Asymptotic Power Law Research Articles

Critical exponents of the explosive percolation transition.

In a new type of percolation phase transition, which was observed in a set of nonequilibrium models, each new connection between vertices is chosen from a number of possibilities by an Achlioptas-like algorithm. This causes preferential merging of small components and delays the emergence of the percolation cluster. First simulations led to a conclusion that a percolation cluster in this irreversible process is born discontinuously, by a discontinuous phase transition, which results in the term "explosive percolation transition." We have shown that this transition is actually continuous (second order) though with an anomalously small critical exponent of the percolation cluster. Here we propose an efficient numerical method enabling us to find the critical exponents and other characteristics of this second-order transition for a representative set of explosive percolation models with different number of choices. The method is based on gluing together the numerical solutions of evolution equations for the cluster size distribution and power-law asymptotics. For each of the models, with high precision, we obtain critical exponents and the critical point.

Abnormal grain growth in nonequilibrium systems: effects of point defect patterning.

We present a mean-field model for abnormal grain growth processes described by competition of both deterministic and stochastic mechanisms with the grain mobility depending on the grain size. The derived approach is applied to study delayed dynamics of grain growth in system of point defects subjected to irradiation according to the swelling rate theory for irradiated materials. We have shown that with the irradiation dose increase, the corresponding grain growth dynamics is slowed down and growth processes with power-law time asymptotics are realized. We discuss behavior of grain size distributions with competing regular and fluctuation mechanisms.

Reliability analysis for sluice gate anti-sliding stability using Lévy stable distributions

This paper is aimed at developing a statistical method to analyze the static and dynamic reliability of sluice gate anti-sliding stability. The proposed method investigates the static reliability by using a modified Monte Carlo method in conjunction with Lévy stable distributions, and the dynamic reliability is analyzed via the temporal scales of water level signal which is detected by the detrended fluctuation analysis. We exemplify this Lévy-Monte Carlo strategy in the evaluation of the anti-sliding stability of the Yongji second sluice gate. The simulated failure probabilities are less conservative than the existing results given by the first-order second-moment method and the maximum entropy principle. We also find that the tail distribution of water level signal obeys an asymptotic power law and its increment process can well be depicted by the linear fractional stable noise model.

Split Fermi seas in one-dimensional Bose fluids

For the one-dimensional repulsive Bose gas (Lieb-Liniger model), we study a special class of highly-excited states obtained by giving a finite momentum to subgroups of particles. These states, which correspond to `splitting' the ground state Fermi sea-like quantum number configuration, are zero-entropy states which display interesting properties more normally associated to ground states. Using a numerically exact method based on integrability, we study these states' excitation spectrum, density correlations and momentum distribution functions. These correlations display power-law asymptotics, and are shown to be accurately described by an effective multicomponent Tomonaga-Luttinger liquid theory whose parameters are obtained from Bethe Ansatz. The non-universal correlation prefactors are moreover obtained from integrability, yielding a completely parameter-free fit of the correlator asymptotics.

Numerical simulations of thermal convection in rotating spherical shells under laboratory conditions

An exhaustive study, based on numerical three-dimensional simulations, of the Boussinesq thermal convection of a fluid confined in a rotating spherical shell is presented. A moderately low Prandtl number fluid (σ=0.1) bounded by differentially-heated solid spherical shells is mainly considered. Asymptotic power laws for the mean physical properties of the flows are obtained in the limit of low Rossby number and compared with laboratory experiments and with previous numerical results computed by taking either stress-free boundary conditions or quasi-geostrophic restrictions, and with geodynamo models. Finally, using parameters as close as possible to those of the Earth’s outer core, some estimations of the characteristic time and length scales of convection are given.

