Closure Approximation Research Articles

Effect of Spin-Dependent Short-Range Correlations on Nuclear Matrix Elements for Neutrinoless Double Beta Decay of 48Ca

Neutrinoless double beta decay is a pivotal weak nuclear process that holds the potential to unveil the Majorana nature of neutrinos and predict their absolute masses. In this study, we delve into examining the impact of spin-dependent short-range correlations (SRCs) on the nuclear matrix elements (NMEs) for the light neutrino-exchange mechanism in neutrinoless double beta (0νββ) decay of 48Ca, employing an extensive interacting nuclear shell model. All computations are performed employing the effective shell model Hamiltonian GXPF1A, encompassing the entire fp model space through the closure approximation. Our investigation examines the NMEs’ dependencies on factors such as the number of intermediate states, coupled spin-parity attributes of neutrons and protons, neutrino momentum, inter-nucleon separation, and closure energy. This scrutiny is performed with respect to both the conventional Jastrow-type approach of SRCs, employing various parameterizations, and the spin-dependent SRC paradigm. Our findings illuminate a discernible distinction in NMEs induced by spin-dependent SRCs, differing by approximately 10–20% from those computed through the conventional Jastrow-type SRCs, incorporating distinct parameterizations.

On fully symmetric implicit closure approximations for fiber orientation tensors

On fully symmetric implicit closure approximations for fiber orientation tensors

Application of Approximate Maximum-Entropy Moment Closure to the Wigner Equation

This paper investigates the potential of moment closures as a possible path to future quantum hydrodynamics models. Moment closures are known to produce accurate predictions of continuum and non-equilibrium classical gas flows while offering modeling and numerical advantages over other commonly used methods. The maximum-entropy hierarchy of moment closures holds the promise of robustly hyperbolic stable moment equations, however, the predicted distribution function in phase space is positive by design, which is in disagreement with the quasi-distribution function described by the Wigner equation. In this paper, predictions made by an interpolative one-dimensional five-moment system are compared to direct numerical solutions of the Wigner equation for a simple low-energy quantum state in unbounded double-well potentials. Numerical solutions for a particle in various potentials described by quartic polynomials are presented. The capacity of moment methods to provide accurate and efficient models for quantum systems is explored.

An orientation corrected shaking method for the microstructure generation of short fiber-reinforced composites with almost planar fiber orientation

We present an algorithm for generating short fiber-reinforced microstructures with almost planar fiber orientation. The orientation corrected shaking (OCS) method achieves a high accuracy regarding the volume fraction, fiber length distribution and fiber orientation state. Additionally, the algorithm is capable of generating microstructures for industrial materials, e.g., for a PA66GF35 material with a volume fraction of 19.3% and an aspect ratio of 33. For typical manufacturing processes, short fiber-reinforced composites show a mainly planar fiber arrangement. Therefore, we extend the two-step shaking algorithm of Li et al. [J. Ind. Text. 51(1), pp. 506–530, 2022] for a user-selected rectangular size of the unit cell and periodic boundary conditions. Additionally, the hidden closure structure of the algorithm is uncovered and a precise realization of the fiber orientation state achieved. We examine the representative volume element size for the OCS method, observing representative errors below 2% even for unit cells with edge lengths smaller than the mean fiber length. Additionally, the influence of different closure approximations on the stiffness is investigated. When applied to an industrial PA66GF35 material with sandwich structure, the OCS method demonstrates differences below 2% and 9% for the computed directional Young’s moduli E1 and E2 compared to experimental data.

Lagrangian model for passive scalar gradients in turbulence

The equation for the fluid velocity gradient along a Lagrangian trajectory immediately follows from the Navier–Stokes equation. However, such an equation involves two terms that cannot be determined from the velocity gradient along the chosen Lagrangian path: the pressure Hessian and the viscous Laplacian. A recent model handles these unclosed terms using a multi-level version of the recent deformation of Gaussian fields (RDGF) closure (Johnson & Meneveau, Phys. Rev. Fluids, vol. 2 (7), 2017, 072601). This model is in remarkable agreement with direct numerical simulations (DNS) data and works for arbitrary Taylor Reynolds numbers $\textit {Re}_\lambda$ . Inspired by this, we develop a Lagrangian model for passive scalar gradients in isotropic turbulence. The equation for passive scalar gradients also involves an unclosed term in the Lagrangian frame, namely the scalar gradient diffusion term, which we model using the RDGF approach. However, comparisons of the statistics obtained from this model with DNS data reveal substantial errors due to erroneously large fluctuations generated by the model. We address this defect by incorporating into the closure approximation information regarding the scalar gradient production along the local trajectory history of the particle. This modified model makes predictions for the scalar gradients, their production rates, and alignments with the strain-rate eigenvectors that are in very good agreement with DNS data. However, while the model yields valid predictions up to $\textit {Re}_\lambda \approx 500$ , beyond this, the model breaks down.

