Boltzmann Operator Research Articles

A spectral study of the linearized Boltzmann operator in $ L^2 $-spaces with polynomial and Gaussian weights

<p style='text-indent:20px;'>The spectrum structure of the linearized Boltzmann operator has been a subject of interest for over fifty years and has been inspected in the space <inline-formula><tex-math id="M2">\begin{document}$ L^2\left( {\mathbb R}^d, \exp(|v|^2/4)\right) $\end{document}</tex-math></inline-formula> by B. Nicolaenko [<xref ref-type="bibr" rid="b27">27</xref>] in the case of hard spheres, then generalized to hard and Maxwellian potentials by R. Ellis and M. Pinsky [<xref ref-type="bibr" rid="b13">13</xref>], and S. Ukai proved the existence of a spectral gap for large frequencies [<xref ref-type="bibr" rid="b33">33</xref>]. The aim of this paper is to extend to the spaces <inline-formula><tex-math id="M3">\begin{document}$ L^2\left( {\mathbb R}^d, (1+|v|)^{k}\right) $\end{document}</tex-math></inline-formula> the spectral studies from [<xref ref-type="bibr" rid="b13">13</xref>,<xref ref-type="bibr" rid="b33">33</xref>]. More precisely, we look at the Fourier transform in the space variable of the inhomogeneous operator and consider the dual Fourier variable as a fixed parameter. We then perform a precise study of this operator for small frequencies (by seeing it as a perturbation of the homogeneous one) and also for large frequencies from spectral and semigroup point of views. Our approach is based on Kato's perturbation theory for linear operators [<xref ref-type="bibr" rid="b22">22</xref>] as well as enlargement arguments from [<xref ref-type="bibr" rid="b25">25</xref>,<xref ref-type="bibr" rid="b19">19</xref>].</p>

Dissipation and Semigroup on $H^k_n$: Non-cutoff Linearized Boltzmann Operator with Soft Potential

In this paper, we find that the linearized collision operator $L$ of the non-cutoff Boltzmann equation with soft potential generates a strongly continuous semigroup on $H^k_n$, with $k,n\in\mathbb{R}$. In the theory of Boltzmann equation without angular cutoff, the weighted Sobolev space plays a fundamental role. The proof is based on pseudo-differential calculus and in general, for a specific class of Weyl quantization, the $L^2$ dissipation implies $H^k_n$ dissipation. This kind of estimate is also known as the G{\aa}rding's inequality.

The resolvent of the linearized Boltzmann operator with a stationary potential

In this paper, we study the linear non-cutoff Boltzmann equation with confining potential, and give a compactness criteria for its resolvent by taking advantage of the symbolic calculation for linearized Boltzmann operator.

Alternative Adjoint Boltzmann Transport Operators in Alternative Hilbert Spaces in Spherical Geometry: Theory and Paradigm Applications

The solution of the customary adjoint Boltzmann equation for linear transport of particles and radiation, referred to as the “adjoint flux,” plays a prominent role in reactor physics, shielding, control, and optimization as a weighting function for cross-section processing, optimization, Monte Carlo acceleration procedures, and sensitivity and uncertainty analyses. All of the textbooks and scientific works published thus far use the same procedure to derive “the” adjoint Boltzmann operator, thereby conveying inadvertently the misleading impression that this traditional procedure is the only way to obtain “the” adjoint Boltzmann operator, and that the form of “the” adjoint operator thus derived is universally unique. None of the works published in the literature thus far touches on the fact that the customary textbook-form of the adjoint Boltzmann operator is actually derived in a particular Hilbert space, which is endowed with a specific inner product that is based on integrating spatially over the domain’s spatial volume such that Gauss’ divergence theorem holds. As this work will show, however, the Hilbert space that has been implicitly used in all of the published works thus far is not the only possible Hilbert space for deriving operators that are adjoint, in the respective Hilbert space, to the forward Boltzmann operator. Alternative Hilbert spaces may be used just as legitimately, and may actually be more suitable than the customary Hilbert space for computing adjoint functions to be used in inner products involving various forward and/or adjoint fluxes and forward and/or adjoint source terms.By presenting paradigm illustrative examples in three-dimensional spherical coordinates, this work shows that although a unique form of the adjoint Boltzmann operator is obtained for each Hilbert space in which the respective adjoint operator is constructed, distinct Hilbert spaces will produce distinct adjoint Boltzmann operators accompanied by distinct forms of the corresponding bilinear concomitants on the respective spatial domain’s boundary. The fundamental practical reason for using alternative Hilbert spaces is to obtain alternative adjoint functions and/or Green’s functions that may be less singular than the customary adjoint function and/or Green’s functions (in the customary Hilbert space) and would consequently be computable numerically. As this work shows, such situations arise when attempting to use the adjoint sensitivity analysis methodology in the conventional Hilbert space for computing sensitivities to cross sections, isotopic number densities, etc., of responses of flux and/or power detectors placed near or at the center of the spherical coordinates. In such sensitivity analysis problems, the singularities of the conventional adjoint Boltzmann equation in the conventional Hilbert space may preclude its use, but the requisite sensitivities can nevertheless be computed efficiently using an alternative adjoint Boltzmann equation in an alternative Hilbert space. The consequences of this powerful breakthrough new concept of using alternative adjoint operators in alternative Hilbert spaces are highlighted by presenting a paradigm benchmark problem that admits a closed-form exact solution. This benchmark problem shows that the customary adjoint equation becomes singular at the sphere’s center, so the conventional adjoint flux is therefore noncomputable there, but the alternative adjoint transport equation in a judiciously chosen alternative Hilbert space is everywhere nonsingular and can therefore be used to compute the requisite sensitivities. By indicating the path for using alternative Hilbert spaces, this work opens new conceptual procedures for solving problems that have hitherto been unsolvable in the framework of the conventional adjoint particle transport equation.

