Correlation Length Research Articles

Not all that is β0 is β-function: the DGLAP resummation and the running coupling in NLO JIMWLK

We reanalyze the origin of the large transverse logarithms associated with the QCD one loop β function coefficient in the NLO JIMWLK Hamiltonian. We show that some of these terms are not associated with the running of the QCD coupling constant but rather with the DGLAP evolution. The DGLAP-like resummation of these logarithms is mandatory within the JIMWLK Hamiltonian, as long as the color correlation length in the projectile is larger than that in the target. This regime in fact covers the whole range of rapidities at which JIMWLK evolution is supposed to be applicable. We derive the RG equation that resums these logarithms to all orders in αs in the JIMWLK Hamiltonian. This is a nonlinear equation for the eikonal scattering matrix S(x). We solve this equation, and perform the DGLAP resummation in two simple cases: the dilute limit, where both the projectile and the target are far from saturation, and the saturated regime, where the target correlation length also determines its saturation momentum.

Biswas–Chatterjee–Sen Model on Solomon Networks with Two Three-Dimensional Lattices

The Biswas–Chatterjee–Sen (BChS) model of opinion dynamics has been studied on three-dimensional Solomon networks by means of extensive Monte Carlo simulations. Finite-size scaling relations for different lattice sizes have been used in order to obtain the relevant quantities of the system in the thermodynamic limit. From the simulation data it is clear that the BChS model undergoes a second-order phase transition. At the transition point, the critical exponents describing the behavior of the order parameter, the corresponding order parameter susceptibility, and the correlation length, have been evaluated. From the values obtained for these critical exponents one can confidently conclude that the BChS model in three dimensions is in a different universality class to the respective model defined on one- and two-dimensional Solomon networks, as well as in a different universality class as the usual Ising model on the same networks.

Dynamic Behavior of Poly(N-isopropylmethacrylamide) in Neat Water and in Water/Methanol Mixtures.

We investigate the collective dynamics of thermoresponsive polymer poly(N-isopropylmethacrylamide) (PNIPMAM) in aqueous solution and in water/methanol mixtures in the one-phase region. In neat water, the polymer concentration c is varied in a wide range around the overlap concentration c*, that is estimated at 23 g L-1. Using dynamic light scattering (DLS), two decays ("modes") are consistently observed in the intensity autocorrelation functions for c = 2-150 g L-1 with relaxation rates which are proportional to the square of the momentum transfer. Below c*, these are attributed to the diffusion of single chains and to clusters from PNIPMAM that are formed due to hydrophobic interactions. Above c*, they are assigned to the diffusion of the chain segments between overlap points and to long-range concentration fluctuations. From the temperature-dependent behavior of the overall scattering intensities and the dynamic correlation lengths of the fast mode, the critical temperatures and the scaling exponents are determined. The latter are significantly lower than the static values predicted by mean-field theory, which may be related to the presence of the large-scale inhomogeneities. The effect of the cosolvent methanol on the dynamics is investigated for polymer solutions having c = 30 g L-1 and methanol volume fractions in the solvent mixtures of up to 60 vol %. The phase diagram was established by differential scanning calorimetry. The slow mode detected by DLS becomes significantly weaker as methanol is added, i.e., the solutions become more homogeneous. Beyond the minimum of the coexistence line, which is located at 40-50 vol % of methanol, the dynamics is qualitatively different from the one at lower methanol contents. Thus, going from the water-rich to the methanol-rich side of the miscibility gap, the change of interaction of the PNIPMAM chains with the two solvents has a severe effect on the collective dynamics.

