Power-law Growth Research Articles

Moving mirrors, OTOCs and scrambling

We explore the physics of scrambling in the moving mirror models, in which a two-dimensional CFT is subjected to a time-dependent boundary condition. It is well-known that by choosing an appropriate mirror profile, one can model quantum aspects of black holes in two dimensions, ranging from Hawking radiation in an eternal black hole (for an “escaping mirror”) to the recent realization of Page curve in evaporating black holes (for a “kink mirror”). We explore a class of OTOCs in the presence of such a boundary and explicitly demonstrate the following primary aspects: First, we show that the dynamical CFT data directly affect an OTOC and maximally chaotic scrambling occurs for the escaping mirror for a large-c CFT with identity block dominance. We further show that the exponential growth of OTOC associated with the physics of scrambling yields a power-law growth in the model for evaporating black holes which demonstrates unitary dynamics in terms of a Page curve. We also demonstrate that, by tuning a parameter, one can naturally interpolate between an exponential growth associated with scrambling and a power-law growth in unitary dynamics. Our work explicitly exhibits the role of higher-point functions in CFT dynamics as well as the distinction between scrambling and Page curve. We also discuss several future possibilities based on this class of models.

Power-law-exponential interaction induced quantum spiral phases

We theoretically predict a kind of power-law-exponential (PLE) dipole-dipole interaction between quantum emitters in a 1D waveguide QED system. This unconventional long-range interaction is the combination of power-law growth and exponential decay couplings. Applying the PLE interaction to a spin model, we uncover the rich many-body phases. Most remarkably, we find that the PLE interaction can induce the ordered and critical spiral phases. These spiral phases emerge from the strong frustration generated by the power-law factor of the PLE interaction; hence they are absent for other types of long-range interaction, e.g., pure exponential and power-law decay interactions. Our work is also applicable for the higher-dimensional systems. It fundamentally broadens the realm of many-body physics and has significant applications in quantum simulation of strongly correlated matter. Published by the American Physical Society 2024

Power-law growth models explain incidences and sizes of pancreatic cancer precursor lesions and confirm spatial genomic findings.

Pancreatic ductal adenocarcinoma is a rare but lethal cancer. Recent evidence suggests that pancreatic intraepithelial neoplasia (PanIN), a microscopic precursor lesion that gives rise to pancreatic cancer, is larger and more prevalent than previously believed. Better understanding of the growth-law dynamics of PanINs may improve our ability to understand how a miniscule fraction makes the transition to invasive cancer. Here, using three-dimensional tissue mapping, we analyzed >1000 PanINs and found that lesion size is distributed according to a power law. Our data suggest that in bulk, PanIN size can be predicted by general growth behavior without consideration for the heterogeneity of the pancreatic microenvironment or an individual's age, history, or lifestyle. Our models suggest that intraductal spread and fusing of lesions drive our observed size distribution. This analysis lays the groundwork for future mathematical modeling efforts integrating PanIN incidence, morphology, and molecular features to understand tumorigenesis and demonstrates the utility of combining experimental measurement with dynamic modeling in understanding tumorigenesis.

Optimal operating conditions for the rapid growth of KH2PO4 crystal by numerical simulation

Numerical simulations of the turbulent hydrodynamics and concentration in the rapid growth of potassium dihydrogen phosphate (KH2PO4 or KDP) crystals have been performed. In this work, a quadratic power-law growth model is applied, concerning the mass transfer occurring at the crystal surface during the growth process under low supersaturations. The mass transfer boundary condition is discretized by using the Over-Relaxed approach. The effects of the growth temperature and rotation rate on surface supersaturation distribution and the diffusion layer are investigated, which are critically involved in the processes of surface morphological instability and inclusion formation. Due to the kinetic limitation, the response of different types of crystal surfaces to temperature varies, with a pyramidal surface about twice as high as a prismatic face. In addition, the contour map of the growth rate combined with the coefficient of variation (C.V) for various temperatures and rotation rates is used. Uniformity of surface supersaturation is provided by lowering the growth temperature and increasing the rotation rate. Furthermore, the criteria of crystal growth strategies are proposed to optimize the process parameters and obtain favorable conditions for crystal growth, thereby facilitating defect-free and high-quality crystal formation at an appropriate growth rate.

