporal random fields.

Modified DFA for a robust discrimination between short-term and long-range correlations in short time series

Since the standard detrended fluctuation analysis (s-DFA) method depends on asymptotic power-law scaling assumption, it has been questioned on its ability in characterizing the correlation structure at the small scales, let alone discriminating the correlation structures of short time series. A modified DFA (m-DFA) with a novel defined fluctuation function is employed in this study to discriminate and characterize the distinct (short-termed or long-ranged) correlation structures in short time series. Detailed results show that the m-DFA is able not only to distinguish the short-termed, or long-ranged correlation, but also to well quantify the correlation strengths in both output from classical models and real-world series (for example, DTR variability over China) of data length as short as 2000. Moreover, m-DFA can work effectively in time series with strong correlation of even shorter length as 500. The correlation structures inferred by m-DFA in short intervals contribute greatly to better understanding of local and evolutionary correlation features of an underlying process, not limited to only global one from s-DFA.

Time-Asymptotic Expansion with Pointwise Remainder Estimates for 1D Viscous Compressible Flow

We construct a time-asymptotic expansion with pointwise remainder estimates for solutions to 1D compressible Navier–Stokes equations. The leading-order term is the well-known diffusion wave and the higher-order terms are a newly introduced family of waves which we call higher-order diffusion waves. In particular, these provide an accurate description of the power-law asymptotics of the solution around the origin x=0, where the diffusion wave decays exponentially. The expansion is valid locally and also globally in the L^p({mathbb {R}})-norm for all 1le ple infty . The proof is based on pointwise estimates of Green’s function.

Anomalous Transport of Energetic Particles Interacting with Dynamic Small-Scale Magnetic Flux Rope Structures

le Roux and Zank [25] showed previously how one can derive from first principles a pitch-angle dependent fractional diffusion-advection kinetic equation to model the anomalous diffusion of energetic particles interacting with small-scale magnetic flux ropes (SMFRs) in the inner heliosphere on the basis of the standard focused transport equation. This equation has the following limitations: (1) The asymptotic power law of a Lévy distribution was specified to model the non-Gaussian statistics of the disturbed energetic particle trajectories generated during energetic particle interaction with numerous SMFRs. The second moment (variance) and higher moments of the Lévy distribution are infinite, indicating over-efficient non-local transport that is scale-free. (2) The theory does not naturally allow for a transition of anomalous transport to normal diffusion, or to a different anomalous diffusion state. An outline of a derivation is presented in which an exponentially truncated Lévy distribution was specified instead, resulting in a tempered fractional diffusion-advection kinetic equation that addresses these two concerns.

Asymptotic Relations in Applied Models of Inhomogeneous Poisson Point Flows

A model of a particle flow forming a copy of some image and the distance between the copy and the image are estimated using a special probability metric. The ability of the flow of balls to cover the surface, when grinding the balls, was investigated using formulas of stochastic geometry. Reconstruction of characteristics of an inhomogeneous Poisson flow by inaccurate observations is analysed using the Poisson flow point colouring theorem. The dependence of the Poisson parameter of the distribution of the number of customers in a queuing system with an infinite number of servers and a deterministic service time on the peak load created by an inhomogeneous input Poisson flow is estimated. All these models consist of an inhomogeneous Poisson flow of points and marks glued to each point of the flow and are characterised by their mass, area, volume, observability (or non-observability), and service time. The presence of an asymptotic power–law relationship between model objective functions and parameters of mark crushing is established. These results may be applied in nanotechnology, powder metallurgy, ecology, and consumer services in the implementation of the “Smart City” program. The proposed approach is phenomenological in nature and is justified by the results of real observations and experiments.

