Linear Boltzmann Research Articles

Homogeneous generalized master equation for an arbitrary initial state of a many-particle classical system

A special projection operator is introduced, which exactly transforms the Nakajima–Zwanzig inhomogeneous Generalized Master Equation (GME) with an irrelevant initial condition term (a source) into a completely closed homogeneous GME (with no initial correlations term) for the relevant part of an N-particle distribution function FN(t) for an arbitrary initial distribution function FN(0) of a system of classical particles. It resolves the problem of the derivation of a closed linear evolution equation from the Liouville equation with no “molecular chaos” type approximation. The obtained equation is equivalent to the completely closed equation for s-particle (s≤N) distribution function Fs(t). The initial correlations are accounted for in this equation in the modified by the introduced projection operator kernel governing the evolution of Fs(t). This equation is further simplified by presenting its kernel in the linear in the particles’ density n approximation. In this approximation the equations for one-particle F1(t) and two-particle F2(t) distribution functions are derived. Special attention is given to analyzing the equation for F1(t) and showed that this equation in the Markov approximation can be written as the linear Boltzmann or Landau (for a small interparticle interaction) equations but with additional terms caused by initial correlations.

Diffusion-approximation in stochastically forced kinetic equations

We derive the hydrodynamic limit of a kinetic equation where the interactions in velocity are modelled by a linear operator (Fokker-Planck or Linear Boltzmann) and the force in the Vlasov term is a stochastic process with high amplitude and short-range correlation. In the scales and the regime we consider, the hydrodynamic equation is a scalar second-order stochastic partial differential equation. Compared to the deterministic case, we also observe a phenomenon of enhanced diffusion.

Magnon theory for the spin Peltier effect

We propose a model based on the magnonic current to describe the spin Peltier effect in ferromagnetic insulator/paramagnetic heavy metal bilayers. Through the linear Boltzmann’s theory, the diffusive regime of magnons into the ferromagnetic layer and the creation/annihilation of magnons at the interface are identified as the main ingredients of the SPE. The dependence of the temperature change at the interface as a function of the charge current, ferromagnetic layer thickness and magnon diffusion length are obtained. The model is applied to the yttrium-iron-garnet/Pt, where a good agreement between theory and recently reported experimental data is achieved.

Traces of nonequilibrium effects, initial condition, bulk dynamics, and elementary collisions in the charm observables

We study how heavy quark dynamics is affected by the nature of the bulk evolution of the QCD matter, the initial condition of the system, and the treatment of elementary interactions between heavy quarks and the surrounding medium. For the same initial condition and the same QGP expansion scenario we discuss the consequences of the assumption of a local equilibrium by comparing the consequences for the nuclear modification factor and the elliptic flows of charm quarks. For this purpose we employ the parton-hadron-string dynamics (PHSD), which is an off-shell microscopic transport approach, as well as the linearized-Boltzmann (LB) scheme obtained by coarse graining the PHSD bulk and assuming local equilibrium for the interactions of the charm quarks with the bulk. The $R_{\rm AA}$ of charm quarks stemming from the later LB approach is also compared to a genuine fluid dynamics evolution initiated by the coarse grained PHSD, which allows to further assess the consequences of reducing the full n-body dynamics. We then proceed to a systematic comparison of PHSD with MC@HQ, another transport model for heavy flavors which also relies on LB approach. In particular, we investigate the consequences for the nuclear modification factor of charm quarks if we vary separately the initial heavy quark distribution function in matter, the expansion dynamics of the QGP and the elementary interactions of heavy quarks of these models. We find that the results for both models vary significantly depending on the details of the calculation. However, both models achieve very similar predictions for key heavy quark observables for certain combinations of initial condition, bulk evolution and interactions. We conclude that this ambiguity limits our ability to determine the different properties of the system based on the current set of observables.

Landau Damping for the Linearized Vlasov Poisson Equation in a Weakly Collisional Regime

In this paper, we consider the linearized Vlasov-Poisson equation around an homogeneous Maxwellian equilibrium in a weakly collisional regime: there is a parameter $\eps$ in front of the collision operator which will tend to $0$. Moreover, we study two cases of collision operators, linear Boltzmann and Fokker-Planck. We prove a result of Landau damping for those equations in Sobolev spaces uniformly with respect to the collision parameter $\eps$ as it goes to $0$.

