Intermediate Power Law Research Articles

Self-similarity of the third type in ultra-relativistic blastwaves

A new type of self-similarity is found in the problem of a plane-parallel, ultra-relativistic blastwave, propagating in a power-law density profile of the form ρ∝z−k. Self-similar solutions of the first kind can be found for k < 7∕4 using dimensional considerations. For steeper density gradients with k > 2, second type solutions are obtained by eliminating a singularity from the equations. However, for intermediate power-law indices 7/4<k<2, the flow does not obey any of the known types of self-similarity. Instead, the solutions belong to a new class in which the self-similar dynamics are dictated by the non-self-similar part of the flow. We obtain an exact solution to the ultra-relativistic fluid equations and find that the non-self-similar flow is described by a relativistic expansion into vacuum, composed of (1) an accelerating piston that contains most of the energy and (2) a leading edge of a fast material that coincides with the interiors of the blastwave and terminates at the shock. The dynamics of the piston itself are self-similar and universal and do not depend on the external medium. The exact solution of the non-self-similar flow is used to solve for the shock in the new class of solutions.

Coupled Fredkin and Motzkin chains from quantum six- and nineteen-vertex models

We generalize the area-law violating models of Fredkin and Motzkin spin chains into two dimensions by building quantum six- and nineteen-vertex models with correlated interactions. The Hamiltonian is frustration free, and its projectors generate ergodic dynamics within the subspace of height configuration that are non negative. The ground state is a volume- and color-weighted superposition of classical bi-color vertex configurations with non-negative heights in the bulk and zero height on the boundary. The entanglement entropy between subsystems has a phase transition as the qq-deformation parameter is tuned, which is shown to be robust in the presence of an external field acting on the color degree of freedom. The ground state undergoes a quantum phase transition between area- and volume-law entanglement phases with a critical point where entanglement entropy scales as a function L\log LLlogL of the linear system size LL. Intermediate power law scalings between L\log LLlogL and L^2L2 can be achieved with an inhomogeneous deformation parameter that approaches 1 at different rates in the thermodynamic limit. For the q>1q>1 phase, we construct a variational wave function that establishes an upper bound on the spectral gap that scales as q^{-L^3/8}q−L3/8.

Stochastic model for the alignment and tumbling of rigid fibers in two-dimensional turbulent shear flow

Non-spherical particles transported by an anisotropic turbulent flow preferentially align with the mean shear and intermittently tumble when the local strain fluctuates. Such an intricate behaviour is here studied for inertialess, rod-shaped particles embedded in a two-dimensional turbulent flow with homogeneous shear. A Lagrangian stochastic model for the rods angular dynamics is introduced and compared to the results of direct numerical simulations. The model consists in superposing a short-correlated random component to the steady large-scale mean shear, and can thereby be integrated analytically. To reproduce the single-time orientation statistics obtained numerically, it is found that one has to properly account for the combined effect of the mean shear, for anisotropic velocity gradient fluctuations, and for the presence of persistent rotating structures in the flow that bias Lagrangian statistics. The model is then used to address two-time statistics. The notion of tumbling rate is extended to diffusive dynamics by introducing the stationary probability flux of the rods unfolded angle. The model is found to reproduce the long-term effects of an average shear on the mean and the variance of the fibres angular increment. Still, it does not reproduce an intricate behaviour observed in numerics for intermediate times: the unfolded angle is there very similar to a L\'evy walk with distributions of increments displaying intermediate power-law tails.

Lateral response of a layered material with interlayer friction

ABSTRACT We investigate the mechanical properties of a layered material with interlayer friction. We propose a model that contains lateral elasticity and interlayer friction to obtain the response function both in the Fourier and real spaces. By investigating how the internal deformation is laterally induced due to the applied surface displacement, we find that it is transmitted into the material with an apparent phase difference. We also obtain the effective complex modulus of the layered material and show that it exhibits an intermediate power-law behavior in the low-frequency regime. Our results can be used to estimate the internal deformation of layered materials that exists on various different scales.

Signatures of the spatial extent of plastic events in the yielding transition in amorphous solids.