Filaments in the OMC-3 cloud and uncertainties in estimates of filament profiles

Context.Filamentary structures are an important part of star-forming interstellar clouds. The properties of filaments hold clues to their formation mechanisms and their role in the star-formation process.Aims.We compare the properties of filaments in the Orion Molecular Cloud 3 (OMC-3), as seen in mid-infrared (MIR) absorption and far-infrared (FIR) dust emission. We also wish to characterise some potential sources of systematic errors in filament studies.Methods.We calculated optical depth maps of the OMC-3 filaments based on the MIR absorption seen inSpitzerdata and FIR dust emission observed withBerscheland the ArTéMiS instrument. We then compared the filament properties extracted from the data. Potential sources of error were investigated more generally with the help of radiative transfer models.Results.The widths of the selected OMC-3 filament segments are in the range 0.03–0.1 pc, with similar average values seen in both MIR and FIR analyses. Compared to the widths, the individual parameters of the fitted Plummer functions are much more uncertain. The asymptotic power-law index has typically valuesp ~3 but with a large scatter. Modelling shows that the FIR observations can systematically overestimate the filament widths. The effect is potentially tens of per cent at column densities aboveN(H2) ~ 1022cm–2but is reduced in more intense radiation fields, such as the Orion region. Spatial variations in dust properties could cause errors of similar magnitude. In the MIR analysis, dust scattering should generally not be a significant factor, unless there are high-mass stars nearby or the dust MIR scattering efficiency is higher than in the tested dust models. Thermal MIR dust emission can be a more significant source of error, especially close to embedded sources.Conclusions.The analysis of interstellar filaments can be affected by several sources of systematic error, but mainly at high column densities and, in the case of FIR observations, in weak radiation fields. The widths of the OMC-3 filaments were consistent between the MIR and FIR analyses and did not reveal any systematic dependence on the angular resolution of the observations.

Precision Measurement of the Return Distribution Property of the Chinese Stock Market Index.

In econophysics, the analysis of the return distribution of a financial asset using statistical physics methods is a long-standing and important issue. This paper systematically conducts an analysis of composite index 1 min datasets over a 17-year period (2005−2021) for both the Shanghai and Shenzhen stock exchanges. To reveal the differences between Chinese and mature stock markets, we precisely measure the property of the return distribution of the composite index over the time scale Δt, which ranges from 1 min to almost 4000 min. The main findings are as follows: (1) The return distribution presents a leptokurtic, fat-tailed, and almost symmetrical shape that is similar to that of mature markets. (2) The central part of the return distribution is described by the symmetrical Lévy α-stable process, with a stability parameter comparable with a value of about 1.4, which was extracted for the U.S. stock market. (3) The return distribution can be described well by Student’s t-distribution within a wider return range than the Lévy α-stable distribution. (4) Distinctively, the stability parameter shows a potential change when Δt increases, and thus a crossover region at 15 <Δt< 60 min is observed. This is different from the finding in the U.S. stock market that a single value of about 1.4 holds over 1 ≤Δt≤ 1000 min. (5) The tail distribution of returns at small Δt decays as an asymptotic power law with an exponent of about 3, which is a widely observed value in mature markets. However, it decays exponentially when Δt≥ 240 min, which is not observed in mature markets. (6) Return distributions gradually converge to a normal distribution as Δt increases. This observation is different from the finding of a critical Δt= 4 days in the U.S. stock market.

Superharmonic instability of stokes waves

AbstractA stability of nearly limiting Stokes waves to superharmonic perturbations is considered numerically in approximation of an infinite depth. Investigation of the stability properties can give one an insight into the evolution of the Stokes wave. The new, previously inaccessible branches of superharmonic instability were investigated. Our numerical simulations suggest that eigenvalues of linearized dynamical equations, corresponding to the unstable modes, appear as a result of a collision of a pair of purely imaginary eigenvalues at the origin, and a subsequent appearance of a pair of purely real eigenvalues: a positive and a negative one that are symmetric with respect to zero. Complex conjugate pairs of purely imaginary eigenvalues correspond to stable modes, and as the steepness of the underlying Stokes wave grows, the pairs move toward the origin along the imaginary axis. Moreover, when studying the eigenvalues of linearized dynamical equations we find that as the steepness of the Stokes wave grows, the real eigenvalues follow a universal scaling law, that can be approximated by a power law. The asymptotic power law behavior of this dependence for instability of Stokes waves close to the limiting one is proposed. Surface elevation profiles for several unstable eigenmodes are made available through http://stokeswave.org website.

