Moment Space Research Articles

Threshold resummation at NNLL for single-particle production in hadronic collisions

We develop threshold resummation for single-particle inclusive cross sections in hadron-hadron collisions to the level of next-to-next-to-leading logarithm, up to full matching with two-loop hard functions. We define and calculate all one-loop soft functions for all partonic channels. This enables us to separate the hard and soft functions at one loop. Along with these results, the one-loop finite parts of jet functions are used to check that the full soft, collinear and virtual corrections are reproduced to one loop for all partonic reactions. We exhibit these NLO results explicitly. NLO expansions of the resummed cross section match the exact NLO results extremely well numerically, and two loop expansions result in substantial corrections over many kinematic configurations. Explicit results are given in Mellin moment space, and a number of options for generating resummed cross sections are discussed.

Determinants of block Hankel matrices for random matrix-valued measures

We consider the moment space M2n+1dn of moments up to the order 2n+1 of dn×dn real matrix measures defined on the interval [0,1]. The asymptotic properties of the Hankel determinant {logdet(Mi+jdn)i,j=0,…,⌊nt⌋}t∈[0,1] of a uniformly distributed vector (M1,…,M2n+1)t∼U(M2n+1) are studied when the dimension n of the moment space and the size of the matrices dn converge to infinity. In particular weak convergence of an appropriately centered and standardized version of this process is established. Mod-Gaussian convergence is shown and several large and moderate deviation principles are derived. Our results are based on some new relations between determinants of subblocks of the Jacobi-beta-ensemble, which are of their own interest and generalize Bartlett decomposition-type results for the Jacobi-beta-ensemble from the literature.

Discrete Boltzmann method for non-equilibrium flows: Based on Shakhov model

A general framework for constructing discrete Boltzmann model for non-equilibrium flows based on the Shakhov model is presented. The Hermite polynomial expansion and a set of discrete velocity with high spatial isotropy are adopted to solve the kinetic moments of discrete equilibrium distribution function. Such a model possesses both an adjustable specific heat ratio and Prandtl number, and can be applied to a wide range of flow regimes including continuous, slip, and transition flows. To recover results for actual situations, the nondimensionalization process is demonstrated. To verify and validate the new model, several typical non-equilibrium flows including the Couette flow, Fourier flow, unsteady boundary heating problem, cavity flow, and Kelvin–Helmholtz instability are simulated. Comparisons are made between the results of discrete Boltzmann model and those of previous models including analytic solution in slip flow, Lattice ES-BGK, and DSMC based on both BGK and hard-sphere models. The results show that the new model can accurately capture the velocity slip and temperature jump near the wall, and show excellent performance in predicting the non-equilibrium flow even in transition flow regime. In addition, the measurement of non-equilibrium effects is further developed and the non-equilibrium strength Dn∗ in the nth order moment space is defined. The non-equilibrium characteristics and the advantage of using Dn∗ in Kelvin–Helmholtz instability are discussed. It concludes that the non-equilibrium strength Dn∗ is more appropriate to describe the interfaces than the individual components of Δn∗. Besides, the D3∗ and D3,1∗ can provide higher resolution interfaces in the simulation of Kelvin–Helmholtz instability.

Equivalent distortion risk measures on moment spaces

We show that maximizing distortion risk measures over the set of distributions with given mean is equivalent to maximizing their concave counterpart. In the case of Value-at-Risk and Tail Value-at-Risk the equivalence also holds when adding information on higher moments.

Radar detection in the moments space of the scattered signal parameters

This paper proposes the theoretical foundations of a novel radar detection method, based on analysis and decision making in the moments space of the scattered signal parameters. Through an adaptive process and from large-size samples of these parameters, several normal-distributed moments are calculated, which allow assigning each resolution cell of the searching region to the background or anomaly classes. Following the Neyman–Pearson criterion two optimal detection algorithms are proposed, one for statistically independent moments and other for correlated moments. Finally, three background-anomaly pairs are simulated, taking the amplitude of the video signal as parameter and two of its moments for decision making. As result, the detection curves for two sizes of the sample are presented, showing the possibilities of the proposed method.

Upper bounds for strictly concave distortion risk measures on moment spaces

The study of worst-case scenarios for risk measures (e.g., Value-at-Risk) when the underlying risk (or portfolio of risks) is not completely specified is a central topic in the literature on robust risk measurement. In this paper, we tackle the open problem of deriving upper bounds for strictly concave distortion risk measures on moment spaces. Building on early results of Rustagi (1957, 1976), we show that in general this problem can be reduced to a parametric optimization problem. We completely specify the sharp upper bound (and corresponding maximizing distribution function) when the first moment and any other higher moment are fixed. Specifically, in the case of a fixed mean and variance, we generalize the Cantelli bound for (Tail) Value-at-Risk in that we express the sharp upper bound for a strictly concave distorted expectation as a weighted sum of the mean and standard deviation.