Effects of soil spatial variability on the behaviour of the embankment supported with a combined retaining structure

In this study, the effects of soil spatial variability on the behaviour of the embankment supported with a combined retaining structure (CRS) were investigated. The numerical model of the CRS embankment was established and validated with the field data. An application programming interface (API) was developed to deal with the data exchanging issue between the numerical model and the spatial variability characterization model. Based on the verified numerical model and the API, the probabilistic analysis with 500 Monte Carlo simulations was automatically computed. Three influencing factors of the retained soil (the mean of the friction angle, the variation of the friction angle and the vertical correlation length of the random field) are analysed by parametric analysis. The results show that the vertical correlation length of the random field is most important in the earth pressure calculation process, while the mean of the friction angle is the factor with least impact. On the whole, the spatial variability of soil properties has minimal impact on the distribution and magnitude of earth pressure behind the retaining structure.

Observable Atrial and Ventricular Fibrillation Episode Durations Are Conformant With a Power Law Based on System Size and Spatial Synchronization.

Atrial fibrillation (AF) and ventricular fibrillation (VF) episodes exhibit varying durations, with some spontaneously ending quickly while others persist. A quantitative framework to explain episode durations remains elusive. We hypothesized that observable self-terminating AF and VF episode lengths, whereby durations are known, would conform with a power law based on the ratio of system size and correlation length ([Formula: see text]. Using data from computer simulations (2-dimensional sheet and 3-dimensional left-atrial), human ischemic VF recordings (256-electrode sock, n=12 patients), and human AF recordings (64-electrode basket-catheter, n=9 patients; 16-electrode high definition-grid catheter, n=42 patients), conformance with a power law was assessed using the Akaike information criterion, Bayesian information criterion, coefficient of determination (R2, significance=P<0.05) and maximum likelihood estimation. We analyzed fibrillatory episode durations and [Formula: see text], computed by taking the ratio between system size ([Formula: see text], chamber/simulation size) and correlation length (xi, estimated from pairwise correlation coefficients over electrode/node distance). In all computer models, the relationship between episode durations and [Formula: see text] was conformant with a power law (Aliev-Panfilov R2: 0.90, P<0.001; Courtemanche R2: 0.91, P<0.001; Luo-Rudy R2: 0.61, P<0.001). Observable clinical AF/VF durations were also conformant with a power law relationship (VF R2: 0.86, P<0.001; AF basket R2: 0.91, P<0.001; AF grid R2: 0.92, P<0.001). [Formula: see text] also differentiated between self-terminating and sustained episodes of AF and VF (P<0.001; all systems), as well as paroxysmal versus persistent AF (P<0.001). In comparison, other electrogram metrics showed no statistically significant differences (dominant frequency, Shannon Entropy, mean voltage, peak-peak voltage; P>0.05). Observable fibrillation episode durations are conformant with a power law based on system size and correlation length.

A mechano-regulation model to design and optimize the surface microgeometry of titanium textured devices for biomedical applications

In a technological context where, thanks to the additive manufacturing techniques, even sophisticated geometries as well as surfaces with specific micrometric features can be realized, we propose a mechano-regulation algorithm to determine the optimal microgeometric parameters of the surface of textured titanium devices for biomedical applications. A poroelastic finite element model was developed including a portion of bone, a portion of a textured titanium device and a layer of granulation tissue separating the bone from the device and occupying the space between them. The algorithm, implemented in the Matlab environment, determines the optimal values of the root mean square and the correlation length that the device surface must possess to maximize bone formation in the gap between the bone and the device. For low levels of compression load acting on the bone, the algorithm predicts low values of root mean square and high values of correlation length. Conversely, high levels of load require high values of root mean square and low values of correlation length. The optimal microgeometrical parameters were determined for various thickness values of the granulation tissue layer. Interestingly, the predictions of the proposed computational model are consistent with the experimental results reported in the literature. The proposed algorithm shows promise as a valuable tool for addressing the demands of precision medicine. In this approach, the device or prosthesis is no longer designed solely based on statistical averages but is tailored to each patient's unique anthropometric characteristics, as well as considerations related to their metabolism, sex, age, and more.