QUANTUM STRUCTURE OF FIELD THEORY AND STANDARD MODEL BASED ON INFINITY-FREE LOOP REGULARIZATION/RENORMALIZATION

To understand better the quantum structure of field theory and standard model in particle physics, it is necessary to investigate carefully the divergence structure in quantum field theories (QFTs) and work out a consistent framework to avoid infinities. The divergence has got us into trouble since developing quantum electrodynamics in 1930s. Its treatment via the renormalization scheme is satisfied not by all physicists, like Dirac and Feynman who have made serious criticisms. The renormalization group analysis reveals that QFTs can in general be defined fundamentally with the meaningful energy scale that has some physical significance, which motivates us to develop a new symmetry-preserving and infinity-free regularization scheme called loop regularization (LORE). A simple regularization prescription in LORE is realized based on a manifest postulation that a loop divergence with a power counting dimension larger than or equal to the space–time dimension must vanish. The LORE method is achieved without modifying original theory and leads the divergent Feynman loop integrals well-defined to maintain the divergence structure and meanwhile preserve basic symmetries of original theory. The crucial point in LORE is the presence of two intrinsic energy scales which play the roles of ultraviolet cutoff Mcand infrared cutoff μsto avoid infinities. As Mccan be made finite when taking appropriately both the primary regulator mass and number to be infinity to recover the original integrals, the two energy scales Mcand μsin LORE become physically meaningful as the characteristic energy scale and sliding energy scale, respectively. The key concept in LORE is the introduction of irreducible loop integrals (ILIs) on which the regularization prescription acts, which leads to a set of gauge invariance consistency conditions between the regularized tensor-type and scalar-type ILIs. An interesting observation in LORE is that the evaluation of ILIs with ultraviolet-divergence-preserving (UVDP) parametrization naturally leads to Bjorken–Drell's analogy between Feynman diagrams and electric circuits, which enables us to treat systematically the divergences of Feynman diagrams and understand better the divergence structure of QFTs. The LORE method has been shown to be applicable to both underlying and effective QFTs. Its consistency and advantages have been demonstrated in a series of applications, which includes the Slavnov–Taylor–Ward–Takahaski identities of gauge theories and supersymmetric theories, quantum chiral anomaly, renormalization of scalar interaction and power-law running of scalar mass, quantum gravitational effects and asymptotic free power-law running of gauge couplings.

Attraction, with boundaries

We study the basin of attraction of static extremal black holes, in the concrete setting of the STU model. By finding a connection to a decoupled Toda-like system and solving it exactly, we find a simple way to characterize the attraction basin via competing behaviors of certain parameters. The boundaries of attraction arise in the various limits where these parameters degenerate to zero. We find that these boundaries are generalizations of the recently introduced (extremal) subtracted geometry: the warp factors still exhibit asymptotic integer power law behaviors, but the powers can be different from one. As we cross over one of these boundaries (‘generalized subttractors’), the solutions turn unstable and start blowing up at finite radius and lose their asymptotic region. Our results are fully analytic, but we also solve a simpler theory where the attraction basin is lower dimensional and easy to visualize, and present a simple picture that illustrates many of the basic ideas.

Superlinear scaling of offspring at criticality in branching processes

For any branching process, we demonstrate that the typical total number rmp(ντ) of events triggered over all generations within any sufficiently large time window τ exhibits, at criticality, a superlinear dependence rmp(ντ)∼(ντ)γ (with γ>1) on the total number ντ of the immigrants arriving at the Poisson rate ν. In branching processes in which immigrants (or sources) are characterized by fertilities distributed according to an asymptotic power-law tail with tail exponent 1<γ⩽2, the exponent of the superlinear law for rmp(ντ) is identical to the exponent γ of the distribution of fertilities. For γ>2 and for standard branching processes without power-law distribution of fertilities, rmp(ντ)∼(ντ)2. This scaling law replaces and tames the divergence ντ/(1-n) of the mean total number R̅t(τ) of events, as the branching ratio (defined as the average number of triggered events of first generation per source) tends to 1. The derivation uses the formalism of generating probability functions. The corresponding prediction is confirmed by numerical calculations, and an heuristic derivation enlightens its underlying mechanism. We also show that R̅t(τ) is always linear in ντ even at criticality (n=1). Our results thus illustrate the fundamental difference between the mean total number, which is controlled by a few extremely rare realizations, and the typical behavior represented by rmp(ντ).