Symmetrized operator split schemes for force and source modeling in cascaded lattice Boltzmann methods for flow and scalar transport.

Operator split forcing schemes exploiting a symmetrization principle, i.e., Strang splitting, for cascaded lattice Boltzmann (LB) methods in two- and three-dimensions for fluid flows with impressed local forces are presented. Analogous scheme for the passive scalar transport represented by a convection-diffusion equation with a source term in a novel cascaded LB formulation is also derived. They are based on symmetric applications of the split solutions of the changes on the scalar field or fluid momentum due to the sources or forces over half time steps before and after the collision step. The latter step is effectively represented in terms of the post-collision change of moments at zeroth and first orders, respectively, to represent the effect of the sources on the scalar transport and forces on the fluid flow. Such symmetrized operator split cascaded LB schemes are consistent with the second-order Strang splitting and naturally avoid any discrete effects due to forces or sources by appropriately projecting their effects for higher-order moments. All the force or source implementation steps are performed only in the moment space and they do not require formulations as extra terms and their additional transformations to the velocity space. These result in particularly simpler and efficient schemes to incorporate forces or sources in the cascaded LB methods unlike those considered previously. Numerical study for various benchmark problems in 2D and 3D for fluid flow problems with body forces and scalar transport with sources demonstrate the validity and accuracy, as well as the second-order convergence rate of the symmetrized operator split forcing or source schemes for the cascaded LB methods.

On the Five-Moment Hamburger Maximum Entropy Reconstruction

We consider the Maximum Entropy Reconstruction (MER) as a solution to the five-moment truncated Hamburger moment problem in one dimension. In the case of five monomial moment constraints, the probability density function (PDF) of the MER takes the form of the exponential of a quartic polynomial. This implies a possible bimodal structure in regions of moment space. An analytical model is developed for the MER PDF applicable near a known singular line in a centered, two-component, third- and fourth-order moment (μ_3, μ_4) space, consistent with the general problem of five moments. The model consists of the superposition of a perturbed, centered Gaussian PDF and a small-amplitude packet of PDF-density, called the outlying moment packet (OMP), sitting far from the mean. Asymptotic solutions are obtained which predict the shape of the perturbed Gaussian and both the amplitude and position on the real line of the OMP. The asymptotic solutions show that the presence of the OMP gives rise to an MER solution that is singular along a line in ( μ_3, μ_4) space emanating from, but not including, the point representing a standard normal distribution, or thermodynamic equilibrium. We use this analysis of the OMP to develop a numerical regularization of the MER, creating a procedure we call the Hybrid MER (HMER). Compared with the MER, the HMER is a significant improvement in terms of robustness and efficiency while preserving accuracy in its prediction of other important distribution features, such as higher order moments.

A-hypergeometric distributions and Newton polytopes

We give a bijection between a quotient space of the parameters and the space of moments for any A-hypergeometric distribution. An algorithmic method to compute the inverse image of the map is proposed utilizing the holonomic gradient method and an asymptotic equivalence of the map and the iterative proportional scaling. The algorithm gives a method for solving a conditional maximum likelihood estimation problem in statistics. The interplay between the theory of hypergeometric functions and statistics allows us to give some new formulas for A-hypergeometric polynomials.

Advanced lattice Boltzmann scheme for high-Reynolds-number magneto-hydrodynamic flows

ABSTRACTIs the lattice Boltzmann method suitable to investigate numerically high-Reynolds-number magneto-hydrodynamic (MHD) flows? It is shown that a standard approach based on the Bhatnagar–Gross–Krook (BGK) collision operator rapidly yields unstable simulations as the Reynolds number increases. In order to circumvent this limitation, it is here suggested to address the collision procedure in the space of central moments for the fluid dynamics. Therefore, an hybrid lattice Boltzmann scheme is introduced, which couples a central-moment scheme for the velocity with a BGK scheme for the space-and-time evolution of the magnetic field. This method outperforms the standard approach in terms of stability, allowing us to simulate high-Reynolds-number MHD flows with non-unitary Prandtl number while maintaining accuracy and physical consistency.

