Power-law Growth Research Articles

Kinetics of domain growth in Ising systems with bond disorder at regularly selected sites

We study the structure, phase behaviour and growth laws in Ising ferromagnets and binary mixtures with bond disorder introduced at regularly selected sites for a critical quench. The results presented here are from extensive Monte Carlo simulations on two-dimensional Ising systems. Domain growth in a ferromagnet is modelled by using nonconserved spin-flip (Glauber) kinetics, and phase separation in a binary (AB) mixture is modelled by conserved spin-exchange (Kawasaki) kinetics. In both cases, we observed that the domain growth law is consistent with the respective power-law growth with the variable growth exponent that depends on the number of disordered sites. The nonconserved Ising system follows dynamical scaling for all the number of disordered sites studied here; however, a small deviation is noticed for $$r\rightarrow 0$$ at a large number of disordered sites. Whereas, for the conserved case, dynamical scaling deviates from the master curve after a reasonable number of disordered lattice sites; we notice the formation of lamellar evolution morphology at a higher number of disordered sites at later times.

The COVID-19 pandemic: growth patterns, power law scaling, and saturation

More and more countries are showing a significant slowdown in the number of new COVID-19 infections due to effective governmentally instituted lockdown and social distancing measures. We have analyzed the growth behavior of the top 25 most affected countries by means of a local slope analysis and found three distinct patterns that individual countries follow depending on the strictness of the lockdown protocols: rise and fall, power law, or logistic. For countries showing power law growth we have determined the scaling exponents. For countries that showed a strong slowdown in the rate of infections we have extrapolated the expected saturation of the total number of infections and the expected final date. Three different extrapolation methods (logistic, parabolic, and cutoff power law) were used. All methods agree on the order of magnitude of saturation and end dates. Global infection rates are analyzed with the same methods. The relevance and accuracy of these extrapolations is discussed.

Interface evolution in phase transformations ruled by nucleation and growth

An analytical model for the evolution of the boundary of the new phase in transformations ruled by nucleation and growth is presented. Both homogeneous and heterogeneous nucleation have been considered: The former includes transformations in 2D and 3D space and the latter nucleation and growth on flat solid substrate. The theory is formulated for the general case of spatially correlated nuclei, arbitrary nucleation rate and power growth law of nuclei. In the case of heterogeneous nucleation, spheroidal nuclei have been assumed and the dependence of the kinetics on contact angle investigated. The validity of the present approach is deemed through comparison with experimental data from literature which also comprise oxide growth by ALD (Atomic Layer Deposition) metal electrodeposition at solid substrate and alloy recrystallization.

A stretched logistic equation for pandemic spreading.

In this brief work we present a novel approach to the logistic dynamics of populations and epidemic spreading that can take into account of the complex nature of such a process in several real situations, where due to different agents the dynamics is no longer characterized by a single characteristic timescale, but conversely by a distribution of time scales, rendered via a time-dependent growth rate. In detail, a differential equation containing a power-law time dependent growth rate is proposed, whose solution, named Stretched Logistic Function, provides a modified version of the usual logistic function. The model equation is inspired by and applied to the recent spreading on COVID-19 disease in Italy, showing how the real dynamics of infection spreading is characterized by a time dependent dynamics. A speculative discussion of the Stretched Logistic Function in relation to diffusion processes is attempted.

Behavior of microbubbles in homogeneous stratified turbulence

Dynamics behavior of rising microbubbles in homogeneous stratified turbulence is investigated by direct numerical simulation. Stratification creates a predominantly horizontal oscillatory motion in a fluid, inducing bubbles' zigzag pattern. Despite this oscillating motion, the horizontal dispersion of a single bubble is rather suppressed. The horizontal separation of paired bubbles displays a power-law growth in the Batchelor and Richardson regimes.