Bending and creep buckling response of viscoelastic functionally graded beam-columns

The viscoelastic creep response of flexural beams and beam-columns made with functionally graded materials is numerically investigated. The paper highlights the challenges associated with the modeling and analysis of such structures, and presents a nonlinear theoretical model for their bending and creep buckling analysis. The model accounts for the viscoelasticity of the materials using differential-type constitutive relations that are based on the linear Boltzmann’s principle of superposition. The model is general in terms of its ability to deal with any material volume faction distribution through the depth of the beam, and with different linear viscoelastic laws, boundary conditions, and loading schemes. The governing equations are solved through time stepping numerical integration, which yields an exponential algorithm following the expansion of the relaxation function into a Dirichlet series. A numerical study that examines the capabilities of the model and quantifies the creep response of functionally graded beam-columns is presented, with special focus on the stresses and strains redistribution over time and on the creep buckling response. The results show that the creep response of such structures can be strongly nonlinear due to the variation of the viscoelastic properties through the depth, along with unique phenomena that are not observed in homogenous structures.

Lattice Boltzmann method for oscillatory Stokes flow with applications to micro- and nanodevices

A lattice Boltzmann (LB) method based on the linearized Boltzmann Bhatnagar-Gross-Krook equation for numerical simulation of oscillatory (unsteady) Stokes flow is proposed. Unlike the conventional (nonlinear) LB method that utilizes the time domain exclusively, the proposed method is formulated in the frequency domain to allow for direct access to the complex-valued stress, force, and velocity field--these parameters are of direct interest in characterizing microelectromechanical systems (MEMS) and nanoelectromechanical systems (NEMS). The proposed method circumvents the requirement for time-dependent boundary velocities, as is needed in the conventional LB method, and convergence of the two methods is compared. Validity of the proposed method is assessed using three classical (unsteady) flows: (1) one-dimensional oscillatory Couette flow between two plates; (2) two-dimensional flow generated by an oscillating circular cylinder; (3) three-dimensional flow generated by an oscillating sphere. The observed excellent numerical performance in all three cases demonstrates that this linear lattice Boltzmann method can be used to study the dynamics of micro- and nanoscale devices of any dimensionality. This is particularly relevant to MEMS and NEMS, where the resonance properties of individual nanomechanical components immersed in fluid can underpin overall device performance.

Multiple Solutions to Linear Kinetic Equations

In the recent years there has been a renewed interest in multiple solutions to kinetic equations prompted by the discovery of multiple solutions for the Boltzmann equation, (Wennberg, B. ([1999]). An example of nonuniqueness for solutions to the homogeneous Boltzmann equation. J. Statist. Phys. 1–2:469–477). In this article we shall show that there exist multiple solutions to a class of kinetic equations describing electron scattering on phonons, (Majorana, A., Milazzo, C. ([2001]). Space homogeneous solutions of the linear semiconductor Boltzmann equation. J. Math. An. Appl. 259:609–629). This demonstrates that the initial value problem for such equations, understood in the pointwise sense, is ill-posed. We shall also briefly discuss some side conditions that ensure the uniqueness of the solution.

Monte Carlo models of electron and ion transport in non-equilibriumplasmas

The fundamentals of the Monte Carlo method for electron and iontransport in plasmas are explained with emphasis on the coupling of theparticle transport with the space charge fields and chemical kinetics. In thefirst three sections the basics of the null-collision Monte Carlo methods andthe particle in cell/Monte Carlo collision method (PIC/MCC) are explained withpractical suggestions for numerical implementation. The fourth section showsthat the Monte Carlo method can be derived from the linear Boltzmann equation.In the fifth section, we show how the charged particle transport can beself-consistently coupled to the chemical kinetics of the gas phase, inparticular to the vibrational kinetics of molecules. In the penultimate section aone-dimensional, self-consistent model for a parallel-plate radio-frequencydischarge in pure hydrogen is presented.

Exact solutions of the Boltzmann equation in the VHP model with removal interaction

The linear and nonlinear Boltzmann equation for very hard particles (VHP) is considered in the case when the collision between two particles may lead not only to elastic scattering, but also to a removal event with the disappearance of the molecules. The extended transport equation is solved for arbitrary initial distributions. The computations are carried out explicitly for a special class of initial distributions and for various removal rates. The results are demonstrated graphically. Finally, source terms fulfilling physically reasonable conditions are introduced into the VHP model, and the time-dependent particle number is calculated.

