Asymptotic Power Law Research Articles

Toward the Characterization of Fractional Stochastic Processes Underlying Methyl Dynamics in Proteins

In this article, we investigate the multiple-scale structure of methyl side chain dynamics in proteins. We show that the orientational correlation functions of CH(3) methyl groups are well described by a fractional Brownian dynamics model. Typical angular correlation functions involved in NMR relaxation were computed from MD simulations performed on two different proteins. These correlation functions were shown to be very well fitted by a fractional Ornstein-Uhlenbeck process in the presence of effective local potentials at the C-H and C-C methyl bonds. In addition, our analysis highlights the presence of the asymptotic power law decay of the waiting time probability density of the stochastic process involved, thereby illustrating the connection between approaches based on fractional diffusion equations and the continuous time random walk.

Thermal creep model of rough fractal surfaces in contact: viscoelastic standard linear solid

Purpose – The purpose of this paper is to construct a continuous time series model to study the thermal creep of rough surfaces in contact.Design/methodology/approach – For normal loading, the contact between rough surfaces can often be modeled as the contact of an effective surface with a rigid fiat surface. A solution for the deformation of such equivalent surface, generated using fractal geometry, can be modified. However, in this study only the case of a single rough surface in contact with a rigid flat surface is considered. In the interface, the material is assumed to follow the idealized constitutive viscoelastic standard linear solid (SLS) model. Fractal geometry, through Cantor set theory, is utilized to model the roughness of the surface.Findings – An asymptotic time series power law is obtained, which associates the creep load, the buck temperature and the creep of the fractal surface.Originality/value – This law is only valid as long as the creep is of the size of the surface roughness. The mo...

Recognition of stable distribution with Lévy indexαclose to 2

We address the problem of recognizing α-stable Lévy distribution with Lévy index close to 2 from experimental data. We are interested in the case when the sample size of available data is not large, thus the power law asymptotics of the distribution is not clearly detectable, and the shape of the empirical probability density function is close to a Gaussian. We propose a testing procedure combining a simple visual test based on empirical fourth moment with the Anderson-Darling and Jarque-Bera statistical tests and we check the efficiency of the method on simulated data. Furthermore, we apply our method to the analysis of turbulent plasma density and potential fluctuations measured in the stellarator-type fusion device and demonstrate that the phenomenon of the L-H transition from low confinement, L mode, to a high confinement, H mode, which occurs in this device is accompanied by the transition from Lévy to Gaussian fluctuation statistics.

Casimir-Polder interaction of fullerene molecules with surfaces

We calculate the thermal Casimir-Polder potential of ${\mathrm{C}}_{60}$ and ${\mathrm{C}}_{70}$ fullerene molecules near gold and silicon nitride surfaces, motivated by their relevance for molecular matter-wave interference experiments. We obtain the coefficients governing the asymptotic power laws of the interaction in the thermal, retarded, and nonretarded distance regimes and evaluate the full potential numerically. The interaction is found to be dominated by electronic transitions and hence independent of the internal temperature of the molecules. The contributions from phonon transitions, which are affected by the molecular temperature, give rise to only a small correction. Moreover, we find that the sizable molecular line widths of thermal fullerenes may modify the nonretarded interaction, depending on the model used. Detailed measurements of the nonretarded potential of fullerene thus allow one to distinguish between different theories of incorporating damping.

Scaling properties of paths on graphs

Let G be a directed graph on finitely many vertices and edges, and assign a positive weight to each edge on G. Fix vertices u and v and consider the set of paths that start at u and end at v, self-intersecting in any number of places along the way. For each path, sum the weights of its edges, and then list the path weights in increasing order. The asymptotic behaviour of this sequence is described, in terms of the structure and type of strongly connected components on the graph. As a special case, for a Markov chain the asymptotic probability of paths obeys either a power law scaling or a weaker type of scaling, depending on the structure of the transition matrix. This generalizes previous work by Mandelbrot and others, who established asymptotic power law scaling for special classes of Markov chains.

A treatise on frequency spectrum of drift ice velocity

Abstract Sea ice dynamics is examined for the frequency spectra of ice velocity using mathematical models and ice motion data. The data are from the Baltic Sea and Sea of Okhotsk. A general spectrum for linear coupled ice–ocean free drift is derived and analysed. Interior dynamics shows singularity in the Coriolis (inertial) frequency and asymptotic high frequency power law of − 2. In the presence of internal friction the spectrum is expected to evenly fall to zero with frictional resistance increasing above yield level. In the observations, the main spectral peak is wide at the synoptic time scales. The Okhotsk Sea spectra show inertial and tidal signals but they are missing from the Baltic Sea due to very weak tides and shallow depth damping inertial oscillations. Above semidiurnal frequency ice velocity spectra fall in power law of − 5/3, inherited from atmospheric and oceanic dynamics, and at very high frequencies, above 0.5 cph (cycles per hour) the fall seems to slow down to power law below − 1. The Eulerian ice velocity spectrum was higher than the Lagrangian spectrum throughout the entire obtained spectrum (0.1–5 cpd (cycles per day)). Theoretical spectra show general agreement with observations except that they have less kinetic energy at very high frequencies and much stronger inertial peak. Thus Eulerian observations are more variable than Lagrangian observations.

