Scaling Regime Research Articles

Instability of mixing layers: Momentum and thermal transport in the continuum breakdown regime.

We examine the momentum and thermal transport in the continuum breakdown regime of a mixing layer flow, which exhibits Kelvin-Helmholtz instability under ideal continuum conditions. The Grad 13 moment model is used as it provides an adequate description of the flow physics (second-order accurate in Knudsen number) in the transition regime. Analytical solutions are developed under breakdown conditions for two-dimensional, compressible, parallel shear flows. It is shown that the deviation of viscous stress and heat flux from the Navier-Stokes-Fourier system follows two different scaling regimes depending upon the Mach number. At low Mach numbers, the departure of all stress and heat-flux components depends only upon the Knudsen number. At high Mach number, the scaling of shear stress and transverse heat flux depends on the product of the Knudsen and Mach numbers. The normal stresses depend individually on the Knudsen and Mach number. The scaling results are verified against numerical simulations of compressible mixing layers performed using the unified gas kinetic scheme for various degrees of rarefaction.

Precise learning curves and higher-order scaling limits for dot-product kernel regression *

As modern machine learning models continue to advance the computational frontier, it has become increasingly important to develop precise estimates for expected performance improvements under different model and data scaling regimes. Currently, theoretical understanding of the learning curves (LCs) that characterize how the prediction error depends on the number of samples is restricted to either large-sample asymptotics () or, for certain simple data distributions, to the high-dimensional asymptotics in which the number of samples scales linearly with the dimension (). There is a wide gulf between these two regimes, including all higher-order scaling relations , which are the subject of the present paper. We focus on the problem of kernel ridge regression for dot-product kernels and present precise formulas for the mean of the test error, bias and variance, for data drawn uniformly from the sphere with isotropic random labels in the rth-order asymptotic scaling regime with held constant. We observe a peak in the LC whenever for any integer r, leading to multiple sample-wise descent and non-trivial behavior at multiple scales. We include a colab (available at: https://tinyurl.com/2nzym7ym) notebook that reproduces the essential results of the paper.

Dynamical symmetries in the non-equilibrium dynamics of the directed spherical model

The dynamical scaling and ageing in the relaxational dynamics of the quenched directed spherical model is analysed. The exact two-time correlation and response functions display new regimes of ballistic or anisotropic ballistic scaling, at larger distances than probed in the usual regime of diffusive scaling. The role of long-ranged initial correlations on the existence of these scaling regimes is clarified. Their dynamical symmetries are described in terms of extensions of the Schrödinger algebra appropriate to non-equilibrium dynamics in that the anisotropic ballistic scaling regime can be interpreted in terms of meta-Schrödinger invariance while the regime of isotropic ballistic scaling is meta-conformally invariant.

Universal scaling of shear thickening transitions

Nearly, all dense suspensions undergo dramatic and abrupt thickening transitions in their flow behavior when sheared at high stresses. Such transitions occur when the dominant interactions between the suspended particles shift from hydrodynamic to frictional. Here, we interpret abrupt shear thickening as a precursor to a rigidity transition and give a complete theory of the viscosity in terms of a universal crossover scaling function from the frictionless jamming point to a rigidity transition associated with friction, anisotropy, and shear. Strikingly, we find experimentally that for two different systems—cornstarch in glycerol and silica spheres in glycerol—the viscosity can be collapsed onto a single universal curve over a wide range of stresses and volume fractions. The collapse reveals two separate scaling regimes due to a crossover between frictionless isotropic jamming and frictional shear jamming, with different critical exponents. The material-specific behavior due to the microscale particle interactions is incorporated into a scaling variable governing the proximity to shear jamming, that depends on both stress and volume fraction. This reformulation opens the door to importing the vast theoretical machinery developed to understand equilibrium critical phenomena to elucidate fundamental physical aspects of the shear thickening transition.