Amorphous solids are yield stress materials that flow when a sufficient load is applied. Their flow consists of periods of elastic loading interrupted by rapid stress drops, or avalanches, coming from microscopic rearrangements known as shear transformations (STs). Here we show that the spatial extent of avalanches in a steadily sheared amorphous solid has a profound effect on the distribution of local residual stresses that in turn determines the stress drop statistics. As reported earlier, the most unstable sites are located in a flat "plateau" region that decreases with system size. While the entrance into the plateau is set by the lower cutoff of the mechanical noise produced by individual STs, the departure from the usually assumed power-law (pseudogap) form of the residual stress distribution comes from far field effects related to spatially extended rearrangements. Interestingly, we observe that the average residual stress of the weakest sites is located in an intermediate power-law regime between the pseudogap and the plateau regimes, whose exponent decreases with system size. Our findings imply a new scaling relation linking the exponents characterizing the avalanche size and residual stress distributions.

A Rational Friction Factor Correlation for Laminar Fully Developed Pipe Flows of Shear-Thinning Non-Newtonian Fluids

Abstract A friction factor correlation for laminar, hydrodynamically fully developed pipe flows of shear-thinning non-Newtonian fluids is derived through analysis and asymptotic considerations. The specific non-Newtonian fluid model used is the extended modified power law model, which is functionally equivalent to the Cross model, and spans the entire shear rate range from the low to the high shear rate Newtonian regions, including the intermediate power law region. The friction factor correlation is a function of three dimensionless parameters and has an explicit form, making it well-suited to design calculations. The overall accuracy of the correlation is 6.6%, though it is much better than this in most cases. Graphical results for the correlation, and deviations with respect to high-accuracy numerical calculations are presented and discussed.

A random walk Monte Carlo simulation study of COVID-19-like infection spread

Recent analysis of early COVID-19 data from China showed that the number of confirmed cases followed a subexponential power-law increase, with a growth exponent of around 2.2 (Maier and Brockmann, 2020). The power-law behavior was attributed to a combination of effective containment and mitigation measures employed as well as behavioral changes by the population. In this work, we report a random walk Monte Carlo simulation study of proximity-based infection spread. Control interventions such as lockdown measures and mobility restrictions are incorporated in the simulations through a single parameter, the size of each step in the random walk process. The step size l is taken to be a multiple of 〈r〉, which is the average separation between individuals. Three temporal growth regimes (quadratic, intermediate power-law and exponential) are shown to emerge naturally from our simulations. For l=〈r〉, we get intermediate power-law growth exponents that are in general agreement with available data from China. On the other hand, we obtain a quadratic growth for smaller step sizes l≲〈r〉∕2, while for large l the growth is found to be exponential. We further performed a comparative case study of early fatality data (under varying levels of lockdown conditions) from three other countries, India, Brazil and South Africa. We show that reasonable agreement with these data can be obtained by incorporating small-world-like connections in our simulations.

Light and heavy particles on a fluctuating surface: Bunchwise balance, irreducible sequences, and local density-height correlations

We study the early time and coarsening dynamics in the Light-Heavy model, a system consisting of two species of particles ($light$ and $heavy$) coupled to a fluctuating surface (described by tilt fields). The dynamics of particles and tilts are coupled through local update rules, and are known to lead to different ordered and disordered steady state phases depending on the microscopic rates. We introduce a generalized balance mechanism in non-equilibrium systems, namely $bunchwise~balance$, in which incoming and outgoing transition currents are balanced between groups of configurations. This allows us to exactly determine the steady state in a subspace of the phase diagram of this model. We introduce the concept of $irreducible~sequences$ of interfaces and bends in this model. These sequences are non-local, and we show that they provide a coarsening length scale in the ordered phases at late times. Finally, we propose a $local$ correlation function ($\mathcal{S}$) that has a direct relation to the number of irreducible sequences, and is able to distinguish between several phases of this system through its coarsening properties. Starting from a totally disordered initial configuration, $\mathcal{S}$ displays an initial linear rise and a broad maximum. As the system evolves towards the ordered steady states, $\mathcal{S}$ further exhibits power law decays at late times that encode coarsening properties of the approach to the ordered phases. Focusing on early time dynamics, we posit coupled mean-field evolution equations governing the particles and tilts, which at short times are well approximated by a set of linearized equations, which we solve analytically. Beyond a timescale set by a lattice cutoff and preceding the onset of coarsening, our linearized theory predicts the existence of an intermediate power-law stretch, which we also find in simulations of the ordered regime of the system.