Matrix continued-fraction representation of the dynamical self-structure factor Ss(q, omega ).

A new approach is presented for determining the dynamical self-structure factor ${S}_{s}$(q,\ensuremath{\omega}) for a multicomponent dilute gas mixture. The starting point is the three-dimensional linear inhomogeneous Boltzmann equation with a Maxwell interaction potential. It is shown that the dynamical self-structure factor can be expressed analytically in terms of a continued fraction of infinite but sparse matrices. Although other methods and approximation schemes for calculating ${S}_{s}$(q,\ensuremath{\omega}) via the Boltzmann equation have been suggested so far, the method and the results presented here are new in the following sense. First, to our knowledge there does not exist an analytical solution of the Boltzmann equation covering the entire (q,\ensuremath{\omega}) range for ${S}_{s}$(q,\ensuremath{\omega}) and yielding in particular the correct dynamical self-structure factor ${S}_{s}$(q,\ensuremath{\omega}) for both the hydrodynamical limit (q\ensuremath{\rightarrow}0) and the free-gas limit (q\ensuremath{\rightarrow}\ensuremath{\infty}). Second, at least the example considered in this paper shows most clearly that the commonly suggested expansion of ${S}_{s}$(q,\ensuremath{\omega}) in a continued fraction of scalars cannot be regarded as a useful expansion method.

Parallel Sn Transport algorithms

Abstract Parallelization of standard multigroup methods used to solve the linear (Boltzmann) tranaport equation in the discrete ordinates (Sn) representation and coupled chaotic iteration schemes are the focus of this analyais. On the Denelcor HEP, we extend two serial iteration schemea, categorize speedup, and contrast ordered and chaotic methods. Ordered and chaotic iteration atrategies, with arid without acceleration, support relatively unreatricted paralleliam and appear to be robust parallel techniques. Parallel modifications and recoding efforts to seriol iteration algorithms are straightforward, actual speedup and efficiency are high, and payoff appears substantial, largely due to the coarse computational granularity of the multigroup technique. Chaotic schemes also converge faster than ordered strategies.

Tensorial diffusion approximation to the integral particle transport theory in the presence of a Lorentz force

A stationary tensorial diffusion approximation for charged particles migrating upon the action of a Lorentz force in an unbounded host medium is derived starting from the linear integral Boltzmann equation. The results for the diffusion coefficient and for the diffusion equation for the total density of the particles considered are compared with the corresponding ones obtained in the literature through aP1-approximation to the linear integro-differential Boltzmann equation. The discrepancies observed are appropriately commented on a physical ground.

Onsager's reciprocal relations in the linear problem of flow of rarefied gas past a heat conducting body

Symmetry of integral transport coefficients is established on the basis of a linear stationary Boltzmann equation for the problem of flow past a heat conducting body. An expression is obtained for the entropy production in the gas-body system, and this determines the thermodynamic fluxes and forces. Bakanov and Roldugin [2] have considered the problem of motion of a heat conducting sphere at small Knusden numbers Kn, using the symmetry of the Onsager coefficients to construct an asymptotic solution as Kn → 0. In the present paper, a general method is proposed for establishing the Onsager relations that does not require the actual construction of a solution to the problem and is applicable for bodies of arbitrary shape and all values of the Knudsen number.

On the spectrum of the linear transport operator in a semi-infinite medium

The linear mono-energetic Boltzmann equation with isotropic scattering is considered for a semi-infinite medium in plane geometry and the spectrum of the corresponding operator under perfectly reflecting, vacuum, generalized or diffusely reflecting boundary conditions is explored in the frame of the 'initial-value problem'. By the Hille-Yosida theorem, the existence and uniqueness of the solutions of these problems are assured. As a common feature, one observes the absence of a true isolated asymptotic eigenmode, the solution displaying, due to the infinite extent of the medium, only 'transient' modes.

Transport theory of a totally reflected parallelepiped reactor

An application of the linear integral Boltzmann equation for determining the stationary total flux distribution for monoenergetic neutrons in a totally reflected parallelepiped reactor — where both core and reflector are made by isotropically scattering and multiplying materials — is here presented. The general properties of the solution for the total flux are discussed in the context of the theory of linear integral equations in a Le-besgue space Lp. A practical version of the solution, holding in the whole reactor, is proposed in terms of a polynomials expansion.