High‐dimensional limit theorems for SGD: Effective dynamics and critical scaling

AbstractWe study the scaling limits of stochastic gradient descent (SGD) with constant step‐size in the high‐dimensional regime. We prove limit theorems for the trajectories of summary statistics (i.e., finite‐dimensional functions) of SGD as the dimension goes to infinity. Our approach allows one to choose the summary statistics that are tracked, the initialization, and the step‐size. It yields both ballistic (ODE) and diffusive (SDE) limits, with the limit depending dramatically on the former choices. We show a critical scaling regime for the step‐size, below which the effective ballistic dynamics matches gradient flow for the population loss, but at which, a new correction term appears which changes the phase diagram. About the fixed points of this effective dynamics, the corresponding diffusive limits can be quite complex and even degenerate. We demonstrate our approach on popular examples including estimation for spiked matrix and tensor models and classification via two‐layer networks for binary and XOR‐type Gaussian mixture models. These examples exhibit surprising phenomena including multimodal timescales to convergence as well as convergence to sub‐optimal solutions with probability bounded away from zero from random (e.g., Gaussian) initializations. At the same time, we demonstrate the benefit of overparametrization by showing that the latter probability goes to zero as the second layer width grows.

Anomalous scaling regime for one-dimensional Mott variable-range hopping

Anomalous scaling regime for one-dimensional Mott variable-range hopping

Liquid−vapor order parameter and coexistence-curve diameter of nitrogen, ethylene, and sulfur hexafluoride: From the triple point to the critical scaling regime

Analytic closed-form expressions are obtained for the liquid and vapor saturation densities defining the coexistence curve. The densities are modeled with broken power laws and Weibull distributions. Specifically, the coexistence curves of nitrogen, ethene and sulfur hexafluoride are derived, without the use of perturbative expansions, based on high-precision data extending from the triple point into the critical regime. The analytic continuation of the vapor branch below the triple point decays exponentially at low temperature. The order parameter and coexistence-curve diameter of the fluids are assembled from the regressed liquid and vapor branches of the coexistence curve, and the critical power-law scaling of these quantities is examined. The scaling exponents of the order parameter and the reduced diameter are regressed and compared with the calculated critical exponents of the 3D Ising universality class. Index functions representing the Log−Log slopes (temperature-dependent effective exponents) of the liquid and vapor densities, order parameter and diameter are used to determine the onset of the ideal power-law scaling regime and to illustrate the slope evolution of these quantities in the subcritical regime.

Symbiotic Antimicrobial Effects of Cellulose-Based Bio-Nanocomposite for Disease Management of Agricultural Crops.

In the present work, a bionanocomposite for plant crop protection was prepared by non-toxic biocompatible & biodegradable nanomaterials (Cellulose & TiO2 ) to utilize its synergistic effects against antimicrobial pathogens. The commercially available microcrystalline cellulose has been reduced to a nanometric scale regime using acid hydrolysis, while the standard TiO2 nano-powder of particle size ~20 nm has been used to prepare their nanocomposite (NC). The antibacterial studies via agar well diffusion method demonstrated that after 72 h of incubation, parent nanomaterials Ncell and TiO2 were not showing any activity against phytopathogens X. campestris pv. campestris, and Clavibacter while the nanocomposite's NC's were still effective depicting both bacteriostatic and bactericidal actions. However, the bacterial growth of biocontrol P. fluorescence was not affected by Ncell, TiO2 NPs and NC after 72 h of incubation. The antifungal testing results via poison food agar assay method suggest that the nanocomposite, along with Ncell and TiO2 NPs, exhibited strong inhibition of fungal growth of Phytophthora Spp at 0.125 mg/ml concentration while for F. graminearum, similar effect was observed at 0.25 mg/ml concentration. The nanocomposite has proved its potential by exhibiting longer & stronger synergistic effects against plant pathogens as a good antimicrobial agent for protection of agricultural crops.

Spatial dynamics of synergistic coinfection in rock-paper-scissors models.