Bounding the number of self-avoiding walks: Hammersley–Welsh with polygon insertion

Let $c_n = c_n(d)$ denote the number of self-avoiding walks of length $n$ starting at the origin in the Euclidean nearest-neighbour lattice $\mathbb{Z}^d$. Let $\mu = \lim_n c_n^{1/n}$ denote the connective constant of $\mathbb{Z}^d$. In 1962, Hammersley and Welsh [HW62] proved that, for each $d \geq 2$, there exists a constant $C > 0$ such that $c_n \leq \exp(C n^{1/2}) \mu^n$ for all $n \in \mathbb{N}$. While it is anticipated that $c_n \mu^{-n}$ has a power-law growth in $n$, the best known upper bound in dimension two has remained of the form $n^{1/2}$ inside the exponential. The natural first improvement to demand for a given planar lattice is a bound of the form $c_n \leq \exp (C n^{1/2 - \epsilon})\mu^n$, where $\mu$ denotes the connective constant of the lattice in question. We derive a bound of this form for two such lattices, for an explicit choice of $\epsilon > 0$ in each case. For the hexagonal lattice $\mathbb{H}$, the bound is proved for all $n \in \mathbb{N}$; while for the Euclidean lattice $\mathbb{Z}^2$, it is proved for a set of $n \in \mathbb{N}$ of limit supremum density equal to one. A power-law upper bound on $c_n \mu^{-n}$ for $\mathbb{H}$ is also proved, contingent on a non-quantitative assertion concerning this lattice's connective constant.

Can India develop herd immunity against COVID-19?

World Health Organization declared the novel coronavirus disease 2019 (COVID-19) outbreak to be a public health crisis of international concern. Further, it provided advice to the global community that countries should place strong measures to detect disease early, isolate and treat cases, trace contacts and promote “social distancing” measures commensurate with the risk. This study analyses the COVID-19 infection data from the top 15 affected countries in which we observed heterogeneous growth patterns of the virus. Hence, this paper applies multifractal formalism on COVID-19 data with the notion that country-specific infection rates follow a power law growth behaviour. According to the estimated generalized fractal dimension curves, the effects of drastic containment measures on the pandemic in India indicate that a significant reduction of the infection rate as its population is concern. Also, comparison results with other countries demonstrate that India has less death rate or more immunity against COVID-19.

COVID-19 Pandemic: Power Law Spread and Flattening of the Curve.

In this paper, we analyze the real-time infection data of COVID-19 epidemic for nine nations. Our analysis is up to May 04, 2020. South Korea, China, Italy, France, Spain, and Germany have either flattened or close to flattening their epidemic curves. USA and Japan have transitioned to a linear regime, while India is still in a power-law phase. We argue that the transition from an exponential regime to a succession of power-law regimes is a good indicator for flattening of the epidemic curve. We also argue that lockdowns, long-term community transmission, and the transmission by asymptomatic carriers traveling long distances may be inducing the power-law growth of the epidemic.

City components–area relationship and diversity pattern: towards a better understanding of urban structure

Optimizing the distribution of urban facilities with scarcity urban land can improve the fairness of residents’ access to services. The lack of assessment on the diversity pattern of city components (i.e., urban facilities of all types of industry related to socioeconomic activities) limits the reasonable planning of emergence polycentric cities. Therefore, we took Shanghai, Hangzhou, and Xi’an City in China as cases. A total of 407,652 city components were analyzed to assess the spatial pattern of components diversity through the nested sampling design and diversity assessment methods of ecology. Results showed that the city components–area relationship (CAR, i.e., the number of component types increases as the sampled area) had a power-law growth with land area, which can be a reference to construct reasonable urban sub-centers with sufficient components diversity. The individual density and components richness decreased on the urban–rural gradients, which significantly positive related to population density, land price, and building cover. Furthermore, the similarity of component types decreased with increasing geographic distance, indicating that there is an agglomeration distribution of industrial diversity. This study provides a new insight into applying ecological methods to explore the urban structure, and further supports the sustainable planning and development of cities.

Phase Separation by Entanglement of Active Polymerlike Worms.

We investigate the aggregation and phase separation of thin, living T. tubifex worms that behave as active polymers. Randomly dispersed active worms spontaneously aggregate to form compact, highly entangled blobs, a process similar to polymer phase separation, and for which we observe power-law growth kinetics. We find that the phase separation of active polymerlike worms does not occur through Ostwald ripening, but through active motion and coalescence of the phase domains. Interestingly, the growth mechanism differs from conventional growth by droplet coalescence: the diffusion constant characterizing the random motion of a worm blob is independent of its size, a phenomenon that can be explained from the fact that the active random motion arises from the worms at the surface of the blob. This leads to a fundamentally different phase-separation mechanism that may be unique to active polymers.