The behavior of solutions of the nonlinear biharmonic equation in an unbounded domain

Periodic (in one variable) solutions in the half-plane of the two-dimensional nonlinear biharmonic equation with exponential nonlinearity on the right-hand side are considered. The power-law and logarithmic asymptotics of the solutions at infinity are obtained.

High frequency energy cascades in inviscid hydrodynamics

With the aim of gaining insight into the notoriously difficult problem of energy and vorticity cascades in high dimensional incompressible flows, we take a simpler and very well understood low dimensional analog and approach it from a new perspective, using the Fourier transform. Specifically, we study, numerically and analytically, how kinetic energy moves from one scale to another in solutions of the hyperbolic or inviscid Burgers equation in one spatial dimension (1D). We restrict our attention to initial conditions which go to zero as x→±∞. The main result we report here is a Fourier analytic way of describing the cascade process. We find that the cascade proceeds by rapid growth of a crossover scale below which there is asymptotic power law decay of the magnitude of the Fourier transform.

Applicability of mode-coupling theory to polyisobutylene: A molecular dynamics simulation study

The applicability of Mode Coupling Theory (MCT) to the glass-forming polymer polyisobutylene (PIB) has been explored by using fully atomistic molecular dynamics simulations. MCT predictions for the so-called asymptotic regime have been successfully tested on the dynamic structure factor and the self-correlation function of PIB main-chain carbons calculated from the simulated cell. The factorization theorem and the time-temperature superposition principle are satisfied. A consistent fitting procedure of the simulation data to the MCT asymptotic power-laws predicted for the α-relaxation regime has delivered the dynamic exponents of the theory-in particular, the exponent parameter λ-the critical non-ergodicity parameters, and the critical temperature T(c). The obtained values of λ and T(c) agree, within the uncertainties involved in both studies, with those deduced from depolarized light scattering experiments [A. Kisliuk et al., J. Polym. Sci. Part B: Polym. Phys. 38, 2785 (2000)]. Both, λ and T(c)/T(g) values found for PIB are unusually large with respect to those commonly obtained in low molecular weight systems. Moreover, the high T(c)/T(g) value is compatible with a certain correlation of this parameter with the fragility in Angell's classification. Conversely, the value of λ is close to that reported for real polymers, simulated "realistic" polymers and simple polymer models with intramolecular barriers. In the framework of the MCT, such finding should be the signature of two different mechanisms for the glass-transition in real polymers: intermolecular packing and intramolecular barriers combined with chain connectivity.

Thermal contact conductance of elastically deforming nominally flat surfaces using fractal geometry

Purpose – This work aims at constructing a continuous mathematical, linear elastic, model for the thermal contact conductance (TCC) of two rough surfaces in contact. Design/methodology/approach – The rough surfaces, known to be physical fractal, are modelled using a deterministic Cantor structure. Such structure shows several levels of imperfections and including, therefore, several scales in the constriction of the flux lines. The proposed model will study the effect of the deformation (approach) of the two rough surfaces on the TCC as a function of the remotely applied load. Findings – An asymptotic power law, derived using approximate iterative relations, is used to express the area of contact and, consequently, the thermal conductance as a function of the applied load. The model is valid only when the approach of the two surface in contact is of the order of the surface roughness. The results obtained using this model, which admits closed form solution, are displayed graphically for selected values of the system parameters; the fractal surface roughness and various material properties. The obtained results showed good agreement with published experimental results both in trend and the numerical values. Originality/value – The model obtained provides further insight into the effect that surface texture has on the heat conductance process. The proposed model could be used to conduct an analytical investigation of the thermal conductance of rough surfaces in contact. This model, although simple (composed of springs), nevertheless works well.