We investigate the spatial dynamics of two-disease epidemics reaching a three-species cyclic model. Regardless of their species, all individuals are susceptible to being infected with two different pathogens, which spread through person-to-person contact. We consider that the simultaneous presence of multiple infections leads to a synergistic amplification in the probability of host mortality due to complications arising from any of the co-occurring diseases. Employing stochastic simulations, we explore the ramifications of this synergistic coinfection on spatial configurations that emerge from stochastic initial conditions. Under conditions of pronounced synergistic coinfection, we identify the emergence of zones inhabited solely by hosts affected by a singular pathogen. At the boundaries of spatial domains dominated by a single disease, interfaces of coinfected hosts appear. The dynamics of these interfaces are shaped by curvature-driven processes and display a scaling behavior reflective of the topological attributes of the underlying two-dimensional space. As the lethality linked to coinfection diminishes, the evolution of the interface network's spatial dynamics is influenced by fluctuations stemming from waves of coinfection that infiltrate territories predominantly occupied by a single disease. Our analysis extends to quantifying the implications of synergistic coinfection at both the individual and population levels Our outcomes show that organisms' infection risk is maximized if the coinfection increases the death due to disease by 30% and minimized as the network dynamics reach the scaling regime, with species populations being maximum. Our conclusions may help ecologists understand the dynamics of epidemics and their impact on the stability of ecosystems.

Impact of fuel properties on the transition of liquid-gas interface dynamics under supercritical pressure

Fuel injection dynamics transition from two-phase theory to single-phase dense fluid dynamics are a crucial phenomenon that markedly affects the mixing process and subsequent combustion under diesel engine-relevant conditions. As the significant effect of fuel properties on the timescale and mechanism of this transition remains elusive yet, this work focuses on the impact of fuel properties on the transition to single-phase dense fluid at supercritical pressure. The detailed thermodynamic structure of non-continuum gas-liquid interfaces at supercritical pressures was constructed through gradient theory, and its evolution was analyzed. The driving mechanism of transition from spray to single-phase dense fluid was discussed. The influences of fuel properties on the timescale and mechanism of breakdown of the two-phase theory were investigated through the comparison of three different alkanes. Theoretical analysis suggests that the transition from spray to single-phase dense fluid is jointly promoted by the thickening interface, the diminished surface tension, and the shorter molecular mean free path. The gas-liquid interface of lighter-alkane/nitrogen binary systems is more inclined to evolve into the continuum length-scale regime. The cause is a combination of two aspects: (1) smaller thermodynamic potential energy density difference ω¯(ϱM)−ω¯s and smaller interfacial tangential pressure across lighter alkanes/nitrogen interfaces lead to broader interfaces and smaller surface tension, (2) the more volatile nature of lighter alkanes results in shorter molecular mean free path. The minimum reduced pressure required for gas-liquid interfaces to reach critical mixing states is lower for lighter and more volatile alkanes at the same reduced temperature. From a time-scale perspective, the non-continuum gas-liquid interfaces evolve into the continuum length scale regime later for heavier alkanes. This suggests that the transition to the diffusive mixing mechanism occurs further downstream for heavier alkane sprays. A conceptual diagram was presented to illustrate the effect of fuel properties on the transition.

Entanglement propagation and dynamics in non-additive quantum systems

The prominent collective character of long-range interacting quantum systems makes them promising candidates for quantum technological applications. Yet, lack of additivity overthrows the traditional picture for entanglement scaling and transport, due to the breakdown of the common mechanism based on excitations propagation and confinement. Here, we describe the dynamics of the entanglement entropy in many-body quantum systems with a diverging contribution to the internal energy from the long-range two body potential. While in the strict thermodynamic limit entanglement dynamics is shown to be suppressed, a rich mosaic of novel scaling regimes is observed at intermediate system sizes, due to the possibility to trigger multiple resonant modes in the global dynamics. Quantitative predictions on the shape and timescales of entanglement propagation are made, paving the way to the observation of these phases in current quantum simulators. This picture is connected and contrasted with the case of local many body systems subject to Floquet driving.

