Set Of Moment Equations Research Articles

Fluid dynamics of out of equilibrium boost invariant plasmas

We establish a set of equations for moments of the distribution function. In the relaxation time approximations, these moments obey a coupled set of equations that can be truncated order-by-order. Solving the equations of moments, we are able to identify an attractor solution that controls a transition from a free streaming fixed point to a hydrodynamic fixed point. In particular, this attractor solution provides a renormalization of the effective value of the shear viscosity to entropy density ratio, η/s, taking into account off-equilibrium effects.

Moment-based availability prediction for bike-sharing systems

We study the problem of predicting the future availability of bikes in a bike station through the moment analysis of a PCTMC model with time-dependent rates. Given a target station for prediction, the moments of the number of available bikes in the station at a future time can be derived by a set of moment equations with an initial set-up given by the snapshot of the current state of all stations in the system. A directed contribution graph is constructed, and a contribution propagation method is proposed to prune the PCTMC so that it only contains stations which have significant contribution to the journey flows to the target station. Once the moments have been derived, the underlying probability distribution of the available number of bikes is reconstructed through the maximum entropy approach. We illustrate our approach on Santander Cycles, the bike-sharing system in London. The model is parameterized using historical data from Santander Cycles. Experimental results show that our model outperforms a time-inhomogeneous Markov queueing model with respect to several performance metrics for bike availability prediction.

A new reduced model for the moments of droplet size distribution in condensing flow

Purpose– The purpose of this paper is to create a computationally efficient reduced model (RM) for the moments of droplet size distribution (DSD) in condensing flow.Design/methodology/approach– The kinetic equation (KE) exactly describes the time dependence of the DSD and can be regarded as the most rigorous representation of a system with condensation. Because of the typical wide range in droplet size, the KE requires excessive computational time and is not attractive for most practical applications. To reduce the overall computational efforts, a novel set of moment equations, derived from the KE has been proposed.Findings– To demonstrate the simplicity and accuracy of the model, the authors employ a typical nucleation pulse experiment for which benchmark KE-solutions have also been computed. Comparison of predicted moments from both the RM and the KE approach reveals that the RM is capable of capturing the evolving feature of moments with reasonable accuracy.Originality/value– The authors have created a novel reduced method for numerical computations of the lower-order moments of the DSD in condensing flow. Unlike the typical method of moments, the RM eliminates the need for assumptions on the shape of the distribution function and could estimate the moments at very low computational cost.

A modified TEMOM model for Brownian coagulation of nanoparticles based on the asymptotic solution of the sectional method

The Taylor-series expansion method of moments (TEMOM) is modified to match the behavior of real self-preserved aerosols by taking advantage of the numerical results obtained by the sectional method for Brownian coagulation in both continuum and free molecular regimes. The newly proposed model is able to predict the evolution of the zeroth and second moments more accurately than the original TEMOM when the aerosol size distribution approaches self-preserving or the coagulation time is sufficiently long. A special kind of coordinate diagram, which describes the relationship between the moment equations and one non-dimensional moment is first used to investigate different methods of moments that only involve the first three moments. The errors produced by different methods of moments can be qualitatively explained by these diagrams. By polynomial fitting, a new set of moment equations for Brownian coagulation in the free molecular regime is proposed in the framework of the log-normal preserving theory.

SOLUTION OF POPULATION BALANCE EQUATIONS IN EMULSION POLYMERIZATION USING METHOD OF MOMENTS

In this work, the method of moments is used for solution of population balance equations appearing in modeling of emulsion polymerization (EP). The zero-one model without coagulation effect and the pseudo-bulk model including coagulation effect are investigated as two common approaches for modeling EP processes. The fixed quadrature method is used to close the set of moment equations, and the maximum entropy approach is applied to reconstruct the particle size distribution from a finite number of its moments. Comparing the results with those obtained by the high-precision finite volume technique indicates that, despite the low computational load of the moment method, it has an acceptable accuracy. These features support use of the moment technique for other applications such as on-line control or optimization in particulate processes.