ON QUIET-TIME SOLAR WIND ELECTRON DISTRIBUTIONS IN DYNAMICAL EQUILIBRIUM WITH LANGMUIR TURBULENCE

A recent series of papers put forth a self-consistent theory of an asymptotically steady-state electron distribution function and Langmuir turbulence intensity. The theory was developed in terms of the κ distribution which features Maxwellian low-energy electrons and a non-Maxwellian energetic power-law tail component. The present paper discusses a generalized κ distribution that features a Davydov-Druyvesteyn type of core component and an energetic power-law tail component. The physical motivation for such a generalization is so that the model may reflect the influence of low-energy electrons interacting with low-frequency kinetic Alfvenic turbulence as well as with high-frequency Langmuir turbulence. It is shown that such a solution and the accompanying Langmuir wave spectrum rigorously satisfy the balance requirement between the spontaneous and induced emission processes in both the particle and wave kinetic equations, and approximately satisfy the similar balance requirement between the spontaneous and induced scattering processes, which are nonlinear. In spite of the low velocity modification of the electron distribution function, it is shown that the resulting asymptotic velocity power-law index α, where fe ~ v –α is close to the average index observed during the quiet-time solar wind condition, i.e., whereas αaverage ~ 6.69, according to observation.

Theory of magnetic small-angle neutron scattering of two-phase ferromagnets

Based on micromagnetic theory we have derived analytical expressions for the magnetic small-angle neutron scattering (SANS) cross section of a two-phase particle-matrix-type ferromagnet. The approach---valid close to magnetic saturation---provides access to several features of the spin structure such as perturbing magnetic anisotropy and magnetostatic fields. Depending on the applied magnetic field and on the magnitude $H_p$ of the magnetic anisotropy field relative to the magnitude $\Delta M$ of the jump in the longitudinal magnetization at the particle-matrix interface, we observe a variety of angular anisotropies in the magnetic SANS cross section. In particular, the model explains the "clover-leaf"-shaped angular anisotropy which was previously observed for several nanostructured magnetic materials, and it provides access to the magnetic interaction parameters such as the average exchange-stiffness constant. It is also shown that the ratio $H_p / \Delta M$ decisively determines the asymptotic power-law exponent and the range of spin-misalignment correlations.

Depth and minimal slope for surface flows of cohesive granular materials on inclined channels

AbstractWe present analytical predictions of the depth and onset slope of the steady surface flow of a cohesive granular material in an inclined channel. The rheology of Jop, Forterre &amp; Pouliquen (Nature, vol. 441, 2006, pp. 727–730) is used, assuming co-axiality between the stress and strain-rate tensors, and a coefficient of friction dependent on the strain rate through the dimensionless inertial number $I$. This rheological law is augmented by a constant stress representing cohesion. Our analysis does not rely on a precise $\mu (I)$ functional, but only on its asymptotic power law in the limit of vanishing strain rates. Assuming a unidirectional flow, the Navier–Stokes equations can be solved explicitly to yield parametric equations of the iso-velocity lines in the plane perpendicular to the flow. Two types of channel walls are considered: rough and smooth, depicting walls whose friction coefficient is respectively larger or smaller than that of the flowing material. The steady flow starts above a critical onset angle and consists of a sheared zone confined between a surface plug flow and a deep dead zone. The details of the flow are discussed, depending on dimensionless parameters relating the static friction coefficient, cohesion strength of the material, incline angle, wall friction, and channel width. The depths of the flow at the centre of the channel and at the walls are calculated by a force balance on the flowing material. The critical angle for the onset of the flow is also calculated, and is found to be strongly dependent on the channel width, in agreement with experimental results on heap stability and in rotating drums. Our results predict the important conclusion that a cohesive material always starts to flow for an incline angle lower than 90° between smooth walls, whereas in a narrow enough channel with rough walls, it may not flow, even if the channel is inclined vertically.