Efficient, long-range correlation from occupied wave functions only

Via continuum mechanics [J. Tao, X. Gao, G. Vignale, and I. V. Tokatly, Phys. Rev. Lett. 103, 086401 (2009)] with direct random phase approximation (dRPA) screening, we develop a numerically efficient general-geometry electronic exchange-correlation energy functional. It gives correct asymptotic power laws for dispersion interactions between insulators or metals. As a numerical example we obtain the full binding energy curves $\overline{\ensuremath{\epsilon}}(D)$ for parallel metal slabs of small but finite thickness: At all separations $D$ our $\overline{\ensuremath{\epsilon}}(D)$ agrees better with full dRPA correlation calculations than does the local density approximation, while being much more efficient than full dRPA correlation.

Gravitational contributions to gauge Green’s functions and asymptotic free power-law running of gauge coupling

We perform an explicit one-loop calculation for the gravitational contributions to the two-, three- and four-point gauge Green's functions with paying attention to the quadratic divergences. It is shown for the first time in the diagrammatic calculation that the Slavnov-Taylor identities are preserved even if the quantum graviton effects are included at one-loop level, such a conclusion is independent of the choice of regularization schemes. We also present a regularization scheme independent calculation based on the gauge condition independent background field framework of Vilkovisky-DeWitt's effective action with focusing on both the quadratic divergence and quartic divergence that is not discussed before. With the harmonic gauge condition, the results computed by using the traditional background field method can consistently be recovered from the Vilkovisky-DeWitt's effective action approach by simply taking a limiting case, and are found to be the same as the ones yielded by the diagrammatic calculation. As a consequence, in all the calculations, the symmetry-preserving and divergent-behavior-preserving loop regularization method can consistently lead to a nontrivial gravitational contribution to the gauge coupling constant with an asymptotic free power-law running at one loop near the Planck scale.

Slip–Stick Mechanism in Training the Superconducting Magnets in the Large Hadron Collider

Superconducting magnets can exhibit training quenches during successive powering to reaching nominal performance. The slip-stick motion of the conductors is considered to be one of the mechanisms of training. In this paper, we present a simple quantitative model where the training is described as a discrete dynamical system matching the equilibrium between the energy margin of the superconducting cable and the frictional energy released during the conductor motion. The model can be explicitly solved in the linearized case, showing that the short sample limit is reached via a power law. Training phenomena have a large random component. A large set of data of the large hadron collider magnet tests is postprocessed according to previously defined methods to extract an average training curve for dipoles and quadrupoles. These curves show the asymptotic power law predicted by the model. The curves are then fit through the model, which has two free parameters. The model shows good agreement over a large range, but it fails to describe the very initial part of the training.

A NEW SPECTRAL ENERGY DISTRIBUTION OF PHOTONS IN THE FIRESHELL MODEL OF GRBS

The analysis of various Gamma-Ray Bursts (GRBs) having a low energetics (an isotropic energy E iso ≲ 1053 ergs) within the fireshell model has shown how the N(E) spectrum of their prompt emission can be reproduced in a satisfactory way by a convolution of thermal spectra. Nevertheless, from the study of very energetic bursts (E iso ≳ 1054 ergs) such as, for example, GRB 080319B, some discrepancies between the numerical simulations and the observational data have been observed. We investigate a different spectrum of photons in the comoving frame of the fireshell in order to better reproduce the spectral properties of GRB prompt emission within the fireshell model. We introduce a phenomenologically modified thermal spectrum: a thermal spectrum characterized by a different asymptotic power-law index in the low energy region. Such an index depends on a free parameter α, so that the pure thermal spectrum corresponds to the case α = 0. We test this spectrum by comparing the numerical simulations with the observed prompt emission spectra of various GRBs. From this analysis it has emerged that the observational data can be correctly reproduced by assuming a modified thermal spectrum with α = -1.8.