Numerical Solution of the Linear Integral Boltzmann Equation

An iterative method is proposed for solving the homogeneous (i.e., critical) or inhomogeneous (i.e., source) linear irtegral Boltzmann equation for general geometry. By using successive approximations, these two classes of problems are shown to be mathematically equivalent. For the homogeneous problem, constraints on the algorithm regarding the existence of eigenvalues and the initial approximation are investigated. The algorithm is as the elementary interaction causing the pinning). For higher defect densities, the agreement with the experiments on niobium is better than with the previous theory. This method of correlations seemed suitable for studying the effect of cutting-off'' the small elementary intenactions and for the replacement of the Gauss distribution function by the Poisson distribution function for the number of defects in the elementary volumes. Both these give negative results with respect to the experiments; so far wrte are therefore not able to explain quartitatively the large please delete the above 9 lines======= applied to isotropically scattering slabs and spheres, and is compared with previously published results as well as an independent extrapolation method. For the inhomogeneous problem, an improvement over the normal successive collision method via the use of a Neumann series expansion is used to allow economic parametric studies. Constraintsmore » on the algorithm and methods of efficiently terminating the infinite Neumann series are investigated. The solution via the proposed method as applied to isotropically scattering slabs and spheres is provided in a compact form for a range of multiplication factors and optical dimensions. The shape of the scalar flux distribution is expiained. Extensions of the method to more complex problems are outlined; in particular, the solution to an energy-dependent problem in general geometry is obtained, and the implications of the results are discussed. (7 figures, 2 tables) (auth)« less

Laser-Induced Rate Processes in Gases: Dynamics, Unimolecular Decay, and Scattered Field

We have recast a linear quantum-mechanical Boltzmann equation, describing the dynamics of a gas of $N$-level atoms absorbing light between an arbitrary pair of levels, into the form of a generalized master equation. The atoms are permitted to undergo both phase-changing and energy-changing collisions with a heat bath of inert molecules. The possibility of unimolecular decay is incorporated into the Boltzmann equation. The solutions of the generalized master equation give an accurate description of the system for short as well as long times and for arbitrary light-field intensity. Eigenvalues obtained from the solution of the $N$-level master equation are shown to have particular significance with regard to the scattered electromagnetic field, with the imaginary parts of the complex eigenvalues providing the location and the real parts of the eigenvalues giving the widths of scattered radiation bands. The dynamics and scattered field for a two-level system is discussed in detail. Two collision parameters giving the frequency of phase-changing and energy-changing collisions are important. For special values of these parameters, our results reduce to scattered field spectra calculated previously. When unimolecular decay is included, one identifies the smallest eigenvalue of the generalized master equation with the macroscopic rate constant. Observation of the scattered field for different incident field intensities would permit separation of radiative and thermal effects in an enhanced reaction rate, without requiring an accurate temperature measurement.

Solution to the boltzmann equation for monoenergetic neutrons in a slab

This paper presents a physical solution to the linear integro-differential Boltzmann equation, which governs the stationary distribution of the flux of monoenergetic neutrons in the interior and at the boundary of an infinite slab of finite thickness. Results for both the angular and the total fluxes are obtained by properly accounting for the boundary conditions of the problem and by a Fourier transform of the original Boltzmann equation. An integral linear Fredholm equation for the Fourier transform of the total flux is thus derived. The solution of such an equation is given two different representations according to the two decompositions devised for the given Fredholm kernel. The solution obtained in each case is finally inverted termwise to yield the sought total flux in the original space. The result for the angular flux follows, then, directly from the result for the total flux.

Complementary Variational Principles and Their Application to Neutron Transport Problems

Several variational principles are developed which give upper and lower bounds for the linear functional (S, ψ), where ψ is the solution of the inhomogeneous equation Hψ = S with H a self-adjoint, positive-definite, linear operator. Some of the principles bound this functional only with respect to small or local variations, whereas others give bounds for arbitrary variations. Several of our results coincide with those of other authors widely scattered throughout the literature, and we show that these principles have a common origin. Other results given are new. Examples of the use of these principles are taken from the field of neutron transport theory, and we use both the linear Boltzmann or transport equation and the diffusion equation. One interesting result is that certain ``exact'' values of the extrapolated endpoint for the Milne problem which have been reported in the literature fall, due to numerical inaccuracies, outside the bounds computed here.