Global hypoelliptic and symbolic estimates for the linearized Boltzmann operator without angular cutoff

In this article we provide global subelliptic estimates for the linearized inhomogeneous Boltzmann equation without angular cutoff, and show that some global gain in the spatial direction is available although the corresponding operator is not elliptic in this direction. The proof is based on a multiplier method and the so-called Wick quantization, together with a careful analysis of the symbolic properties of the Weyl symbol of the Boltzmann collision operator.

Quantum Phase Transition in the Spin-Boson Model: A Multilayer Multiconfiguration Time-Dependent Hartree Study.

The multilayer improved relaxation is applied to study the delocalization-localization transition in the spin-boson model at zero temperature-a well-known example of quantum phase transition. Calculations of energy eigenstates are obtained by iteratively diagonalizing the matrix of the Boltzmann operator in the top layer representation, using a Lanczos/Arnoldi method while relaxing the single particle functions of all layers using the multilayer multiconfiguration time-dependent Hartree imaginary time propagation. Two properties are used to examine the quantum phase transition: the energy splitting for the lowest pair of eigenstates and the magnetic susceptibility. Consistent findings are obtained with appropriate scaling parameters.

A semiclassical initial-value representation for quantum propagator and boltzmann operator.

Starting from the position-momentum integral representation, we apply the correction operator method to the derivation of a uniform semiclassical approximation for the quantum propagator and then extend it to approximate the Boltzmann operator. In this approach, the involved classical dynamics is determined by the method itself instead of given beforehand. For the approximate Boltzmann operator, the corresponding classical dynamics is governed by a complex Hamiltonian, which can be described as a pair of real Hamiltonian systems. It is demonstrated that the semiclassical Boltzmann operator is exact for linear systems. A quantum propagator in the complex time is thus proposed and preliminary numerical results show that it is a reasonable approximation for calculating thermal correlation functions of general systems. © 2018 Wiley Periodicals, Inc.

A Class of Low Dissipative Schemes for Solving Kinetic Equations

We introduce an extension of the fast semi-Lagrangian scheme developed in J Comput Phys 255:680–698 (2013) in an effort to increase the spatial accuracy of the method. The basic idea of this extension is to modify the first-order accurate transport step of the original semi-Lagrangian scheme to allow for a general piecewise polynomial reconstruction of the distribution function. For each discrete velocity, we update the solution not at cell centers, but rather at the extreme points of the spatial reconstruction, the locations of which are different for each discrete velocity and change with time. Several approaches are discussed for evaluating the collision operator at these extreme points using only cell center values by making special assumption on the spatial variation of the collision operator. The result is a class of schemes that preserves the structure of the solution over very long times when compared to existing schemes in the literature. As a proof of concept, the new method is implemented in a one-dimensional setting, using piecewise linear reconstructions of the distribution function together with a related reconstruction of the collision operator. The method is derived both for the relatively simple Bhatnagar–Gross–Krook (BGK) operator as well as for the classical Boltzmann operator. Several numerical tests are used to assess the performance of the implementation, including comparisons with the original method in J Comput Phys 255:680–698 (2013) and with classical semi-Lagrangian methods of first and second order. In convergence tests, we observe uniform second-order accuracy across all flow regimes for the BGK operator and nearly second-order accuracy for the Boltzmann operator. In addition, we observe that the method outperforms the classical semi-Lagrangian approach, in particular when resolving fine solution structures in space.