Conditional hyperbolic quadrature method of moments for kinetic equations

The conditional quadrature method of moments (CQMOM) was introduced by Yuan and Fox (2011) [4] to reconstruct a velocity distribution function (VDF) from a finite set of its integer moments. The reconstructed VDF takes the form of a sum of weighted Dirac delta functions in velocity phase space, and provides a closure for the spatial flux term in the corresponding kinetic equation. The CQMOM closure for the flux leads to a weakly hyperbolic system of moment equations. In subsequent work by Chalons et al. (2010) [8], the Dirac delta functions were replaced by Gaussian distributions, which make the moment system hyperbolic but at the added cost of dealing with continuous distributions. Here, a hyperbolic version of CQMOM is proposed that uses weighted Dirac delta functions. While the moment set employed for multi-Gaussian and conditional HyQMOM (CHyQMOM) are equivalent, the latter is able to access all of moment space whereas the former cannot (e.g. arbitrary values of the fourth-order velocity moment in 1-D phase space with two nodes). By making use of the properties of CHyQMOM in 2-D phase space, it is possible to control a symmetrical subset of the optimal moments from Fox (2009) [24]. Furthermore, the moment sets for 2-D problems are smaller for CHyQMOM than in the original CQMOM thanks to a judicious choice of the velocity abscissas in phase space.

Universality in Random Moment Problems

Let $\mathcal{M} _n(E)$ denote the set of vectors of the first $n$ moments of probability measures on $E\subset \mathbb{R} $ with existing moments. The investigation of such moment spaces in high dimension has found considerable interest in the recent literature. For instance, it has been shown that a uniformly distributed moment sequence in $\mathcal M_n([0,1])$ converges in the large $n$ limit to the moment sequence of the arcsine distribution. In this article we provide a unifying viewpoint by identifying classes of more general distributions on $\mathcal{M} _n(E)$ for $E=[a,b],\,E=\mathbb{R} _+$ and $E=\mathbb{R} $, respectively, and discuss universality problems within these classes. In particular, we demonstrate that the moment sequence of the arcsine distribution is not universal for $E$ being a compact interval. Rather, there is a universal family of moment sequences of which the arcsine moment sequence is one particular member. On the other hand, on the moment spaces $\mathcal{M} _n(\mathbb{R} _+)$ and $\mathcal{M} _n(\mathbb{R} )$ the random moment sequences governed by our distributions exhibit for $n\to \infty $ a universal behaviour: The first $k$ moments of such a random vector converge almost surely to the first $k$ moments of the Marchenko-Pastur distribution (half line) and Wigner’s semi-circle distribution (real line). Moreover, the fluctuations around the limit sequences are Gaussian. We also obtain moderate and large deviations principles and discuss relations of our findings with free probability.

High Order Moment Model for Polydisperse Evaporating Sprays towards Interfacial Geometry Description

In this paper we propose a new Eulerian modeling and related accurate and robust numerical methods, describing polydisperse evaporating sprays, based on high order moment methods in size. The main novelty of this model is its capacity to describe some geometrical variables of the droplet-gas interface, by analogy with the liquid-gas interface in interfacial flows. For this purpose, we use fractional size-moments, where the size variable is taken as the droplet surface. In order to evaluate the evaporation of the polydisperse spray, we use a smooth reconstruction which maximizes the Shannon entropy. However, the use of fractional moments introduces some theoretical and numerical difficulties, which need to be tackled. First, relying on a study of the moment space, we extend the Maximum Entropy (ME) reconstruction of the size distribution to the case of fractional moments. Then, we propose a new accurate and realizable algorithm to solve the moment evolution due to evaporation, which preserves the structure of the moment space. This algorithm is based on a mathematical analysis of the kinetic evolution due to evaporation, where it shown that the evolution of some negative order fractional moments have to be properly predicted, a peculiarity related to the use of fractional moments. The present model and numerical schemes yield an accurate and stable evaluationof the moment dynamics with minimal number of variables, as well as a minimal computational cost as with the EMSM model, but with the very interesting additional capacity of coupling with diffuse interface model and transport equation of averaged geometrical interface variables, which are essential in oder to describe atomization.

Upper Bounds for Concave Distortion Risk Measures on Moment Spaces

The study of worst-case scenarios for risk measures (e.g., Value-at-Risk) when the underlying risk (or portfolio of risks) is not completely specified is a central topic in the literature on robust risk measurement. In this paper, we tackle the open problem of deriving upper bounds for strictly concave distortion risk measures on moment spaces. Building on early results of Rustagi (1957, 1976), we show that in general this problem can be reduced to a parametric optimization problem. We completely specify the sharp upper bound (and corresponding maximizing distribution function) when the first moment and any other higher moment are fixed. Specifically, in the case of a fixed mean and variance, we generalize the Cantelli bound for (Tail) Value-at-Risk in that we express the sharp upper bound for a strictly concave distorted expectation as a weighted sum of the mean and standard deviation.

