Power-law Dispersion Research Articles

Impedance spectroscopy: A useful technique to unveil the mechanism of a CO oxidation reaction

AbstractIn this work, the impedance spectroscopy (IS) technique was used to aid in understanding the reaction mechanism of the CO oxidation reaction using the Pr2Zr2−xFexO7±δ (x = 0 and 0.10) mixed oxide as a heterogeneous catalyst. The catalytic reaction occurred in the same temperature range where there was a decrease in the overall resistivity of the materials. Moreover, it has been observed that within the temperature range where the catalytic process takes place, there is a variation in the A and n parameters of the power law dispersion. Combining IS with catalytic reactions offers a robust approach to enhance the understanding of the mechanisms involved in the CO oxidation reaction. It enables precise analysis of the changes that take place at the solid–gas interface, particularly the generation of reactive oxygen species. Furthermore, it allows examining the relationship between the presence of oxygen vacancies and defects, which directly impact the catalytic process.

Isolation by distance in populations with power-law dispersal.

Limited dispersal of individuals between generations results in isolation by distance, in which individuals further apart in space tend to be less related. Classic models of isolation by distance assume that dispersal distances are drawn from a thin-tailed distribution and predict that the proportion of the genome that is identical by descent between a pair of individuals should decrease exponentially with the spatial separation between them. However, in many natural populations, individuals occasionally disperse over very long distances. In this work, we use mathematical analysis and coalescent simulations to study the effect of long-range (power-law) dispersal on patterns of isolation by distance. We find that it leads to power-law decay of identity-by-descent at large distances with the same exponent as dispersal. We also find that broad power-law dispersal produces another, shallow power-law decay of identity-by-descent at short distances. These results suggest that the distribution of long-range dispersal events could be estimated from sequencing large population samples taken from a wide range of spatial scales.

Dispersal Kernel Type Highly Influences Projected Relationships for Plant Disease Epidemic Severity When Outbreak and At-Risk Populations Differ in Susceptibility.

In silico study of biologically invading organisms provide a means to evaluate the complex and potentially cryptic factors that can influence invasion success in scenarios where empirical studies would be difficult, if not impossible, to conduct. I used a disease event simulation program to evaluate whether the two most frequently used types of plant pathogen dispersal kernels for epidemiological projections would provide complementary or divergent projections of epidemic severity when the hosts in a disease outbreak differed from the hosts in the at-risk population in the degree of susceptibility. Exponential dispersal kernel simulations of wheat stripe rust (Pucciniastriiformis var trittici) predicted a relatively strong and dominant influence of the at-risk population on the end epidemic severity regardless of outbreak disease levels. Simulations using a modified power law dispersal kernel gave projections that varied depending on the amount of disease in the outbreak and some interactions were counter-intuitive and opposite of the exponential dispersal kernel projections. Although relatively straightforward, the disease spread simulations in the present study strongly suggest that a more biologically accurate dispersal kernel generates complexity that would not be revealed by an exponential dispersal gradient and that selecting a less accurate dispersal kernel may obscure important interactions during biological invasions.

Diffusive density response of electrons in anisotropic multiband systems

We explicitly calculate the density-density response function with conserving vertex corrections for anisotropic multiband systems in the presence of impurities including long-range disorder. The direction dependence of the vertex corrections is correctly considered to obtain the diffusion constant which is given by a combination of componentwise transport relaxation times and velocities on the Fermi surface. We also investigate the diffusive density response of various anisotropic systems, propose some empirical rules for the corresponding diffusion constant, and demonstrate that it is crucial to consider the component dependence of the transport relaxation times to correctly interpret the transport properties of anisotropic systems, especially various topological materials with a different power-law dispersion in each direction.

Critical dynamics of relativistic diffusion

We study the dynamics of self-interacting scalar fields with Z2 symmetry governed by a relativistic Israel-Stuart type diffusion equation in the vicinity of a critical point. We calculate spectral functions of the order parameter in mean-field approximation as well as using first-principles classical-statistical lattice simulations in real-time. We observe that the spectral functions are well-described by single Breit-Wigner shapes. Away from criticality, the dispersion matches the expectations from the mean-field approach. At the critical point, the spectral functions largely keep their Breit-Wigner shape, albeit with non-trivial power-law dispersion relations. We extract the characteristic time-scales as well as the dynamic critical exponent z, verifying the existence of a dynamic scaling regime. In addition, we derive the universal scaling functions implied by the Breit-Wigner shape with critical power-law dispersion and show that they match the data. Considering equations of motion for a system coupled to a heat bath as well as an isolated system, we perform this study for two different dynamic universality classes, both in two and three spatial dimensions.