Reliability and landslide consequence analysis of long heterogeneous soil infrastructure slopes: A parallel computing investigation

Long infrastructures, like flood defense levees, built upon diverse soils, often face complex landslide failures. Risk assessments for such structures involve evaluating slope stability reliability and potential landslide consequences. Historically, embankment slope reliability and failure assessments focused on shorter sections due to computational limitations. This study, however, delves into a longer 3D soil slope section, 100 times the height, using the random finite element method to analyze the size effect. It conducts a parametric study varying the soil strength’s horizontal correlation length to understand slope failure initiation in heterogeneous soils. In contrast to prior work, this paper emphasizes several facets: the realized factor of safety (FOS) distribution, quantification of discrete unstable zones, failures, and the size impact on long embankment slopes. Findings indicate the Weibull distribution better fits the realized FOS, aiding in predicting long embankment slope system reliability when combined with the weakest link theory. While multiple failures are possible for very long soil embankment slopes, practical observations suggest a likelihood of a single localized failure despite this potential.

Random Field Ising Model Criticality in a Complex Binary Liquid System.

While Ising criticality in classical liquids has been firmly established both theoretically and experimentally, much less is known about criticality in liquids in which the growth of the correlation length is frustrated by finite-size effects. A theoretical approach for dealing with this issue is the random-field Ising model (RFIM). While experimental critical-exponent values have been reported for magnetic samples (here, we consider γ, ν and η), little experimental information is available for critical fluctuations in corresponding liquid systems. In this paper, we present a study on a binary liquid consisting of 3-methyl pyridine and heavy water in a very light-weight porous gel. We find that the experimental results are in agreement with the theoretical predictions from the RFIM.

Microstructures and Rheological Properties of Short-Side-Chain Perfluorosulfonic Acid in Water/2-Propanol.

The viscosity and viscoelasticity of polyelectrolyte solutions with a single electrostatic interaction have been carefully studied experimentally and theoretically. Despite some theoretical models describe experimental results well, the influence of multiple interactions (electrostatic and hydrophobic) on rheological scaling is not yet fully resolved. Herein, we systematically study the microstructures and rheological properties of short-side-chain perfluorosulfonic acid (S-PFSA), the most promising candidate of a proton exchange membrane composed of a hydrophobic backbone with hydrophilic side-chains, in water/2-propanol. Small-angle X-ray scattering confirms that semiflexible S-PFSA colloidal particles with a length of ~38 nm and a diameter of 1-1.3 nm are formed, and the concentration dependence of the correlation length (ξ) obeys the power law ξ~c-0.5 consistent with the prediction of Dobrynin et al. By combining macrorheology with diffusing wave spectroscopy microrheology, the semidilute unentangled, semidilute entangled, and concentrated regimes corresponding to the scaling relationships ηsp~c0.5, ηsp~c1.5, and ηsp~c4.1 are determined. The linear viscoelasticity indicates that the entanglement concentration (ce) obtained from the dependence of ηsp on the polymer concentration is underestimated owing to hydrophobic interaction. The true entanglement concentration (cte) is obtained by extrapolating the plateau modulus (Ge) to the terminal modulus (Gt). Furthermore, Ge and the plateau width, τr/τe (τr and τe denote reptation time and Rouse time), scale as Ge~c2.4 and τr/τe~c4.2, suggesting that S-PFSA dispersions behave like neutral polymer solutions in the concentrated regime. This work provides mechanistic insight into the rheological behavior of an S-PFSA dispersion, enabling quantitative control over the flow properties in the process of solution coating.