Non-realisability problem with the conventional method of moments in wet-steam flows

The method of moments offers an efficient way to preserve the essence of particle size distribution, which is required in many engineering problems such as modelling wet-steam flows. However, in the context of the finite volume method, high-order transport algorithms are not guaranteed to preserve the moment space, resulting in so-called ‘non-realisable’ moment sets. Non-realisability poses a serious obstacle to the quadrature-based moment methods, since no size distribution can be identified for a non-realisable moment set and the moment-transport equations cannot be closed. On the other hand, in the case of conventional method of moments, closures to the moment-transport equations are directly calculated from the moments themselves; as such, non-realisability may not be a problem. This article describes an investigation of the effects of the non-realisability problem on the flow conditions and moment distributions obtained by the conventional method of moments through several one-dimensional test cases involving systems that exhibited similar characteristics to low-pressure wet-steam flows. The predictions of pressures and mean droplet sizes were not considerably disturbed due to non-realisability in any of the test cases. However, in one case that was characterised by strong temporal and spatial gradients, non-realisability did undermine the accuracy of the predictions of measures for the underlying size distributions, including the standard deviation and skewness.

The analytic structure and the transcendental weight of the BFKL ladder at NLL accuracy

We study some analytic properties of the BFKL ladder at next-to-leading logarithmic accuracy (NLLA). We use a procedure by Chirilli and Kovchegov to construct the NLO eigenfunctions, and we show that the BFKL ladder can be evaluated order by order in the coupling in terms of certain generalised single-valued multiple polylogarithms recently introduced by Schnetz. We develop techniques to evaluate the BFKL ladder at any loop order, and we present explicit results up to five loops. Using the freedom in defining the matter content of the NLO BFKL eigenvalue, we obtain conditions for the BFKL ladder in momentum space at NLLA to have maximal transcendental weight. We observe that, unlike in moment space, the result in momentum space in mathcal{N} = 4 SYM is not identical to the maximal weight part of QCD, and moreover that there is no gauge theory with this property. We classify the theories for which the BFKL ladder at NLLA has maximal weight in terms of their field content, and we find that these theories are highly constrained: there are precisely four classes of theories with this property involving only fundamental and adjoint matter, all of which have a vanishing one-loop beta function and a matter content that fits into supersymmetric multiplets. Our findings indicate that theories which have maximal weight are highly constrained and point to the possibility that there is a connection between maximal transcendental weight and superconformal symmetry.

Hybrid method of moments with interpolation closure–Taylor-series expansion method of moments scheme for solving the Smoluchowski coagulation equation

This paper presents a hybrid method of moments with interpolation closure–Taylor-series expansion method of moments (MoMIC–TEMoM) scheme for solving the Smoluchowski coagulation equation. In the proposed scheme, the exponential function, which arises in the conversion from a particle size distribution space to a space of moments, is expressed in an additive form using the third-order Taylor-series expansion; the implicit moments are approximated using two Lagrange interpolation functions, namely the newly defined normalized moment function and the normalized moment function defined by Frenklach and Harris (1987). The new hybrid scheme allows implementation of the method of moments with an arbitrary type of moment sequence, and it overcomes the shortcomings of the Taylor-series expansion moment method proposed by Frenklach and Harris. The proposed scheme is verified with three aerosol dynamics, namely Brownian coagulation in the free molecular regime, Brownian coagulation in the continuum-slip regime, and turbulence coagulation. The results reveal that the hybrid MoMIC–TEMoM scheme has similar accuracy to currently recognized methods including the quadrature method of moments, MoMIC, and TEMoM, and its accuracy can be further enhanced as the fractional moment sequence type is used for Brownian coagulation in the free molecular regime. Thus, the proposed scheme is a reliable for solving the Smoluchowski coagulation equation.