A thermal creep model for the contact of nominally flat surfaces: Jeffreys' linear visco-elastic model

This work considers the mechanics of contact of thermo-visco-elastic materials. In particular the creep behavior of a nominally flat rough surface in contact with a rigid half space is studied. The rough surface is modeled using fractal geometry. A synthesized profile, a Cantor structure, is utilized to model the surface. Such a profile has two scaling parameters and different heights for each generation of asperities. The effect of temperature will be included through the concept of activation energy using the Arrhenius equation. The objective of this model is to study the normal creep approach of the surface (punch) as a function of the applied creep load, time, and temperature. The material of the punch is assumed to behave according to Jeffreys' model. Such a model is an arrangement of springs and dashpots in parallel and/or in series. The creep approach of linearly visco-elastic materials is explored using elastic–visco-elastic correspondence analysis. An asymptotic power law is obtained, which relates the force and the bulk temperature acting on the punch to its approach. This model is valid only when the approach between the punch and the half space is in the range of the roughness size. The proposed model admits an analytical solution for the case when the deformation is linear thermo-visco-elastic. The obtained model shows a good agreement when compared with experimental results from literature.

Generalized Langevin equation with multiplicative noise: Temporal behavior of the autocorrelation functions

The temporal behavior of the mean-square displacement and the velocity autocorrelation function of a particle subjected to a periodic force in a harmonic potential well is investigated for viscoelastic media using the generalized Langevin equation. The interaction with fluctuations of environmental parameters is modeled by a multiplicative white noise, by an internal Mittag-Leffler noise with a finite memory time, and by an additive external noise. It is shown that the presence of a multiplicative noise has a profound effect on the behavior of the autocorrelation functions. Particularly, for correlation functions the model predicts a crossover between two different asymptotic power-law regimes. Moreover, a dependence of the correlation function on the frequency of the external periodic forcing occurs that gives a simple criterion to discern the multiplicative noise in future experiments. It is established that additive external and internal noises cause qualitatively different dependences of the autocorrelation functions on the external forcing and also on the time lag. The influence of the memory time of the internal noise on the dynamics of the system is also discussed.

On the universal relations for the prefactors in correlation functions of 1D quantum liquids

For various one-dimensional quantum liquids in the framework of the Luttinger model (bosonization) we establish the relations between the coefficients before the power-law asymptotics of the correlators (prefactors) and the formfactors of the corresponding local operators. The derivation of these relations in the framework of the bosonization procedure allows to substantiate the prediction for the formfactors corresponding to the low-lying particle–hole excitations. We also obtain the formulas for the summation over the particle–hole states corresponding to the power-law asymptotics of the correlators.

Slow relaxation and aging kinetics for the driven lattice gas

We numerically investigate the long-time behavior of the density-density autocorrelation function in driven lattice gases with particle exclusion and periodic boundary conditions in one, two, and three dimensions using precise Monte Carlo simulations. In the one-dimensional asymmetric exclusion process on a ring with half the lattice sites occupied, we find that correlations induce extremely slow relaxation to the asymptotic power law decay. We compare the crossover functions obtained from our simulations with various analytic results in the literature and analyze the characteristic oscillations that occur in finite systems away from half-filling. As expected, in three dimensions correlations are weak and consequently the mean-field description is adequate. We also investigate the relaxation toward the nonequilibrium steady state in the two-time density-density autocorrelations, starting from strongly correlated initial conditions. We obtain simple aging scaling behavior in one, two, and three dimensions, with the expected power laws.

Understanding finite size effects in quasi-long-range orders for exactly solvable chain models

In this paper, we investigate how much of the numerical artefacts introduced by finite system size and choice of boundary conditions can be removed by finite size scaling, for strongly correlated systems with quasi-long-range order. Starting from the exact ground-state wave functions of hardcore bosons and spinless fermions with infinite nearest-neighbor repulsion on finite periodic chains and finite open chains, we compute the two-point, density–density, and pair–pair correlation functions, and fit these to various asymptotic power laws. Comparing the finite-periodic-chain and finite-open-chain correlations with their infinite-chain counterparts, we find reasonable agreement among them for the power-law amplitudes and exponents, but poor agreement for the phase shifts. More importantly, for chain lengths on the order of 100, we find our finite-open-chain calculation overestimates some infinite-chain exponents (as did a recent density-matrix renormalization-group (DMRG) calculation on finite smooth chains), whereas our finite-periodic-chain calculation underestimates these exponents. We attribute this systematic difference to the different choice of boundary conditions. Eventually, both finite-chain exponents approach the infinite-chain limit: by a chain length of 1000 for periodic chains, and > 2000 for open chains. There is, however, a misleading apparent finite size scaling convergence at shorter chain lengths, for both our finite-chain exponents, as well as the finite-smooth-chain exponents. Implications of this observation are discussed.