Rigorous biaxial limit of a molecular-theory-based two-tensor hydrodynamics

We consider a two-tensor hydrodynamics derived from the molecular model, where high-order tensors are determined by closure approximation through the maximum entropy state or the quasi-entropy. We prove the existence and uniqueness of local in time smooth solutions to the two-tensor system. Then, we rigorously justify the connection between the molecular-theory-based two-tensor hydrodynamics and the biaxial frame hydrodynamics. More specifically, in the framework of Hilbert expansion, we show the convergence of the solution to the two-tensor hydrodynamics to the solution to the frame hydrodynamics.

Nonrelativistic limit of the Euler‐HMPN approximation models arising in radiation hydrodynamics

In this paper, we are concerned with the nonrelativistic limit of a class of computable approximation models for radiation hydrodynamics. The models consist of the compressible Euler equations coupled with moment closure approximations to the radiative transfer equation. They are first‐order partial differential equations with source terms. As hyperbolic relaxation systems, they are showed to satisfy the structural stability condition proposed by the second author. Based on this, we verify the nonrelativistic limit by combining an energy method with a formal asymptotic analysis.

Testing orientational closure approximations in dilute and non-dilute suspensions with Rheo-SANS

Testing orientational closure approximations in dilute and non-dilute suspensions with Rheo-SANS

Quantifying the stochasticity of policy parameters in reinforcement learning problems.

The stochastic dynamics of reinforcement learning is studied using a master equationformalism. We consider two different problems-Q learning for a two-agent game and the multiarmed bandit problem with policy gradient as the learning method. The master equationis constructed by introducing a probability distribution over continuous policy parameters or over both continuous policy parameters and discrete state variables (a more advanced case). We use a version of the moment closure approximation to solve for the stochastic dynamics of the models. Our method gives accurate estimates for the mean and the (co)variance of policy variables. For the case of the two-agent game, we find that the variance terms are finite at steady state and derive a system of algebraic equationsfor computing them directly.

Variety of Planar Fourth‐Order Fiber Orientation Tensors and Implications on Effective Elastic Stiffnesses

AbstractIn this contribution, selected results from [1–3] are presented in a compact and simplified way. In addition, the variety of fiber orientation tensors is used to determine a maximum deviation of the direction‐dependent Young's modulus, which can arise if only second‐order directional information is included in a specific meanfield homogenization. Focusing on the special case of planar fiber distributions, the variety of fiber orientation tensors identified in [1] is considered as a design space. This design space is completely explored for the orientation‐averaging homogenization following [4], fixed material parameters and fixed fiber volume content. The possible directional dependence of the resulting effective stiffnesses is graphically presented using polar plots of the direction‐dependent Young's modulus. These polar plots are arranged on two‐dimensional slices within the parameter space of planar fourth‐order fiber orientation tensors. This gives a complete representation of the influence of the orientation tensor on the anisotropic stiffness tensor. Consequences of closure approximations, i.e., restriction to second‐order directional information, are demonstrated and motivate measurement of fourth‐order fiber orientation tensors.

On the Phase Space of Fourth-Order Fiber-Orientation Tensors

Fiber-orientation tensors describe the relevant features of the fiber-orientation distribution compactly and are thus ubiquitous in injection-molding simulations and subsequent mechanical analyses. In engineering applications to date, the second-order fiber-orientation tensor is the basic quantity of interest, and the fourth-order fiber-orientation tensor is obtained via a closure approximation. Unfortunately, such a description limits the predictive capabilities of the modeling process significantly, because the wealth of possible fourth-order fiber-orientation tensors is not exploited by such closures, and the restriction to second-order fiber-orientation tensors implies artifacts. Closures based on the second-order fiber-orientation tensor face a fundamental problem – which fourth-order fiber-orientation tensors can be realized? In the literature, only necessary conditions for a fiber-orientation tensor to be connected to a fiber-orientation distribution are found. In this article, we show that the typically considered necessary conditions, positive semidefiniteness and a trace condition, are also sufficient for being a fourth-order fiber-orientation tensor in the physically relevant case of two and three spatial dimensions. Moreover, we show that these conditions are not sufficient in higher dimensions. The argument is based on convex duality and a celebrated theorem of D. Hilbert (1888) on the decomposability of positive and homogeneous polynomials of degree four. The result has numerous implications for modeling the flow and the resulting microstructures of fiber-reinforced composites, in particular for the effective elastic constants of such materials. Based on our findings, we show how to connect optimization problems on fourth-order fiber-orientation tensors to semi-definite programming. The proposed formulation permits to encode symmetries of the fiber-orientation tensor naturally. As an application, we look at the differences between orthotropic and general, i.e., triclinic, fiber-orientation tensors of fourth order in two and three spatial dimensions, revealing the severe limitations inherent to orthotropic closure approximations.

