Problems In Kinetic Theory Research Articles

The Boltzmann equation for plane Couette flow in a finite channel

The dynamics of a rarefied gas in a finite channel with the same temperatures and opposite velocities is a fundamental problem in kinetic theory. The relative motion of the planar boundaries can induce a non-equilibrium state which is referred to as the Couette flow. In this paper, we demonstrate that the unsteady Couette flow for the Boltzmann equation in 3D finite channel time asymptotically converges to the 1D steady state constructed in Duan et al. (2022), we also prove the exponential time decay rate as a byproduct. The validity of the analysis is established for all hard potentials.

Linear half-space problems in kinetic theory: Abstract formulation and regime transitions

In this work, a general formulation, which is based on steady boundary layer problems for the Boltzmann equation, of a half-space problem is considered. The number of conditions on the indata at the interface needed to obtain well-posedness is investigated. The solutions will converge exponentially fast “far away” from the interface. For linearized kinetic half-space problems similar to the one of evaporation and condensation in kinetic theory, slowly varying modes might occur near regime transitions where the number of conditions needed to obtain well-posedness changes (corresponding to transition between evaporation and condensation, or subsonic and supersonic evaporation/condensation), preventing uniform exponential speed of convergence. However, those modes might be eliminated by imposing extra conditions on the indata at the interface. Flow velocities at the far end for which regime transitions occur are presented for Boltzmann equations: for monatomic and polyatomic single species and mixtures; as well as bosons and fermions.

An Adaptive Dynamical Low Rank Method for the Nonlinear Boltzmann Equation

Efficient and accurate numerical approximation of the full Boltzmann equation has been a longstanding challenging problem in kinetic theory. This is mainly due to the high dimensionality of the problem and the complicated collision operator. In this work, we propose a highly efficient adaptive low rank method for the Boltzmann equation, concerning in particular the steady state computation. This method employs the fast Fourier spectral method (for the collision operator) and the dynamical low rank method to obtain computational efficiency. An adaptive strategy is introduced to incorporate the boundary information and control the computational rank in an appropriate way. Using a series of benchmark tests in 1D and 2D, we demonstrate the efficiency and accuracy of the proposed method in comparison to the full tensor grid approach.

Hyperbolic Quadrature Method of Moments for the One-Dimensional Kinetic Equation

A tractable solution is proposed to a classical problem in kinetic theory, namely, given any set of realizable velocity moments up to order $2n$, a closure for the moment of order $2n+1$ is constructed for which the moment system found from the free-transport term in the one-dimensional (1-D) kinetic equation is globally hyperbolic and in conservative form. In prior work, the hyperbolic quadrature method of moments (HyQMOM) was introduced to close this moment system up to fourth order ($n \le 2$). Here, HyQMOM is reformulated and extended to arbitrary even-order moments. The HyQMOM closure is defined based on the properties of the monic orthogonal polynomials $Q_n$ that are uniquely defined by the velocity moments up to order $2n-1$. Thus, HyQMOM is strictly a moment closure and does not rely on the reconstruction of a velocity distribution function with the same moments. On the boundary of moment space, $n$ double roots of the characteristic polynomial $P_{2n+1}$ of the Jacobian matrix of the system are the roots of $Q_n$, while in the interior, $P_{2n+1}$ and $Q_n$ share $n$ roots. The remaining $n+1$ roots of $P_{2n+1}$ bound and separate the roots of $Q_n$. An efficient algorithm, based on the Chebyshev algorithm, for computing the moment of order $2n+1$ from the moments up to order $2n$ is developed. The analytical solution to a 1-D Riemann problem is used to demonstrate convergence of the HyQMOM closure with increasing $n$.

Moving boundary problems in kinetic theory of gases: Spatially one-dimensional problems

Unsteady flows of a rarefied gas in a full space caused by an oscillation of an infinitely wide plate in its normal direction are investigated numerically on the basis of the BhatnagarGross-Krook (BGK) model of the Boltzmann equation. The talk aims at showing properties and difficulties inherent to moving boundary problems in kinetic theory of gases using a simple one-dimensional setting. More specifically, the following two problems are considered: (Problem I) the plate starts a forced harmonic oscillation (forced motion); (Problem II) the plate, which is subject to an external restoring force obeying Hooke’s law, is displaced from its equilibrium position and released (free motion). The physical interest in Problem I lies in the propagation of nonlinear acoustic waves in a rarefied gas, whereas that in Problem II in the decay rate of the oscillation of the plate. An accurate numerical method, which is capable of describing singularities caused by the oscillating plate, is developed on the basis of the method of characteristics and is applied to the two problems mentioned above. As a result, the unsteady behavior of the solution, such as the propagation of discontinuities and some weaker singularities in the molecular velocity distribution function, are clarified. Some results are also compared with those based on the existing method. The talk is based on [1, 2, 3].

