Low-order Moments Research Articles

Fokker-Planck Central Moment Lattice Boltzmann Method for Effective Simulations of Fluid Dynamics

We present a new formulation of the central moment lattice Boltzmann (LB) method based on a minimal continuous Fokker-Planck (FP) kinetic model, originally proposed for stochastic diffusive-drift processes (e.g., Brownian dynamics), by adapting it as a collision model for the continuous Boltzmann equation (CBE) for fluid dynamics. The FP collision model has several desirable properties, including its ability to preserve the quadratic nonlinearity of the CBE, unlike that based on the common Bhatnagar-Gross-Krook model. Rather than using an equivalent Langevin equation as a proxy, we construct our approach by directly matching the changes in different discrete central moments independently supported by the lattice under collision to those given by the CBE under the FP-guided collision model. This can be interpreted as a new path for the collision process in terms of the relaxation of the various central moments to “equilibria”, which we term as the Markovian central moment attractors that depend on the products of the adjacent lower order moments and a diffusion coefficient tensor, thereby involving of a chain of attractors; effectively, the latter are nonlinear functions of not only the hydrodynamic variables, but also the non-conserved moments; the relaxation rates are based on scaling the drift coefficient by the order of the moment involved. The construction of the method in terms of the relevant central moments rather than via the drift and diffusion of the distribution functions directly in the velocity space facilitates its numerical implementation and analysis. We show its consistency to the Navier-Stokes equations via a Chapman-Enskog analysis and elucidate the choice of the diffusion coefficient based on the second order moments in accurately representing flows at relatively low viscosities or high Reynolds numbers. We will demonstrate the accuracy and robustness of our new central moment FP-LB formulation, termed as the FPC-LBM, using the D3Q27 lattice for simulations of a variety of flows, including wall-bounded turbulent flows. We show that the FPC-LBM is more stable than other existing LB schemes based on central moments, while avoiding numerical hyperviscosity effects in flow simulations at relatively very low physical fluid viscosities through a refinement to a model founded on kinetic theory.

The Phase Fractional Lower-Order Moment Based DPD Algorithm of Non-circular Sources for Nested Array Under Impulsive Noise

The Phase Fractional Lower-Order Moment Based DPD Algorithm of Non-circular Sources for Nested Array Under Impulsive Noise

Moment evolution equations for rational random dynamical systems: an increment decomposition method

Abstract Statistical moments are commonly used tools for exploring the ensemble behavior in gene regulation and population dynamics, where the rational vector fields are particularly ubiquitous, but how to efficiently derive the corresponding moment evolution equations was not much involved. Traditional derivation methods rely on fractional reduction and Itô formula, but it may become extremely complicated if the vector field is described by multivariate fractional polynomials. To resolve this issue, we present a novel incremental decomposition method, by which the rational vector field is divided into two parts: (proper) fractional polynomials and non-fractional polynomials. For the non-fractional polynomial part, we deduce the variation rate of a statistical moment by the Itô formula, but for the fractional polynomial part we acquire the corresponding variation rate by a relation analogous to that between the moment generating function and the distinct statistical moments. As application of the novel technique, the resultant infinite-dimensional moment systems associated with two typical examples are truncated with the schemes of derivative matching closure and the Gaussian moment closure. By comparing the lower-order statistical moments obtained from the closed moment systems with the counterparts obtained from direct simulation, the correctness of the proposed technique is verified. The present study is significant in facilitating the development of moment dynamics towards more complex systems.

Extreme spillovers across carbon and energy markets: A multiscale higher-order moment analysis

The complexity of carbon market mechanisms and the uncertainty in market conditions raise questions on how carbon and energy markets interact. The majority of existing studies focused on the lower-order moment spillover across carbon and energy markets, thereby posing limitations on carbon risk management and carbon market efficiency. This paper analyzes multiscale skewness and kurtosis spillovers across carbon and energy markets under different market conditions. It is found that carbon market is a short-term net skewness spillover receiver, but becomes a medium-term risk source under all market conditions, capable of transmitting skewness risk to the natural gas market. The carbon market acts as a short- and medium-term kurtosis risk source for the natural gas and electricity markets in the lower probability of extreme returns, but bears the kurtosis risk from the natural gas and coal markets when the extreme risk is high. These results indicate that policymakers should take measures to adjust carbon prices when facing long-term skewness risk and a lower probability of extreme returns in the carbon market, to prevent further spread of risk to the energy markets.