Competitive Attention, Superstars and the Long Tail

We propose a model of competitive attention based on two key premises: i) People have limited information processing capacities and ii) consideration sets are formed according to relative salience. The equilibrium predictions we obtain can help to understand, and connect, diverse empirical phenomena, such as the Paradox of Choices, the Power Law dispersions of key market data (sales, profits, online clicks,...), the relation between advertising expenditures and market shares, the evolution of market inequality, or why evidence favoring a “Long Tail” effect is mixed at best.

Generalization of Kramers-Krönig relations for evaluation of causality in power-law media

Classical Kramers-Krönig (K–K) relations connect real and imaginary parts of the frequency-domain response of a system. The K–K relations also hold between the logarithm of modulus and the argument of the response, e.g. between the attenuation and the phase shift of a solution to a wave-propagation problem. For square-integrable functions of frequency, the satisfaction of classical K–K relations implies causality in the time domain. On the other hand, when the K–K relations are checked for the logarithm of the system response, the function is not a square integrable one. Then one can employ classical K–K relations with subtractions, but their satisfaction for the logarithm of the system response does not imply causality of the original function. In this paper, the K–K relations are generalized towards functions which are not square-integrable, also allowing for causality evaluation when the logarithm of the system response is considered. That is, we propose generalization of the K–K relations with subtractions, whose validity for the logarithm of the system response and the satisfaction of additional assumptions imply causality of the originally considered function. The derived theory is then applied to electromagnetic media characterized by power-law frequency dispersion, i.e. the media which are described by fractional-order models (FOMs). In this case, the subtraction procedure generates functions which may be not square integrable, or even not locally integrable. However, we can rigorously analyse causality of the media described by FOM using the derived theory, as well as the parameter ranges for which such models are causal.

Power law dispersion from shear wave images

Many normal soft tissues exhibit dispersion of shear wave phase velocity that is consistent with power law behavior. This rheology is consistent with the fractional Zener model under assumptions that appear to be realistic for normal liver and other tissues. The phenomenon can be measured using multi-frequency reverberant shear wave elastography (R-SWE). A simultaneous multi-frequency R-SWE field can be accomplished by applying an array of external sources that operate at multiple frequencies, for example, 50, 100, 150…500 Hz, all contributing to the shear wave field produced in the target organ. With estimates of phase velocity across this frequency band, the R-SWE approach can obtain a 2-D power law coefficient (PLC) image. The clinical feasibility of this method has been analyzed by assessing the shear wave speed (SWS) and PLC in phantoms and in vivo human organs. For in vivo liver cases, mean SWS (1.99 m/s, 2.29 m/s, and 2.38 m/s) and mean PLC (0.23, 0.44, and 0.43) were estimated for a thin patient, and for two obese patients, respectively. For these cases, the PLC results may be an additional parameter that could help to differentiate the viscoelastic properties in liver tissue and enhanced image contrast in cases where lesions or pathologies show an altered viscoelastic response compared with normal tissue.

The influence of packing structure and interparticle forces on ultrasound transmission in granular media

Ultrasound propagation through externally stressed, disordered granular materials was experimentally and numerically investigated. Experiments employed piezoelectric transducers to excite and detect longitudinal ultrasound waves of various frequencies traveling through randomly packed sapphire spheres subjected to uniaxial compression. The experiments featured in situ X-ray tomography and diffraction measurements of contact fabric, particle kinematics, average per-particle stress tensors, and interparticle forces. The experimentally measured packing configuration and inferred interparticle forces at different sample stresses were used to construct spring networks characterized by Hessian and damping matrices. The ultrasound responses of these network were simulated to investigate the origins of wave velocity, acoustic paths, dispersion, and attenuation. Results revealed that both packing structure and interparticle force heterogeneity played an important role in controlling wave velocity and dispersion, while packing structure alone quantitatively explained most of the observed wave attenuation. This research provides insight into time- and frequency-domain features of wave propagation in randomly packed granular materials, shedding light on the fundamental mechanisms controlling wave velocities, dispersion, and attenuation in such systems.

Mitigating offset frequency drift in frequency combs using a customized power law dispersion.