Strong correlations between power-law growth of COVID-19 in four continents and the inefficiency of soft quarantine strategies.

In this work, we analyze the growth of the cumulative number of confirmed infected cases by a novel coronavirus (COVID-19) until March 27, 2020, from countries of Asia, Europe, North America, and South America. Our results show that (i) power-law growth is observed in all countries; (ii) by using the distance correlation, the power-law curves between countries are statistically highly correlated, suggesting the universality of such curves around the world; and (iii) soft quarantine strategies are inefficient to flatten the growth curves. Furthermore, we present a model and strategies that allow the government to reach the flattening of the power-law curves. We found that besides the social distancing of individuals, of well known relevance, the strategy of identifying and isolating infected individuals in a large daily rate can help to flatten the power-laws. These are the essential strategies followed in the Republic of Korea. The high correlation between the power-law curves of different countries strongly indicates that the government containment measures can be applied with success around the whole world. These measures are scathing and to be applied as soon as possible.

Ordering kinetics in a q-state random-bond clock model: Role of vortices and interfaces.

In this article, we present a Monte Carlo study of phase transition and coarsening dynamics in the nonconserved two-dimensional random-bond q-state clock model (RBCM) deriving from a pure clock model [Chatterjee etal., Phys. Rev. E 98, 032109 (2018)10.1103/PhysRevE.98.032109]. Akin to the pure clock model, RBCM also passes through two different phases when quenched from a disordered initial configuration representing at infinite temperature. Our investigation of the equilibrium phase transition affirms that both upper (T_{c}^{1}) and lower (T_{c}^{2}) phase transition temperatures decrease with bond randomness strength ε. Effect of ε on the nonequilibrium coarsening dynamics is investigated following independent rapid quenches in the quasi-long-range ordered (QLRO, T_{c}^{2}<T<T_{c}^{1}) regime and long-range ordered (LRO, T<T_{c}^{2}) regime at temperature T. We report that the dynamical scaling of the correlation function and structure factor is independent of ε and the presence of quenched disorder slows down domain coarsening. Coarsening dynamics in both LRO and QLRO regimes are further characterized by power-law growth with disorder-dependent exponents within our simulation timescales. The growth exponents in the LRO regime decrease from 0.5 in the pure case to 0.22 in the maximum disordered case, whereas the corresponding change in the QLRO regime happens from 0.45 to 0.38. We further explored the coarsening dynamics in the bond-diluted clock model and, in both the models, the effect of the disorder is more significant for the quench in the LRO regime compared to the QLRO regime.

Role of many-body correlation in slow dynamics of glass-forming liquids: intrinsic or perturbative

Upon cooling or densification, a supercooled liquid shows drastic slowing down toward its glass-transition point. The physical mechanism behind this slow glassy dynamics has been a matter of discussion for a long time, but there has still been no consensus on its origin. Recently, we have found that for systems mainly interacting with steric repulsions, glassy structural order (or, angular order) generally develops upon cooling and its correlation length, , grows as ( is the bare correlation length, T is the temperature, T0 is the hypothetical ideal glass transition, and d is the spatial dimensionality). This ordering is difficult to detect by two-body density correlation since it is a consequence of sterically-induced (entropically-driven) many-body correlation that lowers local free energy. Interestingly, the power-law growth of with the exponent of 2/d is reminiscent of the Ising criticality. We also find that the structural relaxation time diverges as (: the microscopic relaxation time, K is a fragility index, is the Boltzmann constant), suggesting that glass transition is a consequence of Ising-like criticality with growing activation energy. Unlike ordinary critical phenomena, the activation energy of particle motion increases in proportion to the root of the correlation volume of , implying that the particle motion is strongly correlated in that volume. This relation indicates that the impact of spatial fluctuations of the order parameter on slow dynamics is not perturbative but intrinsic. Although we need further study to confirm our claim, we hope that the discussion in this article would provide a good starting point for further consideration of the physical nature of glass transition.

Chiral-filament self-assembly on curved manifolds.