Central limit theorem for the number of records in Fα-scheme

Consider the sequence {Xk, k ≥ 1} of independent identically distributed random variables whose distribution function is continuous. Then events of the type {Xi = Xj} have probability 0 if i ≠ j. Let L(1) = 1. For n ≥ 2, we define random variablesL(n) = inf{k > L(n − 1) : Xk > XL(n-1)}assuming that inf ∅ := +∞. The members of the sequence L = {L(n), n ≥ 1} are called moments of records constructed for {Xk, k ≥ 1}. Consider the sequence of random variables μ = {μ(n), n ≥ 1}, defined by the relationμ(n) = #{k : L(k) ≤ n}, n ≥ 1.It is clear that μ(n) – is the number of records that happened up to the moment n inclusive.In the work [10], the so-called Fα-scheme is considered for the first time, which is built using a given distribution function and a sequence of positive numbers {αk}. It is clear that Fαn(x) is the distribution function for each n ≥ 1. The set of independent random variables {Xn} is called the Fα scheme, if the distribution function of the random variable Xn is Fαn(x). If all αn are equal to each other, then the Fα scheme – is a set independent identically distributed random variables. If not all αn are equal to each other, then the Fα scheme – is a generalization of the classical case.This paper examines the assertions related to the fulfillment of the central limit theorem (CLT) for the number of records in the Fα-scheme of records. The method of finding exact asymptotic expressions for mathematical expectation and variance, which can be used to replace the real characteristics in CLT, is given.A specific example of power-law growth of exponents of the Fα-scheme was considered, and CLT is constructed only in terms of the moment of observation and the power of growth.The article contains 4 theorems with complete proof. Theorem 1 relates the mathematical expectation and variance to the accumulated intensity of the Fα-scheme. Theorem 2 establishes the implementation of CLT in general, and theorem 4 – for a specific case.

Bidirectional zigzag growth from clusters of active colloidal shakers

Driven or self-propelling particles moving in viscoelastic fluids recently emerged as a novel class of active systems showing a complex yet rich set of phenomena due to the non-Newtonian nature of the dispersing medium. Here we investigate the one-dimensional growth of clusters made of active colloidal shakers, which are realized by oscillating magnetic rotors dispersed within a viscoelastic fluid and at different concentrations of the dissolved polymer. These magnetic particles when actuated by an oscillating field display a flow profile similar to that of a shaker force dipole, i.e., without any net propulsion. We design a protocol to assemble clusters of colloidal shakers and induce their controlled expansion into elongated zigzag structures. We observe a power law growth of the mean chain length and use theoretical arguments to explain the measured 1/3 exponent. These arguments agree well with both experiments and particle based numerical simulations. Published by the American Physical Society 2024

Stochastic modeling of injection induced seismicity based on the continuous time random walk model

The spatiotemporal evolution of earthquakes induced by fluid injections into the subsurface can be erratic owing to the complexity of the physical process. To effectively mitigate the associated hazard and to draft appropriate regulatory strategies, a detailed understanding of how induced seismicity may evolve is needed. In this work, we build on the well-established continuous-time random walk (CTRW) theory to develop a purely stochastic framework that can delineate the essential characteristics of this process. We use data from the 2003 and 2012 hydraulic stimulations in the Cooper Basin geothermal field that induced thousands of microearthquakes to test and demonstrate the applicability of the model. Induced seismicity in the Cooper Basin shows all the characteristics of subdiffusion, as indicated by the fractional order power-law growth of the mean square displacement with time and broad waiting-time distributions with algebraic tails. We further use an appropriate master equation and the time-fractional diffusion equation to map the spatiotemporal evolution of seismicity. The results show good agreement between the model and the data regarding the peak earthquake concentration close to the two injection wells and the stretched exponential relaxation of seismicity with distance, suggesting that the CTRW model can be efficiently incorporated into induced seismicity forecasting.

Growth kinetics and morphology characterization of binary polymeric fluid under random photo-illumination.

We present a comprehensive study using dissipative particle dynamics simulations to investigate phase separation kinetics (PSK) in three-dimensional (3d) polymeric fluids under random photo-illumination. We consider two scenarios: polymer blends with active radicals at one end of each immiscible chain and block copolymer (BCP) melts with photosensitive bonds linking incompatible blocks. The phase separation (PS) is induced by temperature quench of the initial homogeneously mixed system. Simultaneously, the system experiences random photo-illumination, simulated by two concurrent random events: (a) the recombination of active radicals in polymer blends and (b) the breaking of photosensitive bonds in BCP chains. Variations in the bond-breaking probability, Pb, mimic the change in light intensity. The length scale follows power law growth, R(t) ∼ tϕ, where ϕ represents the growth exponent. Increasing Pb results in a gradual transition in growth kinetics from micro-PS to macro-PS, accompanied by corresponding transition probabilities for both systems. Micro-PSK dominates the evolution process at low Pb values. The scaling functions exhibit data overlap for most scaled distances, indicating the statistical self-similarity of evolving patterns. Our study enhances the understanding of PSK in polymeric fluids, revealing the impact of photosensitive bonds and active radicals. Furthermore, it suggests the potential for designing novel polymeric materials with desired properties.