We introduce a new, to the best of our knowledge, passive technique of mitigating the phase noise in optical frequency combs (FC) by reducing the drift of offset frequency. This can be achieved by customizing the dispersion to attain a power law dependence of the wave vector on frequency, k(ω)∼ωα, ensuring a constant ratio between group and phase velocities. When this condition is maintained, the drift offset frequency is passively mitigated, and phase noise is reduced. Using quantum cascade laser (QCL) FCs as an example, we demonstrate, analytically and numerically, that the desired dispersion can be easily engineered by properly adjusting the thickness of the QCL active region and that stable offset frequency can be combined with low residual group dispersion.

Impacts of typical atmospheric dispersion schemes on source inversion

Abstract Accurate source inversion for sudden air pollution accidents (SAPAs) relies on forward dispersion models, and the dispersion parameter scheme is the key factor for modeling performance. However, the impacts of different schemes on source inversion have yet to be investigated. In this study, the performance of source inversion based on four empirical power-law dispersion schemes—BRIGGS, SMITH, Pasquill-Gifford (P-G), and Chinese National Standard (CNS) schemes—was evaluated using the data set from the 1956 Prairie Grass experiment. The inversion performance of each dispersion scheme was assessed with both single (source strength) and multiple (source strength and location) unknown source parameters under six different atmospheric stability classes (A, B, C, D, E, and F) and the schemes exhibited substantially different performances. With a single unknown source parameter, the four schemes demonstrated excellent robustness. The estimation results of BRIGGS, P-G, and CNS exhibited comparable accuracy to similar average absolute relative deviations (ARD, 33.3–37.5%) and better accuracy than SMITH. The CNS scheme performed the best (37.6%) in unstable atmospheric conditions (A, B, and C), P-G and BRIGGS were comparable and performed the best (26.3% and 25.4%, respectively) in neutral atmospheric conditions (D), and the four schemes exhibited similar performances (28.3–31.3%) in stable atmospheric conditions (E and F). When location was also unknown, the source strength accuracy ranking was P-G (31.6%) > SMITH (43.2%) > CNS (50.6%) > BRIGGS (80.1%); however, the robustness of the P-G, SMITH, and CNS schemes was poor. Further analysis revealed that the differences in the schemes' inversion errors were significantly higher than those of algorithms from previous studies, which implies that scheme selection had a greater influence on source inversion than the algorithm selection. The results of this study can improve our understanding of the factors that influence source inversion.

Comparison of the impacts of empirical power-law dispersion schemes on simulations of pollutant dispersion during different atmospheric conditions

Accurate and rapid predictions of air pollutant dispersion are important for effective emergency responses after sudden air pollution accidents (SAPA). Notably, dispersion parameters (σ) are the key variables that influence the simulation accuracy of dispersion models. Empirical dispersion schemes based on power-law formulas are probably appropriate choices for simulations in SAPA because of the requirement for only routine meteorological data. However, performance comparisons of different schemes are lacking. In this study, the performances during simulations of air pollutant dispersion of four typical empirical parameterised schemes, i.e. BRIGGS, SMITH, Pasquill-Gifford, and Chinese National Standard (CNS), were investigated based on the GAUSSIAN plume model with datasets for the classic Prairie Grass experiments, 1956. The performances when simulating peak and overall concentrations in different Pasquill atmospheric stability classes (A, B, C, D, E, F) were quantitatively analysed through different statistical approaches. Results showed that the performances of four schemes for peak and overall concentrations were basically consistent. Scheme CNS in unstable atmospheric conditions (A, B, and C) performed significantly better than the others according to performance criteria, which included the lowest mean of absolute value of fractional biases, lowest normalised mean square errors, and largest mean values of the fraction within a factor of two when predicting peak and overall concentrations, respectively. Schemes BRIGGS and P-G exhibited slightly better performances during the neutral condition (D) followed by scheme CNS. Schemes SMITH and CNS demonstrated slight merits in predicting concentrations compared to the other schemes during stable conditions (E and F). As a whole, scheme CNS generally performed well for the different atmospheric stability classes. These analysis results can help to fill in the data gaps and improve our understanding of the influence of typical power-law function schemes on simulations of air pollutant dispersion. The results are expected to provide scientific support for air pollution predictions, especially during emergency responses to SAPA.

Fractional Derivatives and Integrals: What Are They Needed For?