Leading order track functions in a hot and dense QGP

We study the modifications to the fragmentation pattern of partons into charged particles in the presence of a hot and dense quark gluon plasma. To this end, we analyze the perturbative renormalization group equations of the track functions, which describe the energy fraction carried by charged hadrons. Focusing on pure Yang-Mills theory, we compute the lowest-order moments of the medium-modified track functions, which are found to be sensitive to the reduced phase space for emissions in the medium and to energy loss. We use the extracted moments to calculate the energy energy correlator (EEC) on tracks in the collinear limit. The EEC on medium-evolved tracks does not differ qualitatively from the EEC on vacuum tracks despite being sensitive to the color decoherence transition and suppressing the distribution due to quenching, as seen in other jet observables. Published by the American Physical Society 2024

Moment dynamics of oligomer formation in protein amyloid aggregation with secondary nucleation

The abnormal aggregation of proteins into amyloid fibrils, usually implemented by a series of biochemical reactions, is associated with various neurodegenerative disorders. Considering the intrinsic stochasticity in the involving biochemical reactions, a general chemical master equation model for describing the process from oligomer production to fibril formation is established, and then the lower-order statistical moments of different molecule species are captured by the derivative matching closed system, and the long-time accuracy is verified using the Gillespie algorithm. It is revealed that the aggregation of monomers into oligomers is highly dependent on the initial number of misfolded monomers; the formation of oligomers can be effectively inhibited by reducing the misfolding rate, the primary nucleation rate, elongation rate, and secondary nucleation rate; as the conversion rate decreases, the number of oligomers increases over a long time scale. In particular, sensitivity analysis shows that the quantities of oligomers are more sensitive to monomer production and protein misfolding; the secondary nucleation is more important than the primary nucleation in oligomer formation. These findings are helpful for understanding and predicting the dynamic mechanism of amyloid aggregation from the viewpoint of quantitative analysis.

Ab initio description of monopole resonances in light- and medium-mass nuclei

The paper is the third of a series dedicated to the ab initio description of monopole giant resonances in mid-mass closed- and open-shell nuclei via the so-called projected generator coordinate method. The present focus is on the computation of the moments mk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m_k$$\\end{document} of the monopole strength distribution, which are used to quantify its centroid energy and dispersion. First, the capacity to compute low-order moments via two different methods is developed and benchmarked for the m1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m_1$$\\end{document} moment. Second, the impact of the angular momentum projection on the centroid energy and dispersion of the monopole strength is analysed before comparing the results to those obtained from consistent quasi-particle random phase approximation calculations. Next, the so-called energy weighted sum rule (EWSR) is investigated. First, the appropriate ESWR in the center-of-mass frame is derived analytically. Second, the intrinsic EWSR is tested in order to quantify the (unwanted) local-gauge symmetry breaking of the presently employed chiral effective field theory (χ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi $$\\end{document}EFT) interactions. Finally, the infinite nuclear matter incompressibility associated with the employed χ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi $$\\end{document}EFT interactions is extracted by extrapolating the finite-nucleus incompressibility computed from the monopole centroid energy.

Probability density and information entropy of machine learning derived intracranial pressure predictions.

Even with the powerful statistical parameters derived from the Extreme Gradient Boost (XGB) algorithm, it would be advantageous to define the predicted accuracy to the level of a specific case, particularly when the model output is used to guide clinical decision-making. The probability density function (PDF) of the derived intracranial pressure predictions enables the computation of a definite integral around a point estimate, representing the event's probability within a range of values. Seven hold-out test cases used for the external validation of an XGB model underwent retinal vascular pulse and intracranial pressure measurement using modified photoplethysmography and lumbar puncture, respectively. The definite integral ±1 cm water from the median (DIICP) demonstrated a negative and highly significant correlation (-0.5213±0.17, p< 0.004) with the absolute difference between the measured and predicted median intracranial pressure (DiffICPmd). The concordance between the arterial and venous probability density functions was estimated using the two-sample Kolmogorov-Smirnov statistic, extending the distribution agreement across all data points. This parameter showed a statistically significant and positive correlation (0.4942±0.18, p< 0.001) with DiffICPmd. Two cautionary subset cases (Case 8 and Case 9), where disagreement was observed between measured and predicted intracranial pressure, were compared to the seven hold-out test cases. Arterial predictions from both cautionary subset cases converged on a uniform distribution in contrast to all other cases where distributions converged on either log-normal or closely related skewed distributions (gamma, logistic, beta). The mean±standard error of the arterial DIICP from cases 8 and 9 (3.83±0.56%) was lower compared to that of the hold-out test cases (14.14±1.07%) the between group difference was statistically significant (p<0.03). Although the sample size in this analysis was limited, these results support a dual and complementary analysis approach from independently derived retinal arterial and venous non-invasive intracranial pressure predictions. Results suggest that plotting the PDF and calculating the lower order moments, arterial DIICP, and the two sample Kolmogorov-Smirnov statistic may provide individualized predictive accuracy parameters.