Understanding partonic energy loss from measured light charged particles and jets in PbPb collisions at LHC energies

We perform a comprehensive study of partonic energy loss reflected in the nuclear modification factors of charged particles and jets measured in PbPb collisions at = 2.76 and 5.02 TeV in wide transverse momentum (p T) and centrality range. The p T distributions in pp collisions are fitted with a modified power law and the nuclear modification factor in PbPb collisions can be obtained using effective shift (Δp T) in the spectrum measured at different centralities. Driven by physics consideration, the functional form of energy loss given by Δp T can be assumed as power law with different power indices in three different p T regions. The power indices and the boundaries of three p T regions are obtained by fitting the measured nuclear modification factor as a function of p T in all collision centralities simultaneously. The energy loss in different collisions centralities are described in terms of fractional power of number of participants. It is demanded that the power law functions in three p T regions and their derivatives are continuous at the p T boundaries. The Δp T for light charged particles is found to increase linearly with p T in low p T region below ∼5–6 GeV c−1 and approaches a constant value in high p T region above ∼22–29 GeV c−1 with an intermediate power law connecting the two regions. The method is also used for jets and it is found that for jets, the Δp T increases approximately linearly even at very high p T.

Differences in power law growth over time and indicators of COVID-19 pandemic progression worldwide

Error analysis and data visualization of positive COVID-19 cases in 27 countries have been performed up to August 8, 2020. This survey generally observes a progression from early exponential growth transitioning to an intermediate power-law growth phase, as recently suggested by Ziff and Ziff. The occurrence of logistic growth after the power-law phase with lockdowns or social distancing may be described as an effect of avoidance. A visualization of the power-law growth exponent over short time windows is qualitatively similar to the Bhatia visualization for pandemic progression. Visualizations like these can indicate the onset of second waves and may influence social policy.

Photon waiting-time distributions: A keyhole into dissipative quantum chaos.

Open quantum systems can exhibit complex states, for which classification and quantification are still not well resolved. The Kerr-nonlinear cavity, periodically modulated in time by coherent pumping of the intracavity photonic mode, is one of the examples. Unraveling the corresponding Markovian master equation into an ensemble of quantum trajectories and employing the recently proposed calculation of quantum Lyapunov exponents [I. I. Yusipov et al., Chaos 29, 063130 (2019)], we identify "chaotic" and "regular" regimes there. In particular, we show that chaotic regimes manifest an intermediate power-law asymptotics in the distribution of photon waiting times. This distribution can be retrieved by monitoring photon emission with a single-photon detector so that chaotic and regular states can be discriminated without disturbing the intracavity dynamics.

Theory for IMPS on Rough and Finite Fractal Dye Sensitized Solar Cell

A generalized theoretical model of intensity modulated photocurrent spectroscopy (IMPS) for the random morphology in a dye sensitized solar cell (DSSC) under uniform illumination is developed. The generalized IMPS expression for the disordered semiconducting/conducting glass interface of a DSSC is obtained in term of power spectral density of roughness. Influence of surface roughness on the dynamic response of DSSC originate due to the coupling of characteristic phenomenological and morphological length scales. A detailed analysis of IMPS response is performed over finite self- affine fractals to highlight roughness induced anomalies and cause of photocurrent enhancement. The IMPS of a rough DSSC exhibit three characteristic frequency regimes: lifetime of charge carrier dependent low frequency regime, surface irregularity dependent intermediate power-law regime and diffusion controlled high frequency regime. Finally, our theory facilitates the understanding of dynamics and kinetics of charge carriers under the influence of ubiquitous surface disorder.

Genetic noise mechanism for power-law switching in bacterial flagellar motors

Switching of the direction of flagella rotations is the key control mechanism governing the chemotactic activity of E. coli and many other bacteria. Power-law distributions of switching times are most peculiar because their emergence cannot be deduced from simple thermodynamic arguments. Recently, it was suggested that by adding finite-time correlations into Gaussian fluctuations regulating the energy height of the barrier between the two rotation states, it is possible to generate switching statistics with an intermediate power-law asymptotics. By using a simple model of a regulatory pathway, we demonstrate that the required amount of correlated ‘noise’ can be produced by finite number fluctuations of reacting protein molecules, a condition common to the intracellular chemistry. The corresponding power-law exponent appears as a tunable characteristic controlled by parameters of the regulatory pathway network such as the equilibrium number of molecules, sensitivities, and the characteristic relaxation time.