Cloud atlas: navigating the multiphase landscape of tempestuous galactic winds

ABSTRACT Galaxies comprise intricate networks of interdependent processes which together govern their evolution. Central among these are the multiplicity of feedback channels, which remain incompletely understood. One outstanding problem is the understanding and modelling of the multiphase nature of galactic winds, which play a crucial role in galaxy formation and evolution. We present the results of three-dimensional magnetohydrodynamical simulations of tall–box interstellar medium (ISM) patches with clustered supernova-driven outflows. Dynamical fragmentation of the ISM during superbubble breakout seeds the resulting hot outflow with a population of cool clouds. We focus on analyzing and modelling the origin and properties of these clouds. Their presence induces large-scale turbulence, which, in turn, leads to complex cloud morphologies. Cloud sizes are well described by a power-law distribution and mass growth rates can be modelled using turbulent radiative mixing layer theory. Turbulence provides significant pressure support in the clouds, while magnetic fields only play a minor role. We conclude that many of the physical insights and analytic scalings derived from idealized small-scale simulations of turbulent radiative mixing layers and cloud–wind interactions are directly translatable and applicable to these larger scale cloud populations. This opens the door to developing effective subgrid recipes for their inclusion in global-scale galaxy models where they are unresolved.

Modeling tumor growth using fractal calculus: Insights into tumor dynamics

Important concepts like fractal calculus and fractal analysis, the sum of squared residuals, and Aikaike’s information criterion must be thoroughly understood in order to correctly fit cancer-related data using the proposed models. The fractal growth models employed in this work are classified in three main categories: Sigmoidal growth models (Logistic, Gompertz, and Richards models), Power Law growth model, and Exponential growth models (Exponential and Exponential-Lineal models)”. We fitted the data, computed the sum of squared residuals, and determined Aikaike’s information criteria using Matlab and the web tool WebPlotDigitizer. In addition, the research investigates “double-size cancer” in the fractal temporal dimension with respect to various mathematical models.

Recovery of low phosphorus iron from steel slag using secondary aluminum dross as the reductant

Iron extraction by carbothermic reduction is a promising technique for the recycling and valorization of steel slag, but phosphorus reduction and dissolution are inevitable and limit the value of the metal products. To solve this issue, in this work, a novel smelting reduction method was proposed to produce low phosphorus iron from steel slag by using secondary aluminum dross (SAD) as the reductant. The roles of modified basicity and SAD/slag in the iron recovery and phosphorus distribution behavior were investigated by various characterization and theoretical approaches, such as ICP-AES, XRD, SEM-EDX, FTIR, and FactSage. Owing to the fluxing action of Al2O3 and the supply of abundant free (O2-), the slag shows a fully melted and highly depolymerized state during reduction even when no slag modification is made, contributing to a high iron recovery (92.8 %). This is an advantage compared to other traditional reduction methods. Besides, the higher basicity under no modification significantly suppresses the reduction of P2O5 by reducing its activity coefficient, resulting in a decrease of 50 % in P content in the separated metals. It is also found that phosphorus recovery presents a power growth law with SAD/slag. The P content in the metals can be as low as 0.16 wt% by decreasing this ratio from 0.28 to 0.26. Although a sacrifice of iron recovery (∼6 %) is required for this purification, it shows great potential in producing high-quality metals from steel slag.

Superdiffusion-like behavior in zero-temperature coarsening of the d=3 Ising model

One key aspect of coarsening following a quench below the critical temperature is domain growth. For the non-conserved Ising model a power-law growth of domains of like spins with exponent alpha = 1/2 is predicted. Including recent work, it was not possible to clearly observe this growth law in the special case of a zero-temperature quench in the three-dimensional model. Instead a slower growth with alpha <1/2 was reported. We attempt to clarify this discrepancy by running large-scale Monte Carlo simulations on simple-cubic lattices with linear lattice sizes up to L=2048 employing an efficient GPU implementation. Indeed, at late times we measure domain sizes compatible with the expected growth law—but surprisingly, at still later times domains even grow superdiffusively, i.e., with alpha > 1/2. We argue that this new problem is possibly caused by sponge-like structures emerging at early times.