Modeling stochastic elastic wave diffraction by the tips of randomly rough defects

Elastic wave scattering from a randomly rough surface of a finite length includes surface reflections and diffractions from the tips. Previous research has focused upon reflection waves with applications in ultrasonic defect detection, seismic wave exploration and phonon boundary transport. However, waves diffracted from the tips/edges have been largely neglected so far for rough defects, despite their importance in engineering applications including ultrasonic defect sizing and imaging for assessment of structural integrity. Currently understanding the statistical nature of elastic wave tip diffraction and the role of roughness is limited due to the lack of theoretical studies. In this article, we develop a statistical geometrical tip diffraction (SGTD) theory to rapidly predict the stochastic properties of tip diffraction amplitude as a function of surface roughness and incident angle. By applying a small slope perturbation to the model, a simplified analytical solution of tip diffraction is obtained. It is found that for defects with small to medium roughness, the diffraction amplitude explicitly follows a Gamma distribution, and its mean and the standard deviation are both proportional to the square of the rms slope. High-fidelity Monte Carlo finite element simulations are then run to evaluate the accuracy of the theoretical model. The range of validity of the analytical solution with respect to the level of roughness and the incident angle is obtained. The SGTD method is accurate when the correlation length is approximately equivalent or larger than one wavelength, for a wide range of angles. It is also applicable for a correlation length as short as half wavelength, but only for small rms values and when the beam angle is larger than 45∘. In addition, at large angles, the tip diffraction is almost not affected by roughness, being very similar to that from a smooth crack. This is explained by the significant dependence on the beam angle factor explicitly shown in the theoretical solution.

Using the AIEM and Radarsat-2 SAR to Retrieve Bare Surface Soil Moisture

Taking the Jiefangzha irrigation area of the Inner Mongolia Autonomous Region as the research area, the response relationships between the backscattering coefficient and radar frequency, radar incidence angle, root-mean-square height, correlation length, and soil water content under different conditions were simulated using advanced integral equations. The backscattering characteristics of exposed surfaces in cold and dry irrigation areas were discussed, and the reasons for the different effects were analyzed. Based on this, surface roughness models and statistical regression moisture inversion models were constructed through co-polarized backscatter coefficients and combined surface roughness. The correlation between the inverted surface roughness values and the measured values was R2 = 0.7569. The correlation between the soil moisture simulation values and the measured values was R2 = 0.8501, with an RMSE of 0.04. The findings showed a strong correlation between the values from the regression simulation and the measured data, indicating that the model can be applied to soil moisture inversion and has a good inversion accuracy. Compared with previous studies in the same area, the inversion model proposed in this paper has a higher accuracy and is more suitable for the inversion of soil moisture in the Jiefangzha irrigation area. These findings can support research on the water cycle and water environment assessment in the region.

Shear viscosity of a near-critical binary system analyzed with variable-temperature atomic force microscopy

Variation in shear viscous drag in a near-critical binary mixture was measured between a hydrophilic probe sphere and a hydrophilic substrate down to a distance range comparable to or smaller than the correlation length ξ using a magnetic excitation method of atomic force microscopy at temperatures T=Tc−∆T with ∆T varying from 0.1 to 0.7 K. As the temperature approached the Tc, the increase in the drag coefficient toward the zero distance was observed to be less pronounced in the distance range below ca. 100 nm. The data was discussed as arising from suppression in a concentration gradient in the intersurface region due to the overlap of the affected layers on similar surfaces.

Emergence of order from chaos through a continuous phase transition in a turbulent reactive flow system.

As the Reynolds number is increased, a laminar fluid flow becomes turbulent, and the range of time and length scales associated with the flow increases. Yet, in a turbulent reactive flow system, as we increase the Reynolds number, we observe the emergence of a single dominant timescale in the acoustic pressure fluctuations, as indicated by its loss of multifractality. Such emergence of order from chaos is intriguing. We perform experiments in a turbulent reactive flow system consisting of flame, acoustic, and hydrodynamic subsystems interacting nonlinearly. We study the evolution of short-time correlated dynamics between the acoustic field and the flame in the spatiotemporal domain of the system. The order parameter, defined as the fraction of the correlated dynamics, increases gradually from zero to one. We find that the susceptibility of the order parameter, correlation length, and correlation time diverge at a critical point between chaos and order. Our results show that the observed emergence of order from chaos is a continuous phase transition. Moreover, we provide experimental evidence that the critical exponents characterizing this transition fall in the universality class of directed percolation. Our paper demonstrates how a real-world complex, nonequilibrium turbulent reactive flow system exhibits universal behavior near a critical point.