Magnetic Helicity Fluxes from Triple Correlators

Fluxes of the magnetic helicity density play an important role in large-scale turbulent dynamos, allowing the growth of large-scale magnetic fields while overcoming catastrophic quenching. We show here, analytically, how several important types of magnetic helicity fluxes can arise from terms involving triple correlators of fluctuating fields in the helicity density evolution equation. For this, we assume incompressibility and weak inhomogeneity, and use a quasi-normal closure approximation: fourth-order correlators are replaced by products of second-order ones, and the effect of the fourth-order cumulants on the evolution of the third moments is modeled by a strong damping term. First, we show how a diffusive helicity flux, until now only measured in simulations, arises from the triple correlation term. This is accompanied by what we refer to as a random advective flux, which predominantly transports magnetic helicity along the gradients of the random fields. We also find that a new helicity flux contribution, in some aspects similar to that first proposed by Vishniac, can arise from the triple correlator. This contribution depends on the gradients of the random magnetic and kinetic energies along the large-scale vorticity, and thus arises in any rotating, stratified system, even if the turbulence is predominantly non-helical. It can source a large-scale dynamo by itself while spatially transporting magnetic helicity within the system.

Planar fiber orientation: Jeffery, non-orthotropic closures, and reconstructing distribution functions

Many models for fiber suspensions use the second-order orientation tensor to represent the orientation state. This requires that the fourth-order orientation tensor be approximated using a closure formula. The prediction of non-linear properties also requires reconstructing an approximate orientation distribution function. All existing fourth-order closures and most reconstructions have orthotropic symmetry, which limits their accuracy. In this paper, closures and reconstruction are explored for planar fiber orientation, using a recent parameterization of fourth-order orientation tensors by Bauer and Böhlke (2021, 2022a, 2022b). We show that the orientation distribution for fibers that follow Jeffery’s equation, together with the associated natural closure, form the central axis of the space of fourth-order orientation tensors. A new family of non-orthotropic planar closure approximations is created by including the rate-of-deformation tensor as an additional independent variable. These closures are shown to be substantially more accurate than any planar orthotropic closure. We also explore the Jeffery, Bingham, ellipse radius, and fourth-order maximum entropy (ME4) functions for reconstructing the orientation distribution. When compared to distributions from the Folgar-Tucker model, ME4 using the exact fourth-order orientation tensor gives the most accurate reconstruction, while ME4 with the fourth-order tensor from the new non-orthotropic closure is nearly as good. • The Jeffery distribution is central to planar fourth-order orientation tensors. • The first-ever non-orthotropic closure approximations are developed. • Non-orthotropic closures are more accurate than any previous planar closure. • Fourth-order maximum entropy distributions give the most accurate reconstruction.

Local thermodynamic consistency for integral equations describing single-component fluids.

A new closure approximation is presented here, and it is based on two thermodynamic relations, namely, a particular case of a local expression together with a global thermodynamic condition. The results obtained from this local approximation are compared with thermodynamic and structural properties determined by using well-established closure approximations as well as numerical simulations for different kinds of interaction potentials. In terms of numerical simulations, the new closure delivers results similar to and sometimes better than the well-known closure relations that are specialized in certain types of interactions.

Weinberg operator contribution to the CP -odd nuclear force in the quark model

The contribution of the CP violating three-gluon interaction, proposed by Weinberg, to the short-range CP-odd nuclear force is evaluated in the nonrelativistic quark model. We first show that the naive leading contribution generated by the quark exchange process vanishes at sufficiently short distance within the resonating group method, by considering the one-loop level gluon exchange CP-odd interquark potential induced by the Weinberg operator with massive quarks and gluons. We then estimate the true leading contribution by evaluating the gluonic correction to the CP-odd interquark potential in the closure approximation. It is found that the resulting irreducible CP-odd nuclear force is comparable to that generated by the chiral rotation of the CP-even short-range nuclear force, where the CP-odd mass calculated with QCD sum rules is used as input. The explicit calculation of the electric dipole moment (EDM) of the $^3$He nucleus yields $d_{^3{\rm He}}^{\rm (irr)} (w) = - 1.5 \, w\, e $ MeV. The total $^3$He EDM, accounting for the intrinsic nucleon EDM, the pion-exchange and the short-range CP-odd nuclear force, is $d_{^3{\rm He}}^{\rm (tot)} (w) = 20_{-11}^{+14} \, w\, e $ MeV, with the dominant effect coming from the intrinsic nucleon EDM.