Development of Drop Size Distribution Model for Dropwise Condensation on a Superhydrophobic Surface

This study presents a mathematical model of drop size distribution during dropwise condensation on a superhydrophobic surface. The model is developed by combining a power law growth model, an exponentially decaying population model, and a Gaussian probability model for growth variations. The model is validated against experiment data, with correlations ranging from 88% to 94%. The growth model is shown to sufficiently describe the growth of drops from 0.02 mm to 0.1 mm but may be extrapolated to describe the growth of even smaller drops. The experiment data show that drop size distribution or frequency distribution of drops of different sizes varies significantly with time and may be considered pseudo-cyclic. The developed model, together with the sweep rate of drops, sufficiently describes this behavior and, consequently, may also be used to better estimate the heat transfer rate due to dropwise condensation.

The dynamics of debris streams from tidal disruption events: exact solutions, critical stream density, and hydrogen recombination

ABSTRACT A star destroyed by a supermassive black hole (SMBH) in a tidal disruption event (TDE) is transformed into a filamentary structure known as a tidally disrupted stellar debris stream. We show that when ideal gas pressure dominates the thermodynamics of the stream, there is an exact solution to the hydrodynamics equations that describes the stream evolution and accounts for self-gravity, pressure, the dynamical expansion of the gas, and the transverse structure of the stream. We analyse the stability of this solution to cylindrically symmetric perturbations, and show that there is a critical stream density below which the stream is unstable and is not self-gravitating; this critical density is a factor of at least 40–50 smaller than the stream density in a TDE. Above this critical density the stream is overstable, self-gravity confines the stream, the oscillation period is exponentially long, and the growth rate of the overstability scales as t1/6. The power-law growth and small power-law index of the overstability implies that the stream is effectively stable to cylindrically symmetric perturbations. We also use this solution to analyse the effects of hydrogen recombination, and suggest that even though recombination substantially increases the gas entropy, it is likely incapable of completely destroying the influence of self-gravity. We also show that the transient produced by recombination is far less luminous than previous estimates.

Hydrodynamic effects in kinetics of phase separation in binary fluids: Critical versus off-critical compositions.

Via hydrodynamics-preserving molecular dynamics simulations we study growth phenomena in a phase-separating symmetric binary mixture model. We quench high-temperature homogeneous configurations to state points inside the miscibility gap, for various mixture compositions. For compositions at the symmetric or critical value we capture the rapid linear viscous hydrodynamic growth due to advective transport of material through tubelike interconnected domains. For state points very close to any of the branches of the coexistence curve, the growth in the system, following nucleation of disconnected droplets of the minority species, occurs via a coalescence mechanism. Using state-of-the-art techniques, we have identified that these droplets, between collisions, exhibit diffusive motion. The value of the exponent for the power-law growth, related to this diffusive coalescence mechanism, has been estimated. While the exponent nicely agrees with that for the growth via the well-known Lifshitz-Slyozov particle diffusion mechanism, the amplitude is stronger. For the intermediate compositions we observe initial rapid growth that matches the expectations for viscous or inertial hydrodynamic pictures. However, at later times these types of growth cross over to the exponent that is decided by the diffusive coalescence mechanism.

Slow dynamics of a mobile impurity interacting with an Anderson insulator

We investigate dynamics of a single mobile impurity immersed in a bath of Anderson localized particles and focus on the regime of relatively strong disorder and interactions. In that regime, the dynamics of the system is particularly slow, suggesting, at short times, an occurrence of many-body localization. Considering longer time scales, we show that the latter is a transient effect and that, eventually, the impurity spreads sub-diffusively and induces a gradual delocalization of the Anderson insulator. The phenomenology of the system in the considered regime of slow dynamics includes a sub-diffusive growth of mean square displacement of the impurity, power-law decay of density correlation functions of the Anderson insulator and a power-law growth of entanglement entropy in the system. We observe a similar regime of slow dynamics also when the disorder in the system is replaced by a sufficiently strong quasi-periodic potential.

Surface premelting and melting of colloidal glasses.

The nature of liquid-to-glass transition is a major puzzle in science. A similar challenge exists in glass-to-liquid transition, i.e., glass melting, especially for the poorly investigated surface effects. Here, we assemble colloidal glasses by vapor deposition and melt them by tuning particle attractions. The structural and dynamic parameters saturate at different depths, which define a surface liquid layer and an intermediate glassy layer. The power-law growth of both layers and melting front behaviors at different heating rates are similar to crystal premelting and melting, suggesting that premelting and melting can be generalized to amorphous solids. The measured single-particle kinetics reveal various features and confirm theoretical predictions for glass surface layer.