Correlation Energy of a Weakly Interacting Fermi Gas with Large Interaction Potential

Recently the leading order of the correlation energy of a Fermi gas in a coupled mean-field and semiclassical scaling regime has been derived, under the assumption of an interaction potential with a small norm and with compact support in Fourier space. We generalize this result to large interaction potentials, requiring only |cdot | hat{V} in ell ^1 (mathbb {Z}^3). Our proof is based on approximate, collective bosonization in three dimensions. Significant improvements compared to recent work include stronger bounds on non-bosonizable terms and more efficient control on the bosonization of the kinetic energy.

Weak Edgeworth expansion for the mean-field Bose gas

We consider the ground state and the low-energy excited states of a system of N identical bosons with interactions in the mean-field scaling regime. For the ground state, we derive a weak Edgeworth expansion for the fluctuations of bounded one-body operators, which yields corrections to a central limit theorem to any order in 1/sqrt{N}. For suitable excited states, we show that the limiting distribution is a polynomial times a normal distribution, and that higher-order corrections are given by an Edgeworth-type expansion.

Scaling Approaches to Quasigeostrophic Theory for Moist, Precipitating Air

Abstract Quasigeostrophic (QG) theory is of fundamental importance in the study of large-scale atmospheric flows. In recent years, there has been growing interest in extending the classical QG plus Ekman friction layer model (QG–Ekman) to systematically include additional physical processes known to significantly contribute to real-life weather phenomena. This paper lays the foundation for combining two of these developments, namely, Smith and Stechmann’s family of precipitating quasigeostrophic (PQG) models on the one hand, and the extension of QG–Ekman for dry air by a strongly diabatic layer (DL) of intermediate height (QG–DL–Ekman) on the other hand. To this end, Smith and Stechmann’s PQG equations for soundproof motions are first corroborated within a general asymptotic modeling framework starting from a full compressible flow model. The derivations show that the PQG model family is naturally embedded in the asymptotic hierarchy of scale-dependent atmospheric flow models introduced by one of the present authors. Particular emphasis is then placed on an asymptotic scaling regime for PQG that accounts for a generic Kessler-type bulk microphysics closure and is compatible with QG–DL–Ekman theory. The detailed derivation of a moist QG–DL–Ekman model is deferred to a future publication.

Universal scaling regimes in rotating fluid turbulence

Scaling arguments for steady turbulence in a rotating fluid are set up. These are used to show that it displays dual scaling regimes: a rotation-dominated regime for the largest scales and a Kolmogorov scaling regime for intermediate scales. For decaying rotating turbulence, it is speculated that the energy spectrum, while carrying a decaying amplitude, contains a large-scale rotation-dominated regime that increases in size over time, and an intermediate-scale Kolmogorov scaling regime that shrinks in time.

Salt-Added Solutions of Markov Polyampholytes: Diagram of States, Antipolyelectrolyte Effect, and Self-Coacervate Dynamics

Field-theoretic and scaling approaches are combined to predict conformational behaviors of single-chain globally neutral polyampholytes (PAs) in the presence of salt, the viscosity of their dilute solutions, and the rheology of condensed self-coacervate phases. To reveal the role of the monomer sequence, we consider PAs with Markov statistics of positive and negative charges. The diagram of globular regimes of single-chain PAs in Θ solvent is constructed with the coordinates of the charge blockiness and salt concentrations. For highly flexible chains, it contains six scaling regimes corresponding to low/high salt concentrations and (i) almost alternating, (ii) random correlated, and (iii) highly blocky sequences of ionic monomers. At increasing salt concentration, PAs of any sequence swell until reaching an ideal-coil size. The amplitude of the salt-induced globule-to-coil transition increases with increasing charge blockiness. So does the change in the viscosity of dilute PA solutions, indicating the sequence specificity in the antipolyelectrolyte effect. The similarity between the internal structure of the globules and macroscopic self-coacervate phases enables prediction of the viscosity of the latter. The respective scaling dependencies for self-coacervates of short unentangled and long entangled PAs are provided. Self-coacervate viscosity is the highest for the most blocky PAs and, for any sequence, decreases with the addition of salt. Our findings serve as important guidelines for understanding the dynamics of intracellular condensates, which form from polyampholytic disordered proteins/regions and exhibit the salt-induced viscosity drop described herein.