The average transmitted wave in random particulate materials

Microwave remote sensing is significantly altered when passing through clouds or dense ice. This phenomenon is not unique to microwaves; for instance, ultrasound is also disrupted when traversing through heterogeneous tissues. Understanding the average transmission in particle-filled environments is central to improve data extraction or even to create materials that can selectively block or absorb certain wave frequencies. Most methods that calculate the average transmitted field assume that it satisfies a wave equation with a complex effective wavenumber. However, recent theoretical work has predicted more than one effective wave propagating even in a material which is statistically isotropic and for scalar waves. In this work we provide the first clear evidence of these predicted multiple effective waves by using high-fidelity Monte-Carlo simulations that do not make any statistical assumptions. To achieve this, it was necessary to fill in a missing link in the theory for particulate materials: we prove that the incident wave does not propagate within the material, which is usually taken as an assumption called the Ewald–Oseen extinction theorem. By proving this we conclude that the extinction length—the distance it takes for the incident wave to be extinct—is equal to the correlation length between the particles.

Quantum scaling for the metal–insulator transition in a two-dimensional electron system

The quantum phase transition observed experimentally in two-dimensional (2D) electron systems has been a subject of theoretical and experimental studies for almost 30 years. We suggest Gaussian approximation to the mean-field theory of the second-order phase transition to explain the experimental data. Our approach explains self-consistently the universal value of the critical exponent 3/2 (found after scaling measured resistivities on both sides of the transition as a function of temperature) as the result of the divergence of the correlation length when the electron density approaches the critical value. We also provide numerical evidence for the stretched exponential temperature dependence of the metallic phase’s resistivities in a wide range of temperatures and show that it leads to correct qualitative results. Finally, we interpret the phase diagram on the density-temperature plane exhibiting the quantum critical point, quantum critical trajectory and two crossover lines. Our research presents a theoretical description of the seminal experimental results.

Pinning-depinning transitions in two classes of discrete elastic-string models in (2+1)-dimensions

The pinning-depinning phase transitions of interfaces for two classes of discrete elastic-string models are investigated numerically. In the (1+1)-dimensions, we revisit these two elastic-string models with slight modification to the growth rule, and compare the estimated values with the previous numerical and experimental results. For the (2+1)-dimensional case, we perform extensive simulations on pinning-depinning transitions in these discrete models with quenched disorder. For full comparisons in the physically relevant spatial dimensions, we also perform numerically two distinct universality classes, including the quenched Edwards–Wilkinson, and the quenched Kardar–Parisi–Zhang equations with and without external driving forces. The critical exponents of these systems in the presence of quenched disorder are numerically estimated. Our results show that the critical exponents satisfy scaling relations well, and these two discrete elastic-string models do not fall into the existing universality classes. In order to visually comparisons of these discrete systems with quenched disorder in the (2+1)-dimensional cases, we present surface morphologies with various external driving forces during the saturated time regimes. The relationships between surface morphologies, scaling exponents and correlation length are also revealed.

Sedimentation dynamics of passive particles in dilute bacterial suspensions: emergence of bioconvection

Microorganisms are ubiquitous in nature and technology. They inhabit diverse environments, ranging from small river tributaries and lakes, to oceans, as well as wastewater treatment plants and food manufacturing. In many of these environments, microorganisms coexist with settling particles. Here, we investigate the effects of microbial activity (swimming E. coli) on the settling dynamics of passive colloidal particles using particle tracking methods. Our results reveal the existence of two distinct regimes in the correlation length scale ( $L_u$ ) and the effective diffusivity of the colloidal particles ( $D_{eff}$ ), with increasing bacterial concentration ( $\phi _b$ ). At low $\phi _b$ , the parameters $L_u$ and $D_{eff}$ increase monotonically with increasing $\phi _b$ . Beyond critical $\phi _b$ , a second regime is found where both $D_{eff}$ and $L_u$ are independent of $\phi _b$ . We demonstrate that the transition between these regimes is characterized by the emergence of bioconvection. We use experimentally measured particle-scale quantities $L_u$ and $D_{eff}$ to predict the critical bacterial concentration for the diffusion–bioconvection transition.