The Fourier transforms for the spatially homogeneous Boltzmann equation and Landau equation

In this paper, we study the Fourier transforms for two equations arising in the kinetic theory. The first equation is the spatially homogeneous Boltzmann equation. The Fourier transform of the spatially homogeneous Boltzmann equation has been first addressed by Bobylev (Sov Sci Rev C Math Phys 7:111–233, 1988) in the Maxwellian case. Alexandre et al. (Arch Ration Mech Anal 152(4):327–355, 2000) investigated the Fourier transform of the gain operator for the Boltzmann operator in the cut-off case. Recently, the Fourier transform of the Boltzmann equation is extended to hard or soft potential with cut-off by Kirsch and Rjasanow (J Stat Phys 129:483–492, 2007). We shall first establish the relation between the results in Alexandre et al. (2000) and Kirsch and Rjasanow (2007) for the Fourier transform of the Boltzmann operator in the cut-off case. Then we give the Fourier transform of the spatially homogeneous Boltzmann equation in the non cut-off case. It is shown that our results cover previous works (Bobylev 1988; Kirsch and Rjasanow 2007). The second equation is the spatially homogeneous Landau equation, which can be obtained as a limit of the Boltzmann equation when grazing collisions prevail. Following the method in Kirsch and Rjasanow (2007), we can also derive the Fourier transform for Landau equation.

On the relativistic large-angle electron collision operator for runaway avalanches in plasmas

Large-angle Coulomb collisions lead to an avalanching generation of runaway electrons in a plasma. We present the first fully conservative large-angle collision operator, derived from the relativistic Boltzmann operator. The relation to previous models for large-angle collisions is investigated, and their validity assessed. We present a form of the generalized collision operator which is suitable for implementation in a numerical kinetic equation solver, and demonstrate the effect on the runaway-electron growth rate. Finally we consider the reverse avalanche effect, where runaways are slowed down by large-angle collisions, and show that the choice of operator is important if the electric field is close to the avalanche threshold.

Sharp bounds for Boltzmann and Landau collision operators

The aim of the work is to provide a stable method to get sharp bounds for Boltzmann and Landau operators in weighted Sobolev spaces and in anisotropic spaces. All the sharp bounds are given for the original Boltzmann and Landau operators. The sharpness means the lower and upper bounds for the operators are consistent with the behavior of the linearized operators. Moreover, we make clear the difference between the bounds for the original operators and those for the linearized ones. According to the Bobylev's formula, we introduce two types of dyadic decompositions performed in both phase and frequency spaces to make full use of the interaction and the cancellation. It allows us to see clearly which part of the operator behaves like a Laplace type operator and which part is dominated by the anisotropic structure. It is the key point to get the sharp bounds in weighted Sobolev spaces and in anisotropic spaces. Based on the geometric structure of the elastic collision, we make a geometric decomposition to capture the anisotropic structure of the collision operator. More precisely, we make it explicit that the fractional Laplace-Beltrami operator really exists in the structure of the collision operator. It enables us to derive the sharp bounds in anisotropic spaces and then complete the entropy dissipation estimates. The structures mentioned above are so stable that we can apply them to the rescaled Boltzmann collision operator in the process of the grazing collisions limit. Then we get the sharp bounds for the Landau collision operator by passing to the limit. We remark that our analysis used here will shed light on the investigation of the asymptotics from Boltzmann equation to Landau equation.

On the rate of convergence to equilibrium for the linear Boltzmann equation with soft potentials

In this work we present several quantitative results of convergence to equilibrium for the linear Boltzmann operator with soft potentials under Grad's angular cut-off assumption. This is done by an adaptation of the famous entropy method and its variants, resulting in explicit algebraic, or even stretched exponential, rates of convergence to equilibrium under appropriate assumptions. The novelty in our approach is that it involves functional inequalities relating the entropy to its production rate, which have independent applications to equations with mixed linear and non-linear terms. We also briefly discuss some properties of the equation in the non-cut-off case and conjecture what we believe to be the right rate of convergence in that case.

Many-body formalism for thermally excited wave packets: A way to connect the quantum regime to the classical regime

Free classical particles have well-defined momentum and position, while free quantum particles have well-defined momentum but a position fully delocalized over the sample volume. We develop a many-body formalism based on wave-packet operators that connects these two limits, the thermal energy being distributed between the state spatial extension and its thermal excitation. The corresponding `mixed quantum-classical' states, which render the Boltzmann operator diagonal, are the physically relevant states when the temperature is finite. The formulation of many-body Hamiltonians in terms of these thermally excited wave-packets and the resulting effective scatterings is provided.