Fragmentation functions beyond fixed order accuracy

We give a detailed account of the phenomenology of all-order resummations of logarithmically enhanced contributions at small momentum fraction of the observed hadron in semi-inclusive electron-positron annihilation and the time-like scale evolution of parton-to-hadron fragmentation functions. The formalism to perform resummations in Mellin moment space is briefly reviewed, and all relevant expressions up to next-to-next-to-leading logarithmic order are derived, including their explicit dependence on the factorization and renormalization scales. We discuss the details pertinent to a proper numerical implementation of the resummed results comprising an iterative solution to the time-like evolution equations, the matching to known fixed-order expressions, and the choice of the contour in the Mellin inverse transformation. First extractions of parton-to-pion fragmentation functions from semi-inclusive annihilation data are performed at different logarithmic orders of the resummations in order to estimate their phenomenological relevance. To this end, we compare our results to corresponding fits up to fixed, next-to-next-to-leading order accuracy and study the residual dependence on the factorization scale in each case.

Realizable second-order finite-volume schemes for the advection of moment sets of the particle size distribution

The accurate description and robust simulation at relatively low cost of a size polydisperse population of fine particles in a carrier fluid is still a major challenge for many applications. For this purpose, moment methods, derived from a population balance equation, represent a very interesting strategy. However, one of the major issues of such methods is the realizability: the numerical schemes have to ensure that the moment sets stay realizable, i.e. that an underlying distribution exists. This issue is all the more crucial that some moment vectors can be at the boundary of the moment space for practical applications, corresponding to a population of particles with only one or a few sizes. It is then investigated here for the advection operator, for which it is particularly significant. Then second order realizable kinetic finite volume schemes are designed, with two strategies for the fluxes evaluation based on the work of Kah et al. [1] and of Vikas et al. [2], which are here completely revisited, extended to take into account the boundary of the moment space and any number of moments, analyzed and compared in a Cartesian mesh context. For a potential easiest generalization to unstructured meshes, simplified but still realizable versions of these schemes are also developed. The high accuracy of all the schemes is then numerically checked on 1D and 2D test cases, with Cartesian meshes, and their robustness is shown, even when some moment vectors are at the boundary of the moment space.

Non-orthogonal multiple-relaxation-time lattice Boltzmann method for axisymmetric thermal flows

Axisymmetric thermal flows in cylindrical systems are widely encountered in engineering practices. Typically, axisymmetric thermal flows belong in three-dimensional (3D) problems. However, taking advantage of the axisymmetric condition, the 3D axisymmetric flows can be reduced to quasi two-dimensional (2D) problems in the meridian plane, which significantly reduces the computational requirements and avoids treating the curved boundary. In recent years, various 2D lattice Boltzmann (LB) models, including single relaxation time LB (SRT-LB, or LBGK) and multiple relaxation time LB (MRT-LB) models, for axisymmetric thermal flows have been proposed. In the LB community, it is well accepted that the MRT-LB is superior to the LBGK in terms of numerical stability. The existing MRT-LB model for axisymmetric thermal flows are developed based on orthogonal basis vectors obtained from the combination of the lattice velocity components, i.e., the transform matrix in the existing MRT-LB is an orthogonal one. Unlike the existing MRT-LB model, in this paper, a non-orthogonal multiple-relaxation-time lattice Boltzmann (MRT-LB) method of simulating axisymmetric thermal flows is proposed. In the proposed MRT-LB method, the velocity field is solved by a D2Q9 discrete velocity set while the temperature by a D2Q5 discrete velocity set. The main advantage of the present MRT-LB model is that the transform matrix of the model is a non-orthogonal one, which is comprised of some proper non-orthogonal basis vectors obtained from the combination of the lattice velocity components. The non-orthogonal transform matrix of the present MRT-LB model contains more zero elements than the classical orthogonal transform matrix, and thus the present MRT-LB model is expected to be more efficient than the existing orthogonal-based MRT-LB model. The equilibrium velocity and temperature moments of the present MRT-LB model are expressed by mapping the equilibrium distribution functions onto their moment spaces through using the non-orthogonal transformation matrix. Also the vectors in the forcing term are modified according to the matrix mapping. Through the Chapman-Enskog analysis, it is demonstrated that the macroscopic governing equations in the cylindrical coordinate can be recovered from the present MRT-LB model. Then several numerical tests, including thermal Womersley flow, Rayleigh-Bnard convection in a vertical cylinder and natural convection in a vertical annulus, are conducted to validate the present model. It is found that the present numerical results are in good agreement with the analytical solutions and/or other numerical results reported in the literature. Numerical stability is also tested, and the results suggest that the present MRT model shows better numerical stability than its LBGK counterpart. Moreover, the numerical results also indicate that the present MRT-LB model is more computationally efficient than the existing MRT-LB model for axisymmetric thermal flow. These findings indicate that the present MRT-LB model can serve as a powerful method of computing the axisymmetric thermal flows.