Low leakage decoder using dual-threshold technique for static random-access memory applications

Decoders are one of the significant peripheral components of static random-access memory (SRAM). As the CMOS technology moves towards nano scale regime, the leakage power starts dominating dynamic power. In this paper, we propose decoders using NAND logic in 32 nm CMOS technology. Leakage power is reduced by employing dual-threshold technique. Dual thresholding is a technique that uses transistors of two different threshold voltages. The technique is implemented in simulation by two methods; first method uses transistors with different threshold voltage and the second method uses substrate biasing to vary the threshold voltage. Row and column decoders are designed and simulated in H-Spice. The leakage power is calculated and compared for both the methods. The NAND gate implemented by Method-1 and Method-2 provides a maximum leakage power savings of 87.67% and 90.81% respectively. The maximum leakage power savings of 96.76% and 98.74% is reported for the row decoder implemented by Method-1 and Method-2 respectively. Similarly, Method-1 gives maximum leakage power savings of 97.09% and Method-2 gives a savings of 99.11% for column decoder. The difference in leakage power savings of Method-1 and Method-at same threshold voltage is 3.14%, 1.98%, and 2.02% for NAND gate, row decoder and column decoder respectively.

Isobaric heat capacity of classical and quantum fluids: extending experimental data sets into the critical scaling regime

Isobaric heat capacity of classical and quantum fluids: extending experimental data sets into the critical scaling regime

Role of predator-prey reversal in rock-paper-scissors models

In this letter we consider a single parameter generalization of the standard three species Rock-Paper-Scissors (RPS) model allowing for predator-prey reversal. This model, which shall be referred to as κRPS model, incorporates bidirectional predator-prey interactions between all the species in addition to the unidirectional predator-prey interactions of the standard RPS model. We study the dynamics of a May-Leonard formulation of the κRPS model using lattice-based spatial stochastic simulations with random initial conditions. We find that if the simulation lattices are sufficiently large for the coexistence of all three species to be maintained, the model asymptotically leads to the formation of spiral patterns whose evolution is qualitatively similar to that of the standard RPS model, albeit with larger characteristic length and time scales. We show that if the likelihood of predator-prey reversal is sufficiently large there are two distinct scaling regimes: one transient curvature dominated regime in which the characteristic length of the population network grows with time and another where it becomes a constant. We also estimate the dependence of the asymptotic value of the characteristic length of the population network on the likelihood of predator-prey reversal and show that if the simulation lattices are not sufficiently large then predator-prey reversal could potentially have a negative impact on coexistence. Finally, we interpret these results by considering the much simpler dynamics of circular domains.

Scaling behavior for the detachment of a self-propelling filament from an attractive surface.

Desorption of a self-propelling filament from an attractive surface is studied by computer simulations and the influence of activity, chain length, and chain rigidity is explored. For the flexible filament, we find three scaling regimes of desorption time vs activity with various scaling exponents. At low activity, the scaling law results from the spiral-like detachment kinetics. And at high activity, by theoretical analysis, the desorption is reminiscent of the escaping mechanism of a super-diffusive blob from a potential well at a short time scale. Additionally, the desorption time decreases first and then increases with chain length at low activity, since it is hard to form a spiral for short filaments due to the limited volume repulsion. For high activities, the desorption time approximately scales with chain length, with a scaling exponent ∼0.5, which can be explained by the theory and numerically fitting scaling law between the end-to-end distance of the "globule-like" filament and chain length. Furthermore, a non-monotonic behavior is observed between the desorption time and the chain stiffness. Desorption time slightly decreases first and then rapidly increases with stiffness due to the opposed effects of increasing rigidity on headiing-up time and leaving-away time. In contrast to traditional polymers, the scaling behavior suggests unique desorption characteristics of active polymers.