The question raised in the title of the article is not philosophical. We do not expect general answers of the form “to describe the reality surrounding us”. The question should actually be formulated as a mathematical problem of applied mathematics, a task for new research. This question should be answered in mathematically rigorous statements about the interrelations between the properties of the operator’s kernels and the types of phenomena. This article is devoted to a discussion of the question of what is fractional operator from the point of view of not pure mathematics, but applied mathematics. The imposed restrictions on the kernel of the fractional operator should actually be divided by types of phenomena, in addition to the principles of self-consistency of mathematical theory. In applications of fractional calculus, we have a fundamental question about conditions of kernels of fractional operator of non-integer orders that allow us to describe a particular type of phenomenon. It is necessary to obtain exact correspondences between sets of properties of kernel and type of phenomena. In this paper, we discuss the properties of kernels of fractional operators to distinguish the following types of phenomena: fading memory (forgetting) and power-law frequency dispersion, spatial non-locality and power-law spatial dispersion, distributed lag (time delay), distributed scaling (dilation), depreciation, and aging.

Solute transport in porous media studied by lattice Boltzmann simulations at pore scale and x-ray tomography experiments.

With the aid of nondestructive microfocus x-ray computed tomography (CT), we performed three-dimensional (3D) tracer dispersion experiments on randomly unconsolidated packed beds. Plumes of nonreactive sodium iodide solution were point injected into a sodium chloride solvent as a tracer for the evaluation of the dispersion process. The asymptotic dispersion coefficient was obtainable within the experimental scale and was summarized over Péclet numbers from 11.7 to ∼860. Then, the lattice Boltzmann method and moment propagation method were used to elucidate the mechanisms embedded in the dispersion phenomenon. The methods were rigorously verified against the classical Taylor dispersion problem and extended to simulate fluid flow and tracer dispersion in high-resolution 3D digital porous structures from CT. The method of moments, Lagrangian velocity correction function, and dilution index were thoroughly analyzed to evaluate the dispersion behaviors. Numerical simulations revealed ballistic and superdiffusive regimes at the transient times, whereas asymptotic dispersion behaviors appear at longer characteristic times. Besides, the observed transient times unanimously persist over convective length scales of around 12 particles transversely and 16 particles longitudinally. The estimated dispersion coefficients from simulation are in consistence with the experimental result. Furthermore, the simulation also enabled the identification of regimes, including diffusive, power law, and mechanical dispersion. Thus, the proposed experimental and computational schemes are of practical means to study dispersion behaviors by direct pore scale imaging and modeling.

Multifractality at the Weyl-semimetal–diffusive-metal transition for generic disorder

A Weyl semimetal is a three dimensional topological gapless phase. In the presence of strong enough disorder it undergoes a quantum transition towards a diffusive metal phase whose universality class depends on the range of disorder correlations. Similar to other quantum transitions driven by disorder, the critical wave functions at the semimetal-diffusive metal transition exhibit multifractality. Using renormalization group methods we study the corresponding multifractal spectrum as a function of the range of disorder correlations for generic disorder including random scalar and vector potentials. We also discuss the relation between the geometric fluctuations of critical wave functions and the broad distribution of the local density of states (DOS) at the transition. We derive a new scaling relation for the typical local DOS and argue that it holds for other disorder-driven transitions in which both the average and typical local DOS vanish on one side of the transition. As an illustration we apply it to the recently discussed unconventional quantum transition in disordered semiconductors with power-law dispersion relation near the band edge.

An Empirical Mass–Radius Relation for Cool Giant Planets

Probabilistic relationships between the mass and radius of planets are useful for a variety of purposes, including estimating the yields of planet discovery efforts and the radii of discovered planets given their masses. Previous work on giant planets often do not make the important distinction between cool and (inflated) hot giants. This can lead to needlessly large uncertainties and overestimated radii for the cooler objects. Using hierarchical bayesian modeling and the AIC model selection criterion, we identify an empirical mass-radius relationship for cool giant planets above 15 Earth masses. It takes the form of a quadratic function in log mass, with a power-law dispersion. Parametric uncertainties from MCMC are provided, and we compare our results with the outputs of interior structure models and the radii of the solar system giant planets, which fit comfortably within our relation.