A Conservative, Entropic Multispecies BGK Model

We derive a conservative multispecies BGK model that follows the spirit of the original, single species BGK model by making the specific choice to conserve species masses, total momentum, and total kinetic energy and to satisfy Boltzmann’s $$\mathcal {H}$$ -Theorem. The derivation emphasizes the connection to the Boltzmann operator which allows for direct inclusion of information from higher-fidelity collision physics models. We also develop a complete hydrodynamic closure via the Chapman-Enskog expansion, including a general procedure to generate symmetric diffusion coefficients based on this model. We numerically investigate velocity and temperature relaxation in dense plasmas and compare the model with previous multispecies BGK models and discuss the trade-offs that are made in defining and using them. In particular, we demonstrate that the BGK model in the NRL plasma formulary does not conserve momentum or energy in general.

Accelerating equilibrium isotope effect calculations. I. Stochastic thermodynamic integration with respect to mass

Accurate path integral Monte Carlo or molecular dynamics calculations of isotope effects have until recently been expensive because of the necessity to reduce three types of errors present in such calculations: statistical errors due to sampling, path integral discretization errors, and thermodynamic integration errors. While the statistical errors can be reduced with virial estimators and path integral discretization errors with high-order factorization of the Boltzmann operator, here we propose a method for accelerating isotope effect calculations by eliminating the integration error. We show that the integration error can be removed entirely by changing particle masses stochastically during the calculation and by using a piecewise linear umbrella biasing potential. Moreover, we demonstrate numerically that this approach does not increase the statistical error. The resulting acceleration of isotope effect calculations is demonstrated on a model harmonic system and on deuterated species of methane.

Gevrey Smoothing for Weak Solutions of the Fully Nonlinear Homogeneous Boltzmann and Kac Equations Without Cutoff for Maxwellian Molecules

It has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties as a fractional Laplacian. This has led to the hope that the homogenous Boltzmann equation enjoys similar regularity properties as the heat equation with a fractional Laplacian. In particular, the weak solution of the fully nonlinear non-cutoff homogenous Boltzmann equation with initial datum in $L^1_2(\mathbb{R}^d)\cap L\log L(\mathbb{R}^d)$, i.e., finite mass, energy and entropy, should immediately become Gevrey regular for strictly positive times. We prove this conjecture for Maxwellian molecules.

The Maxwell–Stefan diffusion limit for a kinetic model of mixtures with general cross sections

In this article, we derive the Maxwell–Stefan formalism from the Boltzmann equation for mixtures for general cross-sections. The derivation uses the Hilbert asymptotic method for systems at low Knudsen and Mach numbers. We also formally prove that the Maxwell–Stefan coefficients can be linked to the direct linearized Boltzmann operator for mixtures. That allows to compute the values of the Maxwell–Stefan diffusion coefficients with explicit and simple formulae with respect to the cross-sections. We also justify the specific ansatz we use thanks to the so-called moment method.

High order path integrals made easy.

The precise description of quantum nuclear fluctuations in atomistic modelling is possible by employing path integral techniques, which involve a considerable computational overhead due to the need of simulating multiple replicas of the system. Many approaches have been suggested to reduce the required number of replicas. Among these, high-order factorizations of the Boltzmann operator are particularly attractive for high-precision and low-temperature scenarios. Unfortunately, to date, several technical challenges have prevented a widespread use of these approaches to study the nuclear quantum effects in condensed-phase systems. Here we introduce an inexpensive molecular dynamics scheme that overcomes these limitations, thus making it possible to exploit the improved convergence of high-order path integrals without having to sacrifice the stability, convenience, and flexibility of conventional second-order techniques. The capabilities of the method are demonstrated by simulations of liquid water and ice, as described by a neural-network potential fitted to the dispersion-corrected hybrid density functional theory calculations.

Quantum Tunneling Rates of Gas-Phase Reactions from On-the-Fly Instanton Calculations.

The instanton method obtains approximate tunneling rates from the minimum-action path (known as the instanton) linking reactants to the products at a given temperature. An efficient way to find the instanton is to search for saddle-points on the ring-polymer potential surface, which is obtained by expressing the quantum Boltzmann operator as a discrete path-integral. Here we report a practical implementation of this ring-polymer form of instanton theory into the Molpro electronic-structure package, which allows the rates to be computed on-the-fly, without the need for a fitted analytic potential-energy surface. As a test case, we compute tunneling rates for the benchmark H + CH4 reaction, showing how the efficiency of the instanton method allows the user systematically to converge the tunneling rate with respect to the level of electronic-structure theory.

Stability of global equilibrium for the multi-species Boltzmann equation in $L^\infty$ settings

We prove the stability of global equilibrium in a multi-species mixture, where the different species can have different masses, on the $3$-dimensional torus. We establish stability estimates in $L^\infty_{x,v}(w)$ where $w=w(v)$ is either polynomial or exponential, with explicit threshold. Along the way we extend recent estimates and stability results for the mono-species Boltzmann operator not only to the multi-species case but also to more general hard potential and Maxwellian kernels.