Mechanism-based modeling of solute strengthening: Application to thermal creep in Zr alloy

In this work, a crystallographic thermal creep model is proposed for Zr alloys that accounts for the hardening contribution of solutes via their time-dependent pinning effect on dislocations. The core-diffusion model proposed by Soare and Curtin (2008a) is coupled with a recently proposed constitutive modeling framework (Wang et al., 2017, 2016) accounting for the heterogeneous distribution of internal stresses within grains. The Coble creep mechanism is also included. This model is, in turn, embedded in the effective medium crystallographic VPSC framework and used to predict creep strain evolution of polycrystals under different temperature and stress conditions. The simulation results reproduce the experimental creep data for Zircaloy-4 and the transition between the low (n∼1), intermediate (n∼4) and high (n∼9) power law creep regimes. This is achieved through the dependence on local aging time of the solute-dislocation binding energy. The anomalies in strain rate sensitivity (SRS) are discussed in terms of core-diffusion effects on dislocation junction strength.The mechanism-based model captures the primary and secondary creep regimes results reported by Kombaiah and Murty (2015a, 2015b) for a comprehensive set of testing conditions covering the 500–600 °C interval, stresses spanning 14–156 MPa, and steady state creep rates varying between 1.5·10−9s−1 to 2·10−3s−1. There are two major advantages to this model with respect to more empirical ones used as constitutive laws for describing thermal creep of cladding: 1) specific dependences on the nature of solutes and their concentrations are explicitly accounted for; 2) accident conditions in reactors, such as RIA and LOCA, usually take place in short times, and deformation takes place in the primary, not the steady-state creep stage. As a consequence, a model that accounts for the evolution with time of microstructure is more reliable for this kind of simulation.

Control of properties of nanocomposites bio-based collagen and cellulose nanocrystals

Collagen is an important biomaterial because it has many applications in the biomedical sector. However, the high hydrophilicity of collagen (COL) leads to easy swelling. Thus, controlling this property is highly desirable. In this work, cellulose nanocrystals (CNCs) dispersed in glycerol (GLI) were incorporated in the matrix collagen to tailor the hydrophilicity and mechanical properties. Study of the hydrophilicity of the bio-based nanocomposite was evaluated by contact angle measurement and thermogravimetric analysis. Mechanical analyses showed that CNCs are excellent reinforcing fillers to the collagen matrix. Synchrotron small-angle X-ray scattering was employed to investigate the nanostructures of COL/GLI/CNC nanocomposites and CNC water dispersion. CNC in concentrations up to 1 wt% presents an intermediate shape between a rod and a plane with a 9.34-nm radius of gyration (R g). Bio-based nanocomposites present two different structural levels with two types of particles with very different R gs. At the intermediate power-law regime, a large-scale mass fractal aggregate is observed. In the high-power-law regime, it is observed scattering from primary particles smaller than 1 nm. As the CNC concentration increases, the original particle distorts from a rod to a plate. The cytotoxicity assay indicates that the collagen and nanocomposites did not affect the cell viability of rat calvarial cells in vitro.

Reversible and irreversible aggregation of magnetic liposomes.

Understanding stabilization and aggregation in magnetic nanoparticle systems is crucial to optimizing the functionality of these systems in real physiological applications. Here we address this problem for a specific, yet representative, system. We present an experimental and analytical study on the aggregation of superparamagnetic liposomes in suspension in the presence of a controllable external magnetic field. We study the aggregation kinetics and report an intermediate time power law evolution and a long time stationary value for the average aggregate diffusion coefficient, both depending on the magnetic field intensity. We then show that the long time aggregate structure is fractal with a fractal dimension that decreases upon increasing the magnetic field intensity. By scaling arguments we also establish an analytical relation between the aggregate fractal dimension and the power law exponent controlling the aggregation kinetics. This relation is indeed independent on the magnetic field intensity. Despite the superparamagnetic character of our particles, we further prove the existence of a population of surviving aggregates able to maintain their integrity after switching off the external magnetic field. Finally, we suggest a schematic interaction scenario to rationalize the observed coexistence between reversible and irreversible aggregation.

Melting of Three-Sublattice Order in Easy-Axis Antiferromagnets on Triangular and Kagome Lattices.

When the constituent spins have an energetic preference to lie along an easy axis, triangular and kagome lattice antiferromagnets often develop long-range order that distinguishes the three sublattices of the underlying triangular Bravais lattice. In zero magnetic field, this three-sublattice order melts either in a two-step manner, i.e., via an intermediate phase with power-law three-sublattice order controlled by a temperature-dependent exponent η(T)∈(1/9,1/4), or via a transition in the three-state Potts universality class. Here, I predict that the uniform susceptibility to a small easy-axis field B diverges as χ(B)∼|B|^{-[(4-18η)/(4-9η)]} in a large part of the intermediate power-law ordered phase [corresponding to η(T)∈(1/9,2/9)], providing an easy-to-measure thermodynamic signature of two-step melting. I also show that these two melting scenarios can be generically connected via an intervening multicritical point and obtain numerical estimates of multicritical exponents.