Transfer reactions between odd-odd and even-even nuclei by using the interacting boson fermion fermion model and 0νββ decay in closure approximation

Spectroscopic amplitudes (SAs) in the interacting boson fermion fermion model (IBFFM) are necessary for the computation of $0\ensuremath{\nu}\ensuremath{\beta}\ensuremath{\beta}$ decay but also for cross sections of heavy-ion reactions, in particular, double charge exchange reactions for the NUMEN Collaboration, if one does not want to use the closure limit. We present for the first time (i) the formalism and operators to compute in a general case the spectroscopic amplitudes in the IBFFM scheme from even-even to odd-odd nuclei, in a way suited to be used in reaction code, i.e., extracting the contribution of each orbital; (2) the odd-odd nuclei as described by the old IBFFM are obtained for the first time with the new implementation of machine learning (ML) techniques for fitting the parameters, getting a more realistic description. The one-body transition densities for $^{116}\mathrm{Cd}\ensuremath{\rightarrow}^{116}\mathrm{In}$ and $^{116}\mathrm{In}\ensuremath{\rightarrow}^{116}\mathrm{Sn}$ are part of the experimental program of the NUMEN experiment, which aims to find constraints on $0\ensuremath{\nu}\ensuremath{\beta}\ensuremath{\beta}$ decay matrix elements. As a first application of the spectroscopic amplitudes we present the calculation of the $0\ensuremath{\nu}\ensuremath{\beta}\ensuremath{\beta}$ nuclear matrix elements in the IBFFM formalism in closure approximation.

A Molecular-Based Q-Tensor Hydrodynamic Theory of Smectic Liquid Crystals

The Doi–Onsager molecular theory is capable of providing a rather accurate description of the local behavior of molecules; however, its computation is extremely time-consuming, since some higher-dimensional variables are typically involved. Therefore, establishing a computable reduced model that can capture essential physical properties is an important issue. In this work, we derived a reduced Q-tensor hydrodynamic theory that described smectic phases with density variations from the Doi–Onsager molecular theory using the Bingham closure approximation. The coefficients in the tensor model were derived from those in the molecular model. The energy dissipation law was inherited from the tensor model. Some special cases for the model were also discussed.

Novel closure approximation for prediction of the effective elastic properties of composites with discontinuous reinforcement

The paper introduces a novel closure approximation method for the determination of the fourth-order orientation tensor in terms of known second-order tensor. The fourth-order orientation tensor must be known to predict the effective elastic properties of composites with misaligned fibres but very often only the second-order orientation is available. The proposed method operates on a set of outcomes of optimization based on a genetic algorithm. In consequence, it leads to obtaining distributions of effective elastic constants instead of deterministic quantities. Results obtained by using the novel approach have been compared with results based on exact fourth-order orientation tensor, tensor known from experimental data and determined by using popular closure approximations from literature. Furthermore, the obtained results have been compared with results of finite element analysis based on geometrical models representing the microstructure which contains several hundred fibres. The new method provides high accuracy of elastic constant predictions for all studied fibre orientation distributions, which distinguishes it from the closure approximation methods known from the literature. Moreover, results obtained in form of distributions give an insight into the uncertainty of the predictions.

On the Generalizability of Machine-Learning-Assisted Anisotropy Mappings for Predictive Turbulence Modelling

Several machine learning frameworks for augmenting turbulence closure models have been recently proposed. However, the generalizability of an augmented turbulence model remains an open question. We investigate this question by systematically varying the training and test sets of several models. An optimal three-term tensor basis expansion is used to develop a model-agnostic data-driven turbulence closure approximation. Then, hyperparameter optimization is performed for a random forest, a neural network, and an eXtreme Gradient Boosting (XGBoost) model. We recommend XGBoost for data-driven turbulence closure modelling owing to its low-tuning cost and good performance. We also find that machine learning models generalize well to new parametric variations of flows seen in the training dataset, but lack generalizability to new flow types. This generalizability gap suggests that machine learning methods are most suited for developing specialized models for a given flow type, a problem often encountered in industrial applications.

A comparative study between fiber orientation closure approximations and a new orthotropic closure

AbstractIn injection and compression molding simulation, orientation tensors provide an efficient way with less computational effort to calculate flow‐induced fiber orientations. In these flow calculations, the solution of any even‐ordered orientation tensor needs the following even higher‐ordered orientation tensor. Therefore, a closure is used in order to approximate the higher‐ordered orientation tensor as a function of the components of the lower‐ordered orientation tensors. There exist many closures of the fourth‐order orientation tensor in terms of the second‐order orientation. This paper gives a review with mathematical details for different closure approximations including simple closures, composite closures, eigenvalue‐ based orthotropic closures, invariant‐based closures, and neural‐network‐ based closures. Moreover, a new closure approximation that assumes an orthotropic stiffness tensor was suggested in this work. The results of the closure approximations were validated through the comparison of Young moduli between the closure approximations and the experimental results for both short‐ and long‐fiber‐reinforced composites.