Exact surface-wave spectrum of a dilute quantum liquid

We consider a dilute gas of bosons with repulsive contact interactions, described on the mean-field level by the Gross-Pitaevski\ifmmode \check{i}\else \v{i}\fi{} equation, and bounded by an impenetrable ``hard'' wall (either rigid or flexible). We solve the Bogoliubov-de Gennes equations for excitations on top of the Bose-Einstein condensate analytically, by using matrix-valued hypergeometric functions. This leads to the exact spectrum of gapless Bogoliubov excitations localized near the boundary. The dispersion relation for the surface excitations represents for small wave numbers $k$ a ripplon mode with fractional power law dispersion for a flexible wall, and a phonon mode (linear dispersion) for a rigid wall. For both types of excitation we provide, for the first time, the exact dispersion relations of the dilute quantum liquid for all $k$ along the surface, extending to $k\ensuremath{\rightarrow}\ensuremath{\infty}$. The small wavelength excitations are shown to be bound to the surface with a maximal binding energy $\mathrm{\ensuremath{\Delta}}=\frac{1}{8}{(\sqrt{17}\ensuremath{-}3)}^{2}m{c}^{2}\ensuremath{\simeq}0.158\phantom{\rule{0.16em}{0ex}}m{c}^{2}$, which both types of excitation asymptotically approach, where $m$ is mass of bosons and $c$ bulk speed of sound. We demonstrate that this binding energy is close to the experimental value obtained for surface excitations of helium II confined in nanopores, reported in Phys. Rev. B 88, 014521 (2013).

Group versus Phase Velocity of Shear Waves in Soft Tissues.

Across the varieties of waves that have been studied in physics, it is well established that group velocities can be significantly greater than or less than phase velocities measured within comparable frequency bands, depending on the particular mechanisms involved. The distinction between group and phase velocities is important in elastography, because diagnoses are made based on shear wave speed estimations from a variety of techniques. We review the general definitions of group and phase velocity and examine their specific relations within an important general class of rheological models. For the class of tissues and materials exhibiting power law dispersion, group velocity is significantly greater than phase velocity, and simple expressions are shown to interrelate the commonly measured parameters. Examples are given from phantoms and tissues.

Focus expansion and stability of the spread parameter estimate of the power law model for dispersal gradients.

Empirical and mechanistic modeling indicate that pathogens transmitted via aerially dispersed inoculum follow a power law, resulting in dispersive epidemic waves. The spread parameter (b) of the power law model, which is an indicator of the distance of the epidemic wave front from an initial focus per unit time, has been found to be approximately 2 for several animal and plant diseases over a wide range of spatial scales under conditions favorable for disease spread. Although disease spread and epidemic expansion can be influenced by several factors, the stability of the parameter b over multiple epidemic years has not been determined. Additionally, the size of the initial epidemic area is expected to be strongly related to the final epidemic extent for epidemics, but the stability of this relationship is also not well established. Here, empirical data of cucurbit downy mildew epidemics collected from 2008 to 2014 were analyzed using a spatio-temporal model of disease spread that incorporates logistic growth in time with a power law function for dispersal. Final epidemic extent ranged from 4.16 ×108 km2 in 2012 to 6.44 ×108 km2 in 2009. Current epidemic extent became significantly associated (P < 0.0332; 0.56 < R2 < 0.99) with final epidemic area beginning near the end of April, with the association increasing monotonically to 1.0 by the end of the epidemic season in July. The position of the epidemic wave-front became exponentially more distant with time, and epidemic velocity increased linearly with distance. Slopes from the temporal and spatial regression models varied with about a 2.5-fold range across epidemic years. Estimates of b varied substantially ranging from 1.51 to 4.16 across epidemic years. We observed a significant b ×time (or distance) interaction (P < 0.05) for epidemic years where data were well described by the power law model. These results suggest that the spread parameter b may not be stable over multiple epidemic years. However, b ≈ 2 may be considered the lower limit of the distance traveled by epidemic wave-fronts for aerially transmitted pathogens that follow a power law dispersal function.

Multifractality at non-Anderson disorder-driven transitions in Weyl semimetals and other systems

Systems with the power-law quasiparticle dispersion ϵk∝kα exhibit non-Anderson disorder-driven transitions in dimensions d>2α, as exemplified by Weyl semimetals, 1D and 2D arrays of ultracold ions with long-range interactions, quantum kicked rotors, and semiconductor models in high dimensions. We study the wavefunction structure in such systems and demonstrate that at these transitions they exhibit fractal behaviour with an infinite set of multifractal exponents. The multifractality persists even when the wavefunction localisation is forbidden by symmetry or topology and occurs as a result of elastic scattering between all momentum states in the band on length scales shorter than the mean free path. We calculate explicitly the multifractal spectra in semiconductors and Weyl semimetals using one-loop and two-loop renormalisation-group approaches slightly above the marginal dimension d=2α.