Active Brownian particle under stochastic orientational resetting

We employ renewal processes to characterize the spatiotemporal dynamics of an active Brownian particle under stochastic orientational resetting. By computing the experimentally accessible intermediate scattering function (ISF) and reconstructing the full time-dependent distribution of the displacements, we study the interplay of rotational diffusion and resetting. The resetting process introduces a new spatiotemporal regime reflecting the directed motion of agents along the resetting direction at large length scales, which becomes apparent in an imaginary part of the ISF. We further derive analytical expressions for the low-order moments of the displacements and find that the variance displays an effective diffusive regime at long times, which decreases for increasing resetting rates. At intermediate times the dynamics are characterized by a negative skewness as well as a non-zero non-Gaussian parameter.

Generalized inverse transformation method via representative points in statistical simulation

Discrete approximations of continuous random variables play a crucial role in various areas of research and application, offering advantages in computational efficiency, interpretability, and modeling flexibility. This paper investigates discrete representations of continuous random variables using the mean-squared error criterion (MSE-RPs) and the inverse transformation method. We introduce a novel discrete approximation to the normal distribution that surpasses conventional MSE-RPs obtained from the normal density, particularly in matching lower-order moments. Furthermore, we propose a two-step generalized inverse transformation method to generate approximate MSE-RPs of random variables, inspired by the remarkable performance of the inverse transformation method in statistical simulation. Overall, the generalized inverse transformation method offers a more efficient and reliable alternative for obtaining discrete approximations to target continuous distributions, especially in scenarios where explicit computation and derivation of density functions are challenging or computationally expensive. Moreover, we extend our investigation to the case where the target distribution is a convolution of two random variables, thereby expanding the applicability of our proposed method. Furthermore, our findings hold potential applications in Monte Carlo simulation and resampling techniques.

A Novel DOA Estimation Algorithm Based on Robust Mixed Fractional Lower-Order Correntropy in Impulsive Noise

The estimation of direction of arrival (DOA) is paramount in the realm of practical array signal processing systems. Nevertheless, traditional estimation methods often rely heavily on the Gaussian noise assumption, rendering them ineffective in achieving high-precision estimates in environments plagued by strong impulsive noise. To address this challenge, this paper introduces a novel DOA estimation algorithm that leverages mixed fractional lower-order correntropy (MFLOCR) in the context of Alpha-stable distributed impulsive noise. Correntropy is used as a measure of the similarity of the signals, using a Gaussian function to smooth extreme values and provide greater robustness against impulsive noise. By utilizing diverse kernel lengths to jointly regulate the kernel function, the concept of correntropy is expanded and implemented in the fractional lower-order moment (FLOM) algorithm for received signals. Subsequently, the MFLOCR is derived by adjusting the resulting form of correntropy. Finally, an enhanced DOA estimation algorithm is proposed that combines the MFLOCR operator with the MUSIC algorithm, specifically tailored for impulsive noise environments. Furthermore, a proof of boundedness is provided to validate the effectiveness of the proposed approach in such noisy conditions. Simulation experiments confirmed that the proposed method outperforms existing DOA estimation methods in the context of intense impulsive noise, a low generalized signal-to-noise ratio (GSNR), and a smaller number of snapshots.

Entropic timescales of dynamic heterogeneity in supercooled liquid.

Non-Gaussian displacement distributions are universal predictors of dynamic heterogeneity in slowly varying environments. Here, we explore heterogeneous dynamics in supercooled liquid using molecular dynamics simulations and show the efficiency of the relative-entropy based measure, negentropy, in quantifying dynamic heterogeneity over the widely used non-Gaussian parameter. Our analysis shows that the heterogeneity quantified by the negentropy is significantly different from the one obtained using the conventional moment-based definition that considers deviation from Gaussianity up to lower-order moments. We extract the timescales of dynamic heterogeneity using the two methods and show that the differential changes diverge as the system experiences strong intermittency near the glass transition. Further, we interpret the entropic timescales and discuss the general implications of our work.