On chaos in the oscillations of galaxies

This paper describes a search for chaotic behaviour in a low-dimensional model of the oscillations of a galaxy. The governing equations of the model are the complete set of moment equations of the second order associated with the collisionless Boltzmann equation. The moment equations reduce to the tensor virial equations and equations governing the evolution of the kinetic energy tensor of the system. The classical integrals of motion of a gravitational many-body system supplement the governing equations. Closure of the system of moment equations requires a specification of the density distribution in the system and the velocity field describing the mean motions of the stars. In this paper, we close the system for a galaxy possessing triplanar symmetry in the case that the planes of symmetry remain fixed in an inertial frame of reference. We model the galaxy as a heterogeneous ellipsoid with an arbitrary stratification of the density distribution, and we represent the mean motions of the stars in terms of a velocity field that sustains that density distribution consistently with the equation of continuity. The evolution of the model is quasi-homologous in the sense that the semi-axes of an ellipsoid characterizing the size and shape of the galaxy are functions of the time, whereas the stratification of the density does not change. With the aid of additional, non-classical integrals of the motion, which were discovered by Som Sunder & Kochhar, we eliminate the kinetic energy tensor from the virial equations and reduce the virial equations to a closed system of equations of motion that govern the semi-axes of the characteristic ellipsoid. Those equations of motion represent the dynamics of a galaxy as the dynamics of an oscillator with just three degrees of freedom, and they admit of a Hamiltonian formulation. We investigate chaotic behaviour in the model for axisymmetric galaxies. The oscillator that represents the dynamics of an axisymmetric galaxy has only two degrees of freedom, and the constancy of the energy integral enables an investigation of the motion with the aid of Poincare return maps. Our criteria for chaotic behaviour are dissolution of invariant curves in the return maps and, more generally, sensitivity of the motion to small changes in the initial conditions. We survey axisymmetric oscillations of a spherically symmetric model and of oblate and prolate, axisymmetric models with the geometries of an elliptical galaxy of Hubble type E5. Oscillations with excitation energies up to the order of 40 per cent of the binding energy of an equilibrium configuration are found to be regular. Substantial families of chaotic oscillations appear at excitation energies of the order of 40-50 per cent of the binding energy. Lyapunov times of these chaotic oscillations are generally in the range of 50-150 dynamical units of time. For galaxies with typical sizes and masses, the Lyapunov times would be of the order of 1.5-4.5 Gyr.

Kinetic description of the expansion of a supersonic pulsed jet to vacuum: II. Mixtures of monoatomic gases

A model for the expansion of a pulsed jet of a mixture of monoatomic gases was advanced on the basis of the Boltzmann kinetic equation. The Grad method was used to solve the Boltzmann equation. The set of moment equations was analyzed by matching asymptotic expansions. The behavior of the velocity slip and temperature difference was analyzed as a function of the conditions in the source and the component-gas concentrations in the mixture. A functional relationship between these quantities and the form of the interaction potential has been established.

Unicité dans la méthode des moments pour les mélanges de deux distributions normales

By studying uniqueness, we show that Pearson's method of moments for mixtures of two normal distributions must be completed. We then invert a second set of moment equations, which is constructed to complete the classical system of moments. We are thus able to stabilize the method. To cite this article: E. Monfrini, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

KINETIC THEORY OF A CHEMICALLY REACTING GAS WITH INELASTIC TRANSITIONS

The problem of a gas mixture of four multilevel molecules undergoing a reversible bimolecular chemical reaction plus mechanical encounters, leading to elastic or inelastic scattering, is addressed in the frame of extended kinetic theory. When elastic scattering is the dominant microscopic interaction, an approximate closed set of moment equations is derived for the evolution of the macroscopic observables in a suitable asymptotic limit. It is shown that the resulting set of PDE's describes correctly the main features of the original integrodifferential kinetic equations, namely conservation equations, equilibria (mass action law), and H-theorem.