Scaling behavior of an airplane-boarding model

An airplane-boarding model, introduced earlier by Frette and Hemmer [Phys. Rev. E 85, 011130 (2012)], is studied with the aim of determining precisely its asymptotic power-law scaling behavior for a large number of passengers N. Based on Monte Carlo simulation data for very large system sizes up to N=2(16)=65536, we have analyzed numerically the scaling behavior of the mean boarding time <t(b)> and other related quantities. In analogy with critical phenomena, we have used appropriate scaling Ansätze, which include the leading term as some power of N (e.g., [proportionality]N(α) for <t(b)>), as well as power-law corrections to scaling. Our results clearly show that α=1/2 holds with a very high numerical accuracy (α=0.5001±0.0001). This value deviates essentially from α=/~0.69, obtained earlier by Frette and Hemmer from data within the range 2≤N≤16. Our results confirm the convergence of the effective exponent α(eff)(N) to 1/2 at large N as observed by Bernstein. Our analysis explains this effect. Namely, the effective exponent α(eff)(N) varies from values about 0.7 for small system sizes to the true asymptotic value 1/2 at N→∞ almost linearly in N(-1/3) for large N. This means that the variation is caused by corrections to scaling, the leading correction-to-scaling exponent being θ≈1/3. We have estimated also other exponents: ν=1/2 for the mean number of passengers taking seats simultaneously in one time step, β=1 for the second moment of t(b), and γ≈1/3 for its variance.

Hierarchy of temporal responses of multivariate self-excited epidemic processes

We present the first exact analysis of some of the temporal properties of multivariate self-excited Hawkes conditional Poisson processes, which constitute powerful representations of a large variety of systems with bursty events, for which past activity triggers future activity. The term "multivariate" refers to the property that events come in different types, with possibly different intra- and inter-triggering abilities. We develop the general formalism of the multivariate generating moment function for the cumulative number of first-generation and of all generation events triggered by a given mother event (the "shock") as a function of the current time $t$. This corresponds to studying the response function of the process. A variety of different systems have been analyzed. In particular, for systems in which triggering between events of different types proceeds through a one-dimension directed or symmetric chain of influence in type space, we report a novel hierarchy of intermediate asymptotic power law decays $\sim 1/t^{1-(m+1)\theta}$ of the rate of triggered events as a function of the distance $m$ of the events to the initial shock in the type space, where $0 < \theta <1$ for the relevant long-memory processes characterizing many natural and social systems. The richness of the generated time dynamics comes from the cascades of intermediate events of possibly different kinds, unfolding via a kind of inter-breeding genealogy.

Geometric preferential attachment in non-uniform metric spaces

We investigate the degree sequences of geometric preferential attachment graphs in general compact metric spaces. We show that, under certain conditions on the attractiveness function, the behaviour of the degree sequence is similar to that of the preferential attachment with multiplicative fitness models investigated by Borgs et al. When the metric space is finite, the degree distribution at each point of the space converges to a degree distribution which is an asymptotic power law whose index depends on the chosen point. For infinite metric spaces, we can show that for vertices in a Borel subset of S of positive measure the degree distribution converges to a distribution whose tail is close to that of a power law whose index again depends on the set.

Universality versus nonuniversality in asymmetric fluid criticality

Critical phenomena in real fluids demonstrate a combination of universal features caused by the divergence of long-range fluctuations of density and nonuniversal (system-dependent) features associated with specific intermolecular interactions. Asymptotically, all fluids belong to the Ising-model class of universality. The asymptotic power laws for the thermodynamic properties are described by two independent universal critical exponents and by two independent nonuniversal critical amplitudes; other critical amplitudes can be obtained by universal relations. The nonuniversal critical parameters (critical temperature, pressure, and density) can be absorbed in the property units. Nonasymptotic critical behavior of fluids can be divided into two parts, symmetric ("Ising-like") and asymmetric ("fluid-like"). The symmetric nonasymptotic behavior contains a new universal exponent (Wegner exponent) and the system-dependent crossover scale (Ginzburg number) associated with the range of intermolecular interactions, while the asymmetric features are generally described by an additional universal exponent and by three nonasymptotic amplitudes associated with mixing of the physical fields into the scaling fields.

Survival Exponents for Some Gaussian Processes

The problem is a power-law asymptotics of the probability that a self-similar process does not exceed a fixed level during long time. The exponent in such asymptotics is estimated for some Gaussian processes, including the fractional Brownian motion (FBM) in , and the integrated FBM in , .