A Sequential Folding Model Predicts Length-Independent Secondary Structure Properties of Long ssRNA

We introduce a simple model for folding random-sequence RNA molecules, arguing that it provides a direct route to predicting and rationalizing several average properties of RNA secondary structures. The first folding step involves identifying the longest possible duplex, thereby dividing the molecule into a pair of daughter loops. Successive steps involve identifying similarly the longest duplex in each new pair of daughter loops, with this process proceeding sequentially until the loops are too small for a viable duplex to form. Approximate analytical solutions are found for the average fraction of paired bases, the average duplex length, and the average loop size, all of which are shown to be independent of sequence length for long enough molecules. Numerical solutions to the model provide estimates for these average secondary structure properties that agree well with those obtained from more sophisticated folding algorithms. We also use the model to derive the asymptotic power law for the dependence of the maximum ladder distance on chain length.

Recent Advancements in Fractal Geometric‐Based Nonlinear Time Series Solutions to the Micro‐Quasistatic Thermoviscoelastic Creep for Rough Surfaces in Contact

To understand the tripological contact phenomena, both mathematical and experimental models are needed. In this work, fractal mathematical models are used to model the experimental results obtained from literature. Fractal geometry, using a deterministic Cantor structure, is used to model the surface topography, where recent advancements in thermoviscoelastic creep contact of rough surfaces are introduced. Various viscoelastic idealizations are used to model the surface materials, for example, Maxwell, Kelvin‐Voigt, Standard Linear Solid and Jeffrey media. Such media are modelled as arrangements of elastic springs and viscous dashpots in parallel and/or in series. Asymptotic power laws, through hypergeometric series, were used to express the surface creep as a function of remote forces, body temperatures and time. The introduced models are valid only when the creep approach of the contact surfaces is in the order of the size of the surface roughness. The obtained results using such models, which admit closed‐form solutions, are displayed graphically for selected values of the systems′ parameters; the fractal surface roughness and various material properties. Results obtained showed good agreement with published experimental results, where the utilized methodology can be further extended to the utilization for the contact of surfaces within micro‐ and nano‐electronic devices, circuits and systems.

Betweenness centrality of an edge in tree-like components with finite size

Using the generating function formalism, we propose an exact formula for estimating betweenness centrality of an edge in finite tree-like components of random networks with arbitrary degree distributions. As a function of the degrees of two vertices interconnected by the edge and the size of the component to which the edge belongs, the formula can give an estimation of the mean value of the edge betweenness. We confirm the formula by simulations for Poisson and power law distributions respectively. We also study the asymptotic behavior of the formula and find that there is an asymptotic power law relation with an exponent 1.5 between the edge betweenness and the size of the component to which the edge belongs.

Reptation of a semiflexible polymer through porous media

We study the motion of a single stiff semiflexible filament of length S through an array of topological obstacles. By means of scaling arguments and two-dimensional computer simulations, we show that the stiff chain kinetics follows the reptation picture, albeit with kinetic exponents (for the central monomer) different from those for flexible chain reptation. At early times when topological constraints are irrelevant, the chain kinetics is the anisotropic dynamics of a free filament. After the entanglement time tau(e) transverse modes are equilibrated under the topological constraints, but the chain is not yet correlated over its whole length. During the relaxation of longitudinal modes, both the longitudinal fluctuation of the central monomer and the longitudinal correlation length grow as approximately sqrt t. After time tau(r) approximately S(2) chain ends are correlated, the chain then diffuses globally along the tube and tube renewal takes place. In the reptation regime, the longitudinal fluctuation of the central monomer grows like approximately t(1). The opening of the intermediate approximately sqrt t regime, absent for a free filament, is a signature of the reptation process. Although the underlying physics is quite different, the intermediate regime is reminiscent of the internal Rouse mode relaxation found for reptating flexible chains. In most cases asymptotic power laws from scaling could be complemented by prefactors calculated analytically. Our results are supported by two-dimensional Langevin simulations with fixed obstacles via evaluation of the mean squared displacement of the central monomer. The scaling theory can be extended to long semiflexible polymers adopting random-walk equilibrium configurations and should also apply in three dimensions for porous media with pore diameter smaller than the persistence length of the filament.

Universal power-law and partial condensation in aggregation-chipping processes

The asymptotic behaviour of a distribution function P(X) for X clusters is investigated in aggregation-chipping processes, where aggregation and chipping off of a finite unit of size less than L take place simultaneously. Numerical simulations show that above a certain threshold <X>c of an average cluster size, the system exhibits partial condensation where one condensed cluster coexists with a universal power-law distribution with the exponent -5/2 . The critical value <X>c is calculated and turns out to increase monotonously with L . The z-transform technique is used to analyze the case L=2 in detail. Obtained results agree well with numerical ones. Finally, universality of the asymptotic power law is discussed for general cases. It becomes evident that universality holds as long as the size of chipped off unit is finite.