Modelling epidemic growth models for lumpy skin disease cases in Thailand using nationwide outbreak data, 2021-2022.

Lumpy skin disease (LSD) is a transboundary disease affecting cattle and has a detrimental effect on the cattle industries in numerous countries in Africa, Europe and Asia. In 2021, LSD outbreaks have been reported in almost all of Thailand's provinces. Indeed, fitting LSD occurrences using mathematical models provide important knowledge in the realm of animal disease modeling. Thus, the objective of this study is to fit the pattern of daily new LSD cases and daily cumulative LSD cases in Thailand using mathematical models. The first- and second-order models in the forms of Lorentzian, Gaussian and Pearson-type VII models are used to fit daily new LSD cases whereas Richard's growth, Boltzmann sigmoidal and Power-law growth models are utilized to fit the curve of cumulative LSD cases. Based on the root-mean-squared error (RMSE) and Akaike information criterion (AIC), results showed that both first and second orders of Pearson-type VII models and Richard's growth model (RGM) were fit to the data better than other models used in the present study. The obtained models and their parameters can be utilized to describe the LSD outbreak in Thailand. For disease preparedness purposes, we can use the first order of the Pearson-type VII model to estimate the time of maximum infected cases occurring when the growth rate of infected cases starts to slow down. Furthermore, the period when the growth rate changes at a slower rate, known as the inflection time, obtained from RGM allows us to anticipate when the pandemic has peaked and the situation has stabilized. This is the first study that utilizes mathematical methods to fit the LSD epidemics in Thailand. This study offers decision-makers and authorities with valuable information for establishing an effective disease control strategy.

Effect of sample size on prognostic genes analysis in non-small cell lung cancer.

The identification of prognostic genes can help in the clinical management of non-small cell lung cancer (NSCLC). However, there is little overlap in the prognostic genes identified in different NSCLC studies. One reason for this may be the inadequate sample size. Here, the effect of sample size on prognostic genes analysis was investigated based on 515 stage II/III NSCLC cases from two cohorts detected by whole-exome sequencing. Prognostic genes analysis was repeatedly performed 100 times for each sample size level using random resampling methods. In stage II lung adenocarcinoma (LUAD) and lung squamous cell carcinoma (LUSC) cases from the TCGA Pan-Lung Cancer cohort, the number of statistically significant prognostic genes first increased with sample size in a power law, then fluctuated steadily, and finally decreased slightly. The power law growth curves were also observed in stage III LUAD and LUSC cases from the TCGA Pan-Lung Cancer cohort and stage III Chinese LUAD cases from the OncoSG cohort. The correlation R2 of the fitted power law growth curves were all greater than 0.99. In addition, at the sample size level where the number of prognostic genes peaked, the mean proportion of true prognostic genes in patients with stage II LUAD and LUSC was 28.32% and 23.12%, which could partly explain the little overlap in prognostic genes between reports. In conclusion, the number of prognostic genes takes a power law growth with the sample size in NSCLC, independent of histopathological subtype, race, and stage. These results also show how sample size affects the reliability of prognostic genes and will aid trial design for genomic mutation-based prognostic studies in NSCLC.

Basal boundary conditions for granular surface flows over fragile and brittle erodible beds

Many granular surface flows occur as shear flows of finite thickness, over erodible beds composed of the same granular material. Such beds may be fragile, and offer no more resistance to erosion than to sustained shear. Or they may be brittle, and offer instead an excess resistance to erosion. To take this contrast into account, new basal boundary conditions are proposed. Their implications for parallel flows down infinite slopes are then examined for three different cases: stationary flows; starting; and stopping transients. For all three cases, flow behaviour is altered significantly when beds present an excess resistance to erosion. For stationary flows, non-unique velocity profiles are obtained, implying hysteresis or history-dependence. For starting transients, a power law growth of the flow thickness is predicted, instead of the jump to finite or infinite depth that would otherwise occur. For stopping transients, flows start to decelerate with a finite basal shear rate, even over erodible substrates. Analytical solutions to the corresponding free and moving boundary problems are obtained, and checked against numerical results. Model predictions are then compared with experimental measurements. Overall, good agreement is obtained. In particular, the model describes well the very different erosional responses observed for fragile and brittle beds.