Multi-dimensional solidification with internal radiation and temperature dependent properties

In this study, a two-dimensional radial and axial analytical model was developed and solved to determine the transient temperature distribution and interface location of a semi-transparent, gray, absorbing, emitting and isotropically scattering phase change medium bounded by two finite concentric cylinders when internal energy transfer occurs simultaneously by conduction and thermal radiation. The phase change medium employed in this study was a fluoride salt with a weight composition of LiF — 41.27%, MgF 2 — 48.76% and KF — 8.95%. The principle of conservation of energy where enthalpy and temperature are the dependent variables was coupled with a set of moment equations that were derived from radiative transfer equations and Marshak type boundary conditions by applying the spherical harmonics or P-1 differential approximation. An iterative finite difference numerical scheme was used to solve the non-linear difference equations. Results that show the effects of internal radiation on the temperature distribution, the rate of solidification and the heat extracted are presented. It is seen that internal radiation tends to accelerate the solidification process.

ENERGY CONSERVING DISCRETE BOLTZMANN EQUATION FOR NONIDEAL SYSTEMS

The BBGKY formalism is utilized to obtain a set of moment equations to be satisfied by the collision operator in an energy conserving discrete Boltzmann equation for the case of a nonlocal interaction potential. A modified BGK form of the collision operator consistent with these moment equations is described. In the regime of isothermal flows, a previous proposed nonideal gas model is recovered. Other approaches to constructing the collision operator are discussed. Numerical implementation of the modified BGK form, using a thermal lattice Boltzmann model, is illustrated as an example. The time dependence of the density autocorrelation function was studied for this model and found, at early times, to be strongly affected by the constraint of total energy conservation. The long time behavior of the density autocorrelation function was consistent with the theory of hydrodynamic fluctuations.

Self‐Similar Collapse and the Structure of Dark Matter Halos: A Fluid Approach

We explore the dynamical restrictions on the structure of dark matter halos through a study of cosmological self-similar gravitational collapse solutions. A fluid approach to the collisionless dynamics of dark matter is developed, and the resulting closed set of moment equations are solved numerically, including the effect of halo velocity dispersions (both radial and tangential), for a range of spherically averaged initial density profiles. Our results highlight the importance of tangential velocity dispersions for obtaining density profiles shallower than 1/r2 in the core regions, and for retaining a memory of the initial density profile, in self-similar collapse. For an isotropic core velocity dispersion, only a partial memory of the initial density profile is retained. If tangential velocity dispersions in the core are constrained to be less than the radial dispersion, a cuspy core density profile shallower than 1/r cannot hold in self-similar collapse.

A transient inverse scattering problem for acoustic waveguides

This paper deals with the transient inverse scattering problem for determining a compactly-supported velocity inhomogeneity embedded in a uniform acoustic waveguide. It is assumed that the sound velocity within the inhomogeneity is a small perturbation of the background velocity, so that a linerized formulation based on the modal expansion of the pressure field applies. If the inhomogeneity is both depth-independent and axisymmetric, we reduce the reconstructionproblem to a uniquely solvable Volterra integral equation of the second kind. However, if the inhomogeneity is not axisymmetric we are able only to derive an infinite set of moment equations for the unknown refraction index.

Projected moments in relativistic kinetic theory

In this paper a new set of moment equations in relativistic kinetic theory is presented. The moments under consideration are the projections of particle 4-flux and energy momentum tensor with respect to the Eckart velocity or the Landau–Lifshitz velocity, alternatively. The moment equations follow from integrations of the relativistic Boltzmann equation in which the interactions of the particles are described by the relativistic BGK model for reasons of simplicity. The projected moment formalism is extended to an arbitrary number of moments and moment equations and it is shown that the non-relativistic limit of moments and moments equations leads to the so-called central moments of non-relativistic theory. The moment equations may be closed by means of the entropy maximum principle. After this method has been outlined, the closure is performed for the case of 14 moments, i.e. the projections of particle 4-flux and energy momentum tensor. Moreover local thermal equilibrium is considered where the projected moment formalism is used for the derivation of the relativistic Navier–Stokes and Fourier laws. Different choices of moment equations for this task are compared and it is shown that the proper choice of moment equations depends on the interaction term in the relativistic Boltzmann equation.