A note from the Editorial Board

Robert Glassey (Bob) has decided to withdraw from the Editorial Board of KRM. Bob has been a pioneer of mathematical kinetic theory in the 80's andone of its leading figures since then. His seminal papers on the relativistic Vlasov-Poisson and Vlasov-Maxwell systems and their asymptotic stability have been a source of inspiration for many of us. His book, `The Cauchy problem in kinetic theory' has become a must for all young researchers entering the field. Bob has been involved in the Editorial Board of KRM since the beginning of the journal and has contributed to the edition of many papers. On behalf of the whole editorial board, we express our deep gratitude to him for having joined us in this adventure and contributed to the success of the journal.

Moving boundary problems for a rarefied gas: Spatially one-dimensional case

Unsteady flows of a rarefied gas in a full space caused by an oscillation of an infinitely wide plate in its normal direction are investigated numerically on the basis of the Bhatnagar–Gross–Krook (BGK) model of the Boltzmann equation. The paper aims at showing properties and difficulties inherent to moving boundary problems in kinetic theory of gases using a simple one-dimensional setting. More specifically, the following two problems are considered: (Problem I) the plate starts a forced harmonic oscillation (forced motion); (Problem II) the plate, which is subject to an external restoring force obeying Hooke’s law, is displaced from its equilibrium position and released (free motion). The physical interest in Problem I lies in the propagation of nonlinear acoustic waves in a rarefied gas, whereas that in Problem II in the decay rate of the oscillation of the plate. An accurate numerical method, which is capable of describing singularities caused by the oscillating plate, is developed on the basis of the method of characteristics and is applied to the two problems mentioned above. As a result, the unsteady behavior of the solution, such as the propagation of discontinuities and some weaker singularities in the molecular velocity distribution function, are clarified. Some results are also compared with those based on the existing method.

The Limit of the Boltzmann Equation to the Euler Equations for Riemann Problems

The convergence of the Boltzmann equation to the compressible Euler equations when the Knudsen number tends to zero has been a long-standing open problem in kinetic theory. In the setting of a Riemann solution that contains the generic superposition of shock, rarefaction wave, and contact discontinuity to the Euler equations, we succeed in justifying this limit by introducing hyperbolic waves with different solution backgrounds to capture the extra masses carried by the hyperbolic approximation of the rarefaction wave and the diffusion approximation of contact discontinuity.

An exact solution of the boundary value problem on thermal creep for a quantum Fermi gas

We solve analytically the boundary value problem on the thermal creep of a quantum Fermi-gas along a plane surface in a half-space. The solution uses the τ model Boltzmann equation. The mass velocity profile and the distribution functions on the half-space and on the wall are constructed. The first problem for the τ -model Boltzmann equation that was successfully solved analytically was Kramers’ problem about isothermal creep of a gas [1]. In [1], Cercignani initiated the study of exact solutions to boundary value problems in kinetic theory. Then, Loyalka and Cippola [2] found a solution to the problem about thermal creep of a gas. Latyshev and Yushkanov [3] solved Kramers’ problem for quantum Fermi gases. In the present paper, we continue the study and solve the thermal creep problem for quantum Fermi gases.

A remark on the finite number of particles effect in Monte Carlo methods for kinetic equations

AbstractMonte Carlo methods are the most popular methods for solving problems in kinetic theory [2, 5]. In this short remark we emphasize some of the side effects due to the use of conservative methods over a finite number of statistical samples (particles) in the simulation. The most relevant aspect is that the steady states of the system are compactly supported and thus they cannot be Maxwellian (or any other non compactly supported statistics) unless the number of particles goes to infinity. These aspects are studied numerically with the help of a simple one‐dimensional space homogeneous kinetic model. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Scientific Realism in Action: Molecular Models and Boltzmann’s Bildtheorie

This paper approaches the scientific realism question from a naturalistic perspective. On the basis of a historical case study of the work of James Clerk Maxwell and Ludwig Boltzmann on the kinetic theory of gases, it shows that scientists’ views about the epistemological status of theories and models typically interact with their scientific results. Subsequently, the implications of this result for the current realism debate are analysed. The case study supports Giere’s moderately realist view of scientific models and theories, based on the notion of similarity, and it highlights the crucial role of model users. The paper concludes with a discussion of Boltzmann’s Bildtheorie, the sophisticated form of realism that he developed in response to the scientific problems of kinetic theory.

The Method of Singular Equations in Boundary Value Problems in Kinetic Theory

We develop a new effective method for solving boundary value problems in kinetic theory. The method permits solving boundary value problems for mirror and diffusive boundary conditions with an arbitrary accuracy and is based on the idea of reducing the original problem to two problems of which one has a diffusion boundary condition for the reflection of molecules from the wall and the other has a mirror boundary condition. We illustrate this method with two classical problems in kinetic theory: the Kramers problem (isothermal slip) and the thermal slip problem. We use the Bhatnagar-Gross-Krook equation (with a constant collision frequency) and the Williams equation (with a collision frequency proportional to the molecular velocity).