Efficient dimension-by-dimension adaptive TENO-based limiter and troubled-cell indicator for nodal-based high-order spectral difference method

A new targeted essentially non-oscillatory (TENO) limiter with adaptive dissipation has been developed for the 3rd- and 4th-order spectral difference method. This limiter can potentially be useful for other nodal-based discontinuous high-order methods as well. Unlike traditional WENO limiters for these methods which limit low-order moments one by one, the new limiter is based on a direct convex combination of reconstruction polynomial candidates. This strategy ensures the good computational efficiency for high order reconstructions. The reconstruction stencils only involve nodal values from the target cell and its neighbors sharing faces with it. The reconstruction employs Hermite polynomials, making the new limiter compact. It can be implemented in a dimension-by-dimension manner for multi-dimensional problems, which is consistent with the spectral difference method. To further enhance efficiency, a TENO-based troubled-cell indicator is also developed to only activate limiters in troubled cells. Extensive one-dimensional and two-dimensional numerical experiments validate the performance of the new TENO-based limiter and indicator. In summary, the limiter is much less dissipative than WENO limiters and can resolve richer small-scale flow structures on coarse meshes. The indicator precisely marks out troubled cells, slightly improving over the KXRCF indicator.

In-situ measurement of particle length distribution by FBRM with application to needle- or rod-shape crystallization processes

In this paper, an in-situ measurement method is proposed for detecting needle- or rod-shape particle length distribution (PLD) during a crystallization process, by using the instrument named focused beam reflectance measurement (FBRM). A data pretreatment strategy is firstly presented to modify the measured chord length distribution (CLD) of particles by FBRM for proper analysis, which could guarantee measurement accuracy when the particle length are smaller than 150 μm. Based on the pretreated CLD, a moment conversion algorithm is established for estimating the moments of particle population density, so as to further improve the measurement accuracy for particles possessing larger sizes. Correspondingly, the particle growth-nucleation model parameters are efficiently estimated by using the interior point method (IPM) in combination with the homotopy analysis method (HAM). Then the PLD is reconstructed based on these estimated low-order moments, through a modified cubic-spline interpolation algorithm with variable interpolation grids. Experimental results on the seeded cooling crystallization process of β form L-glutamic acid (LGA) well demonstrate the effectiveness of the proposed method for in-situ measurement.

Extreme statistics and extreme events in dynamical models of turbulence.

We present a study of the intermittent properties of a shell model of turbulence with statistics of ∼10^{7} eddy turn over time, achieved thanks to an implementation on a large-scale parallel GPU factory. This allows us to quantify the inertial range anomalous scaling properties of the velocity fluctuations up to the 24th-order moment. Through a careful assessment of the statistical and systematic uncertainties, we show that none of the phenomenological and theoretical models previously proposed in the literature to predict the anomalous power-law exponents in the inertial range are in agreement with our high-precision numerical measurements. We find that at asymptotically high-order moments, the anomalous exponents tend toward a linear scaling, suggesting that extreme turbulent events are dominated by one leading singularity. We found that systematic corrections to scaling induced by the infrared and ultraviolet (viscous) cutoffs are the main limitations to precision for low-order moments, while high orders are mainly affected by the finite statistical samples.. The high-fidelity numerical results reported in this work offer an ideal benchmark for the development of future theoretical models of intermittency in dynamical systems for either extreme events (high-order moments) or typical fluctuations (low-order moments). For the latter, we show that we achieve a precision in the determination of the inertial range scaling exponents of the order of one part over ten thousand (fifth significant digit), which may be considered a record for out-of-equilibrium fluid-mechanics systems and models.

Constructing efficient strategies for the process optimization by restart.

Optimization of the mean completion time of random processes by restart is a subject of active theoretical research in statistical physics and has long found practical application in computer science. Meanwhile, one of the key issues remains largely unsolved: how to construct a restart strategy for a process whose detailed statistics are unknown to ensure that the expected completion time will reduce? Addressing this query here we propose several constructive criteria for the effectiveness of various protocols of noninstantaneous restart in the mean completion time problem and in the success probability problem. Being expressed in terms of a small number of easily estimated statistical characteristics of the original process (MAD, median completion time, low-order statistical moments of completion time), these criteria allow informed restart decision based on partial information.