Linear systems with polynomials of filtered Poisson processes

Moment equations are calculated exactly for the response of linear systems subjected polynomials of filtered Poisson processes. The Itô formula for stochastic differential equations driven by Poisson white noise is applied to derive moment equations. It is shown that the set of moment equations is closed. The proposed method is used to calculate moments up to the fourth order for the response of two linear systems subjected to quadratic forms of filtered Poisson processes. Results by Monte Carlo simulations are also presented for comparison.

A method for suppressing the irregular frequencies from integral equations in water wave-structure. Interaction problems

It is well known that at certain frequencies the boundary integral equation describing water wave-structure interactions does not have a unique solution. Large numerical errors occur in a frequency bandwidth close to each “irregular” frequency and the fact that these frequencies are not a priori known for a given body adds to the severity of the problem. The present work describes a new method to suppress the influence of the irregular frequencies. The boundary integral equation is supplemented with a set of moment equations—the null-field equations. The resulting overdetermined system is then solved by a least squares procedure. It is shown that this procedure will produce a unique solution. Numerical results for a circular cylinder and a box barge demonstrate the effectiveness of the method.

Thermodynamic derivation of the hydrodynamical model for charge transport in semiconductors.

A thermodynamic derivation is given of the hydrodynamical model describing charge-carrier transport in semiconductors. The set of moment equations is closed by utilizing the principles of extended thermodynamics. The method of Maxwellian iteraton yields, then, hydrodynamical-like equations with constitutive equations for the heat flux and anisotropic stresses.

Laguerre polynomial solution of the nonlinear Boltzmann equation for the hard-sphere model

We consider a spatially homogeneous and isotropic gas consisting of hard-sphere molecules. A vector representation of the scattering kernel is used to adapt the original Boltzmann equation to the idealized geometrical situation. By means of an expansion of the distribution function in terms of Laguerre polynomials this scalar Boltzmann equation is transformed to a set of moment equations. All “algebraized” collision integrals can be evaluated analytically. We discuss the truncation of the moment equations necessary for the practical application of this method. The eigenvalues of the linearized relaxation problem show a good convergence with respect to the truncation index.

Photon statistics of nonclassical radiation in second harmonic generation

The Fokker-Planck equation approach to the second harmonic generation is adopted to obtain a set of moment equations for this process. This set of equations is truncated in consistence with a model of generalized superposition of coherent fields and quantum noise, which provides equations of motion for unknown quantities in the model. Numerical solution of these equations provides at once the photon number distribution and its factorial moments including fully quantum features of radiation such as its sub-Poisson behaviour and squeezing of vacuum fluctuations. Limits of this approximation are discussed and numerical results are provided.

On the time-dependent radiative transfer in photospheric plasmas. I. The analytical approach

This paper is the first in a series investigating time-dependent radiative transfer processes of X-rays in photospheric plasmas at non-relativistic temperatures. Two parallel tracks are followed and each aspect is studied with an analytical and a Monte Carlo method. In this first paper, an analytical random walk model, describing the diffusion of photons in a homogeneous spherical geometry, is presented. The starting point is a set of moment equations derived earlier from the relativistic Boltzmann equation. Green functions for every individual scattering order are derived. The total Green function is the sum over all scattering orders and it solves the hyperbolic diffusion equation. An important element of flexibility arises from the introduction of the so-called characteristic parameter chi which is basically the inverse of the characteristic velocity. For high scattering orders, chi is equal to square root 3. By adjusting chi for the low scattering orders, agreement with numerical simulations is obtained.