X-Ray Crystallographic Analysis of Pr Modified BaTiO3 Ceramics with a Theoretical Perspective for Potential Applications in Microelectronics

By examining the diffraction data, the microstructure of the ball-milled synthesized powder sample ([Formula: see text][Formula: see text]TiO3) was characterized. Through the application of Rietveld’s powder structure refinement approach to analyze X-ray powder diffraction data of ball-milled powder mixes, structural and microstructural alterations in terms of lattice defects have been assessed in the formation of single-phase praseodymium-modified BaTiO3. Bond lengths, bond angles, and generated locations of the [Formula: see text]Pr0[Formula: see text]TiO3 ceramic have been discussed about its generated structure. X-ray diffraction with respect to the first sharp diffraction peak (FSDP) is used to study the nanostructure of void species. General equations were utilized to determine various FSDP characteristics such as interlayer separation, correlation length, nanovoid diameter and scattering vector magnitude. Although the literature reports on many properties of Pr-modified barium titanate ceramic, this sample’s quantitative analysis and nanovoid determination are presented for the first time.

Two distinct charge density waves in the quasi-one-dimensional metal Sr0.95NbO3.37 revealed by resonant soft X-ray scattering

The interplay of electron-electron and electron-lattice interactions plays an important role in determining exotic properties in strongly correlated electron systems. Of particular interest is quasi-one-dimensional SrNbOx metals, which are perovskite-related layered Carpy-Galy phases. Quasi-one-dimensional metals often exhibit a charge density wave (CDW) accompanied by lattice distortion; however, to date, the presence of a CDW in a quasi-one-dimensional metallic Carpy-Galy phase has not been detected. Here, we report the discovery of two distinct and simultaneous commensurate CDWs in Sr0.95NbO3.37 using resonant soft X-ray scattering (RSXS), namely, an electronic-(001) superlattice below ~ 200 K and an electronic-(002) Bragg peak. We also observe a non-electronic-(002) Bragg peak showing lattice distortion below ~ 150 K. Through the temperature dependence and resonance profile of these CDWs and the lattice distortion, as well as the relationship between the wavelength and charge density, these CDWs are determined to be Wigner crystals and Peierls-like instabilities, respectively. The electron‒electron interaction is strong and dominant even up to 350 K, and upon cooling, it drives the electron–lattice interaction. The correlation length of the electronic-(001) superlattice is surprisingly larger than that of the electronic-(002) Bragg peak, and the superlattice is highly anisotropic. Supported by theoretical calculations, the CDWs are determined by the charge anisotropy and redistribution between the O-2p and Nb-4d orbitals, and the strength of the electronic-(001) superlattice is within the strong coupling limit.

Elastic-plastic collision mechanism of micron-sized particle impacting rough surfaces

The interaction between micron-sized particles and rough surfaces represents a common natural phenomenon. Diverse Gaussian rough surfaces are reconstructed by utilizing inverse Fourier transformation, and the impact of surface roughness on the collision of micron-sized particles is investigated, scrutinized through the sizes of RMS roughness and correlation length from 0.025 μm to 0.2 μm. Given the relevance of surface roughness to particle size, it focuses on 1 μm particles collision—a dimension representative size ingested in the turbine blade cooling passage. The results indicate that RMS roughness primarily influences the height of asperities, whereas correlation length predominantly affects the density of asperities. COR diminishes with increasing roughness, and does not exhibit a decline with escalating speed, when the correlation length is small. The maximum contact area remains relatively constant with the larger roughness. Rotational velocity and rebound angle are predominantly impacted by the RMS roughness, less influence of the correlation length.