Computing hydrodynamic eigenmodes of channel flow with slip—A highly accurate algorithm

AbstractThe transient start‐up flow solution with slip is a useful tool to verify computational fluid dynamics (CFD) simulations. However, a highly accurate, open‐source black box solution does not seem to be available. Our method provides a fast, automated, and rigorously verified open‐source implementation that can compute the hydrodynamic eigenmodes of a two‐dimensional channel flow beyond the standard floating‐point precision. This allows for a very accurate computation of the corresponding Fourier series solution. We prove that all roots are found in all special cases for the general flow problem with different slip lengths on the channel walls. The numerical results confirm analytically derived asymptotic power laws for the leading hydrodynamic eigenmode and the characteristic timescale in the limiting cases of small and large slip. The code repository including test cases is publicly available (DOI: 10.5281/zenodo.6806351). The Navier slip boundary condition for numerical simulations in OpenFOAM is also publicly available (DOI: 10.5281/zenodo.7037712).

Errors in approximate ionization energies due to the one-electron space truncation of the EKT eigenproblem.

Unless the approximate wavefunction of the parent system is expressed in terms of explicitly correlated basis functions, the finite size of the generalized Fock matrix is unlikely to be the leading source of the truncation error in the ionization energy E produced by the EKT (extended Koopmans' theorem) formalism. This conclusion is drawn from a rigorous analysis that involves error partitioning into the parent- and ionized-system contributions, the former being governed by asymptotic power laws when the underlying wavefunction is assembled from a large number of spinorbitals and the latter arising from the truncation of the infinite-dimensional matrix V whose elements involve the 1-, 2-, and 3-matrices of the parent system. Quite surprisingly, the decay of the second contribution with the number n of the natural spinorbitals (NOs) employed in the construction of the truncated V turns out to be strongly system-dependent even in the simplest case of the 1S states of two-electron systems, following the n-5 power law for the helium atom while exhibiting an erratic behavior for the H- anion. This phenomenon, which stems from the presence of the so-called solitonic natural spinorbitals among the NOs, renders the extrapolation of the EKT approximates of E to the complete-basis-set limit generally unfeasible. However, attaining that limit is not contingent upon attempted reproduction of the ill-defined one-electron function known as "the removal orbital," which does not have to be invoked in the derivation of EKT and whose expansion in terms of the NOs diverges.

The arrest and recession dynamics of a deflating hydraulic fracture in a permeable elastic medium in a state of plane strain

We investigate the deflation dynamics of a fluid-driven fracture in a permeable elastic medium in a state of plane strain after fluid injection has ceased. Depending on the leak-off characteristics of the porous medium and the volume of injected fluid retained in the fracture at the time of shut-in, the fracture may start to recede almost immediately or continue to propagate till it arrests when the stress intensity factor drops below the fracture toughness. While occupying the arrest footprint, the fracture continues to deflate while the stress intensity factor decreases due to fluid loss to the porous medium. When the stress intensity factor drops to zero, the fracture starts the process of recession, which continues until it finally collapses. To establish a rigorous numerical scheme to explore the deflation dynamics of plane strain hydraulic fractures, we use recently established vertex and multiscale tip asymptotes for arrested and receding hydraulic fractures (Peirce and Detournay, 2022), including the r-vertex linear tip aperture asymptote wˆ∼xˆ for a receding hydraulic fracture and the stationary g-vertex asymptote wˆ∼xˆ3/4. Numerical experiments demonstrate that the multiscale asymptotes are required in order to achieve solutions that remain smooth through the arrest–recession transition point. In contrast, numerical solutions, obtained by only using vertex solutions to model the arrest and recession, exhibit jump discontinuities through this transition point. However, once the transients from these jump discontinuities have decayed the numerical schemes that use vertex and multiscale asymptotes yield almost identical solutions. A scaling analysis shows the existence of asymptotic power law behaviour for various quantities, such as the arrest time, in terms of two new dimensionless parameters. Finally, numerical solutions explore the dependence on the two dimensionless parameters of the arrest time, the duration of the arrest period, and the duration of the recession phase and confirm their asymptotic power law behaviours.