Small-scale turbulent characteristics in transcritical wall-bounded flows

Considerable efforts have been devoted to the understanding of the small-scale characteristics in turbulent flows. While the universality of small-scale quantities has been established for incompressible flows, their extension to high-pressure transcritical flows remains an open area of research. To address this question, we investigate the real-fluid thermodynamic effects on small-scale velocity statistics of high-pressure transcritical wall-bounded turbulence. We show that in the locally isotropic region for transcritical flows, low-order moments of small-scale statistics collapse for all cases and Kolmogorov's (1941) theory holds. However, real-fluid thermodynamic effects introduce deviations in the tails of the probability density function for the velocity derivative and, consequently, high-order moments of velocity gradients and dissipation rate in transcritical flows cannot collapse in the locally isotropic region. Analysis of the intermittency shows that the low-order structure functions in transcritical flows follow extended self-similarity, while the dependence of the intermittency factor on real-fluid effects is observed for high-order structure functions. The real-fluid effects on intermittency are explained by turbulent structures related to rare events. Additionally, the dissipation rate moments for transcritical flows follow a universal scaling with Reynolds number, and the scaling exponents are different from those of incompressible flows. These results extend the small-scale universality in incompressible flows (Schumacher et al., Proc. Natl Acad. Sci. USA, vol. 111, 2014, pp. 10961–10965) to realistic flows with significant changes in thermodynamic properties, and provide a physical underpinning of the scaling laws of small-scale statistics at transcritical conditions.

Statistical learning based blind image watermarking approach

Image watermarking technology poses a significant challenge, requiring a delicate balance between robustness, imperceptibility, and capacity. To solve the balancing problem, a statistical learning based blind image watermarking technique is proposed in this paper. The idea of the proposed method to solve the balancing problem is to enhance imperceptibility and robustness while accommodating sufficient watermarks. Firstly, the local low-order pseudo-Zernike moments (PZM) magnitude in the discrete non-separable Shearlet transform (DNST) domain is constructed as the embedding domain. To ensure stability in embedding positions, we propose an edge embedding strategy. Secondly, by analyzing the statistical characteristics and multi-correlations of DNST-PZM magnitudes, a multi-correlation vector model based on the skew student's-t mixture (SSM) and hidden Markov tree (HMT) is designed. Finally, leveraging the vector SSM-HMT model and the maximum likelihood (ML) criterion, we derive the closed-form decoder expression. Comparative analysis with state-of-the-art methods demonstrates the superior imperceptibility and robustness of our proposed approach in accommodating the same capacity watermark.

Research on the Data-Driven Differential Equation-Solving Algorithm Based on Artificial Intelligence

Data-driven DEs has gained popularity in the past few years. This work proposes the new framework, named Adam Gannet Optimization Algorithm (AdamGOA), that combines Adam Optimization and Gannet Optimization Algorithm (GOA) to improve a stability, solve higher order Differential Equations (DE) and accuracy of DE. Adam is a first-order gradient-based methods, optimizes stochastic objectives using adaptive lower-order moments. In contrast, GOA represents a different distinct action of a gannets mathematically during foraging and is employed to facilitate exploitation and exploration. In addition, a Shepard Convolutional Neural Network (ShCNN) processed data to construct meta-data and estimate derivatives. After that, the unified integral form is established to determine optimal structure. Heterogeneous parameters are used to estimate and are labeled as constants or variables. Furthermore, the experimental findings showed that the AdamGOA_ ShCNN beat leading models in Accuracy, Convergence, and Mean Square Error (MSE), with values of 0.989, 4, and 0.539, respectively.

Direction of Arrival Estimation Method Based on Eigenvalues and Eigenvectors for Coherent Signals in Impulsive Noise

In this paper, a Toeplitz construction method based on eigenvalues and eigenvectors is proposed to combine with traditional denoising algorithms, including fractional low-order moment (FLOM), phased fractional low-order moment (PFLOM), and correntropy-based correlation (CRCO) methods. It can improve the direction of arrival (DOA) estimation of signals in impulsive noise. Firstly, the algorithm performs eigenvalue decomposition on the received covariance matrix to obtain eigenvectors and eigenvalues, and then the Toeplitz matrix is created according to the eigenvectors corresponding to its eigenvalues. Secondly, the spatial averaging method is used to obtain an unbiased estimate of the Toeplitz matrix, which is then weighted and added based on the corresponding eigenvalues. Next, the noise subspace of the Toeplitz matrix is reconstructed to obtain the one that has less angle information. Finally, the DOA of the coherent signal is estimated using the Multiple Signal Classification (MUSIC) algorithm. The improved method based on the Toeplitz matrix can not only suppress the effect of impulsive noise but can also solve the problem of aperture loss due to its decoherence. A series of simulations have shown that they have better performances than other algorithms.