Rod-like and banana-shaped proteins, like BAR-domain proteins and MreB proteins, adsorb on membranes and regulate the membrane curvature. The formation of large filamentous complexes of these proteins plays an important role in cellular processes like membrane trafficking, cytokinesis and cell motion. We propose a simplified model to investigate such curvature-dependent self-assembly processes. Anisotropic building blocks, modeled as trimer molecules, which have a preferred binding site, interact via pair-wise Lennard-Jones potentials. When several trimers assemble, they form an elastic ribbon with an intrinsic curvature and twist, controlled by bending and torsional rigidity. For trimer self-assembly on the curved surface of a cylindrical membrane, this leads to a preferred spatial orientation of the ribbon. We show that these interactions can lead to the formation of helices with several windings around the cylinder. The emerging helix angle and pitch depend on the rigidities and the intrinsic curvature and twist values. In particular, a well-defined and controllable helix angle emerges in the case of equal bending and torsional rigidity. The dynamics of filament growth is characterized by three regimes, in which filament length increases with the power laws tz in time, with z≃ 3/4, z = 1/2, and z = 0 for short, intermediate, and long times, respectively. A comparison with the solutions of the Smoluchowski aggregation equation allows the identification of the underlying mechanism in the short-time regime as a crossover from size-independent to diffusion-limited aggregation. Thus, helical structures, as often observed in biology, can arise by self-assembly of anisotropic and chiral proteins.

Power Law Growth and Delayed Feedbacks in Socio‐Hydrological Systems

Water infrastructure dynamics result from coupled social and physical hydrological processes embedded in “socio‐hydrological systems” (SHSs). Freshwater fuels socioeconomic activity, which in turn exerts pressure on water resources through increased water demand and water quality degradation. Many factors emphasize the need for quantitative tools to predict the future of these systems, including SHS failures due to growing water scarcity from population growth and climate change. However, unfolding the interactions between social and hydrological factors continues to resist theoretical treatment, impeding progress toward a predictive framework. To resolve this issue, we propose and evaluate time delays as surrogates for the social dynamics in SHS models. This approach permits the development of models that describe SHS dynamics in terms of observable, physical, hydrological state variables. We apply this approach to two case studies: (a) reservoir storage capacity growth in the world and the U.S. Water Resources Regions and (b) global water withdrawals. Substantial variability was identified in empirical estimates of growth rates and time delays. Reservoir construction typically follows a saturating, logistic curve with periodic, punctuated equilibria, separated by delays ranging from 5 to 20 years both globally and regionally. In contrast, global water withdrawal data display faster‐than‐exponential growth, characteristic of a positive feedback through which water use drives further water development. Historical analysis suggests that growth trends in water resources systems are superimposed by recurring periods of innovation and inactivity which are indicative of slow memory dissipation and delayed effects of past water use on current water infrastructure decisions.

Power-law entanglement growth from typical product states

Generic quantum many-body systems typically show a linear growth of the entanglement entropy after a quench from a product state. While entanglement is a property of the wave function, it is generated by the unitary time evolution operator and is therefore reflected in its increasing complexity as quantified by the operator entanglement entropy. Using numerical simulations of a static and a periodically driven quantum spin chain, we show that there is a robust correspondence between the entanglement entropy growth of typical product states with the operator entanglement entropy of the unitary evolution operator, while special product states, e.g. $\sigma_z$ basis states, can exhibit faster entanglement production. In the presence of a disordered magnetic field in our spin chains, we show that both the wave function and operator entanglement entropies exhibit a power-law growth with the same disorder-dependent exponent, and clarify the apparent discrepancy in previous results. These systems, in the absence of conserved densities, provide further evidence for slow information spreading on the ergodic side of the many-body localization transition.