Dynamics of PPI Dendrimers: A Study by Dielectric and2H NMR Spectroscopy and by Field-Cycling1H NMR Relaxometry

We investigate bulk poly(propyleneimine) dendrimers of generation (G) 2–5 by dielectric spectroscopy (DS), solid-state 2H NMR, and field-cycling 1H NMR relaxometry (FC 1H NMR) in a large temperature range (120–400 K). Three relaxation processes are identified by DS: a main (α-) relaxation (T > Tg) and two secondary processes (T < Tg). The α-process exhibits a super-Arrhenius temperature dependence typical of glass-forming liquids and changes only weakly with G, yielding Tg ∼ 200 K. The temperature dependence of the secondary relaxations is governed by an Arrhenius law. While one secondary process exhibits features characteristic for glasses, the other is atypical. Its time constant is virtually independent of G, and its spectral width does not increase with lowering temperature as is usually observed for sub-Tg relaxations. Regarding FC 1H NMR probing the dispersion of the spin–lattice rate R1 in the frequency range 200 Hz–30 MHz, transformation to the susceptibility representation, χ″(ω) ≡ ωR1(ω), and applying frequency–temperature superposition, an effective frequency range of 9 decades is covered by a master curve χ″(ωτα). In addition to the segmental time τα(T), which complements the results from DS up to high temperatures, a longer terminal relaxation τt(T) is identified. In between, an intermediate power-law regime is observed in χ″(ωτα) with an exponent of about 0.8. The broad relaxation spectrum is attributed to local dynamics, breathing modes, and overall tumbling and diffusion of the dendrimer molecule. In the low-frequency limit, R1(ω) is determined by intermolecular relaxation from which the molar mass dependence of the translational diffusion coefficient can be estimated. We find D(M) ∝ M–1.2±0.2.

Stochastic Wilson–Cowan models of neuronal network dynamics with memory and delay

We consider a simple Markovian class of the stochastic Wilson–Cowan type models of neuronal network dynamics, which incorporates stochastic delay caused by the existence of a refractory period of neurons. From the point of view of the dynamics of the individual elements, we are dealing with a network of non-Markovian stochastic two-state oscillators with memory, which are coupled globally in a mean-field fashion. This interrelation of a higher-dimensional Markovian and lower-dimensional non-Markovian dynamics is discussed in its relevance to the general problem of the network dynamics of complex elements possessing memory. The simplest model of this class is provided by a three-state Markovian neuron with one refractory state, which causes firing delay with an exponentially decaying memory within the two-state reduced model. This basic model is used to study critical avalanche dynamics (the noise sustained criticality) in a balanced feedforward network consisting of the excitatory and inhibitory neurons. Such avalanches emerge due to the network size dependent noise (mesoscopic noise). Numerical simulations reveal an intermediate power law in the distribution of avalanche sizes with the critical exponent around −1.16. We show that this power law is robust upon a variation of the refractory time over several orders of magnitude. However, the avalanche time distribution is biexponential. It does not reflect any genuine power law dependence.

Localization phase diagram of two-dimensional quantum percolation

We examine quantum percolation on a square lattice with random dilution up to q = 38% and energy 0.001 ≤ E ≤ 1.6 (measured in units of the hopping matrix element), using numerical calculations of the transmission coefficient at a much larger scale than previously. Our results confirm the previous finding that the two dimensional quantum percolation model exhibits localization-delocalization transitions, where the localized region splits into an exponentially localized region and a power-law localization region. We determine a fuller phase diagram confirming all three regions for energies as low as E = 0.1, and the delocalized and exponentially localized regions for energies down to E = 0.001. We also examine the scaling behavior of the residual transmission coefficient in the delocalized region, the power law exponent in the power-law localized region, and the localization length in the exponentially localized region. Our results suggest that the residual transmission at the delocalized to power-law localized phase boundary may be discontinuous, and that the localization length is likely not to diverge with a power-law at the exponentially localized to power-law localized phase boundary. However, further work is needed to definitively assess the characters of the two phase transitions as well as the nature of the intermediate power-law regime.