Electrified cone formation in perfectly conducting viscous liquids: Self-similar growth irrespective of Reynolds number

Above a critical field strength, the free surface of an electrified, perfectly conducting viscous liquid, such as a liquid metal, is known to develop an accelerating protrusion resembling a cusp with a conic tip. Field self-enhancement from tip sharpening is reported to generate divergent power law growth in finite time of the forces acting in that region. Previous studies have established that tip sharpening proceeds via a self-similar process in two distinguished limits—the Stokes regime and the inviscid regime. Using finite element simulations to track the shape and forces acting at the tip of an electrified protrusion in a perfectly conducting Newtonian liquid, we demonstrate that the conic tip always undergoes self-similar growth irrespective of the Reynolds number. The blowup exponents at the conic apex for all terms in the Navier-Stokes equation and the normal stress boundary condition at the moving interface reveal the dominant forces at play as the Reynolds number increases. Rescaling of the tip shape by the power law representing the divergence in capillary stress at the apex yields an excellent collapse onto a universal cone shape with an interior half-angle dependent on the Maxwell stress. The rapid acceleration of the liquid interface also generates a thin interfacial boundary layer characterized by a significant rate of strain. Additional details of the modeled flow, applicable to cone growth in systems such as liquid metal ion sources, help dispel prevailing misconceptions that dynamic cones resemble conventional Taylor cones or that viscous stresses at a finite Reynolds number can be neglected.

Dendrite growth and grain “coarsening” in an undercooled CoNi equiatomic alloy

The CoNi equiatomic alloy was undercooled and spontaneously solidified using the glass fluxing method. The dendrite growth behaviors were recorded by a high-speed high-resolution camera. Three dendrite growth stages (i.e., the power law growth stage when undercooling ΔT<120 K, the plateau growth stage when 120 K<ΔT<135 K and the linear law growth stage when ΔT>135 K) were found and well predicted by a dendrite growth model. For the as-solidified microstructures, the average grain size increases with undercooling, indicating that unique grain “coarsening” instead of common spontaneous grain refinement occurs. Such a grain “coarsening” phenomenon was ascribed to stress accumulation upon rapid solidification. Furthermore, a plenty of sub-grains were observed within the abnormal coarse grains. From the electron back-scattering diffraction results, the angle difference between neighbouring sub-grains was less than 6°. The current work might be helpful for not only theoretically understanding dendrite growth but also practically controlling the non-equilibrium microstructures.

Precursors of the spin glass transition in three dimensions

We study energy landscape and dynamics of the three-dimensional Heisenberg spin glass model in the paramagnetic phase, i.e. for temperature T larger than the critical temperature . The landscape is non-trivially related to the equilibrium states even in the high-temperature phase, and reveals an onset of non-trivial behavior at a temperature , which is also seen through the behavior of the thermoremanent magnetization. We also find a power-law growth of the relaxation times far from the spin-glass transition, indicating a dynamical crossover at a temperature , . The arising picture is reminiscent of the phenomenology of supercooled liquids, and poses questions on which mean-field models can describe qualitatively well the phenomenology in three dimensions. On the technical side, local energy minima are found with the Successive Overrelaxation algorithm, which reveals very efficient for energy minimization in this kind of models.

Standard and fractional Ornstein-Uhlenbeck process on a growing domain

We study normal diffusive and subdiffusive processes in a harmonic potential (Ornstein-Uhlenbeck process) on a uniformly growing or contracting domain. Our starting point is a recently derived fractional Fokker-Planck equation, which covers both the case of Brownian diffusion and the case of a subdiffusive continuous-time random walk (CTRW). We find a high sensitivity of the random walk properties to the details of the domain growth rate, which gives rise to a variety of regimes with extremely different behaviors. At the origin of this rich phenomenology is the fact that the walkers still move while they wait to jump, since they are dragged by the deterministic drift arising from the domain growth. Thus, the increasingly long waiting times associated with the aging of the subdiffusive CTRW imply that, in the time interval between two consecutive jumps, the walkers might travel over much longer distances than in the normal diffusive case. This gives rise to seemingly counterintuitive effects. For example, on a static domain, both Brownian diffusion and subdiffusive CTRWs yield a stationary particle distribution with finite width when a harmonic potential is at play, thus indicating a confinement of the diffusing particle. However, for a sufficiently fast growing or contracting domain, this qualitative behavior breaks down, and differences between the Brownian case and the subdiffusive case are found. In the case of Brownian particles, a sufficiently fast exponential domain growth is needed to break the confinement induced by the harmonic force; in contrast, for subdiffusive particles such a breakdown may already take place for a sufficiently fast power-law domain growth. Our analytic and numerical results for both types of diffusion are fully confirmed by random walk simulations.