LATTICE-GAS AUTOMATA FOR THE PROBLEM OF KINETIC THEORY OF GAS DURING FREE EXPANSION

The lattice-gas method has been applied to solve the problem of kinetic theory of gas in the Gay–Lussac–Joule experiment. Numerical experiments for a two-dimensional gas were carried out to determine the number of molecules in one vessel (Nr), the ratio between the mean square values of the components of molecule velocity [Formula: see text], and the change in internal energy (ΔU) as a function of time during free expansion. These experiments were repeated for different sizes of an aperture in the partition between the two vessels. After puncturing the partition, the curve for the particle number in one vessel shows a damped oscillation for about half of the total number. The oscillations do not vanish after a sampling over different initial configurations. The system is in nonequilibrium due to the pressure equilibration, and here the flow is actually compressible. The equilibration time (in time steps) decreases with decreased size of aperture in the partition. For very small apertures (equal or less than [Formula: see text] lattice units), the number of molecules in one vessel changes with time in a smooth way until it reaches half of the total number; their curves obey the analytical solution for quasi-static processes. The calculations on [Formula: see text] and ΔU also support the results that the equilibration time decreases with decreased size of aperture in the partition.

Symmetry of the nonlinear collision operator matrix and new prospects in the moment method for solving the Boltzmann equation

Traditionally, the moment method has been used in kinetic theory to calculate transport coefficients. Its application to the solution of more complicated problems runs into enormous difficulties associated with calculating the matrix elements of the collision operator. The corresponding formulas for large values of the indices are either lacking or are very cumbersome. In this paper relations between matrix elements are derived from very general principles, and these can be employed as simple recurrence relations for calculating all the nonlinear and linear anisotropic matrix elements from assigned linear isotropic matrix elements. Efficient programs which implement this algorithm are developed. The possibility of calculating the distribution function out to 8–10 thermal velocities is demonstrated. The results obtained open up prospects for solving many topical problems in kinetic theory.

Smoothed particle hydrodynamics techniques for the solution of kinetic theory problems: Part 2. The effect of flow perturbations on the simple shear behavior of LCPs

The effect of slight perturbations to simple shear flow of liquid-crystalline polymers (LCPs) is explored by using the SPH technique to solve the unapproximated orientation distribution function equation arising from the Doi Theory. First, the case of simple shear flow is outlined, and it is shown that skewed distributions play an important role in the transition from periodic to steady behavior as the shear rate is increased. Next, we consider perturbations to flows that are slightly more extensional than simple shear, parametrized by the flow type parameter α. They are shown to eliminate all periodic director behavior (tumbling and wagging), even when the relative increment in flow type is small. At lower shear rates (or more properly, lower Peclet number Pe based upon the rotational diffusivity), the elimination occurs through a homoclinic bifurcation, the transition being rather abrupt as the flow type is changed. At higher Pe, periodic behavior is suppressed more gradually through a Hopf bifurcation, with tumbling being replaced by wagging and negative θ flow-aligning, where θ is the angle of the director in the shear plane. The effect of these perturbations on rheological behavior is also explored. As the flow is made slightly more extensional, the zero-shear rate limiting value of the generalized viscosity η decreases dramatically, due to the slowing down of tumbling as the system approaches a homoclinic orbit; as Pe is increased, the viscosity rises again before falling, due to the induction of wagging behavior where tumbling would normally prevail in simple shear. Finally, it is found when the flow type is changed sufficiently, the interesting, non-monotonic behavior of rheological functions seen in simple shear of LCPs is replaced by monotonic behavior, even though the flow is still relatively close to simple shear.

Test Particle Diffusion and the Failure of Integration along Unperturbed Orbits

This paper discusses a new effect in the kinetic theory of single species plasmas in the guidingcenter regime rc ? lD . If two guiding centers collide once, their velocities parallel to the magnetic field are eventually reversed through interactions with surrounding particles, and the same two guiding centers then collide several times. These multiple collisions are ignored in the method of integration along unperturbed orbits (IUO) that is usually applied in plasma kinetic theory problems. This effect leads to a factor of 3 enhancement in the predicted rate of collisional test particle diffusion as compared to theory based on IUO. The new prediction is in agreement with recent experiments. [S0031-9007(97)04177-X] The collisional diffusion of test particles across a magnetic field has recently been measured in a single species ion plasma confined in a Penning-Malmberg trap in the regime rc ? lD (where rc is the cyclotron radius and lD is the Debye length) [1]. In this paper we present a calculation of the collisional test particle diffusion coefficient D , and compare it to the measurement. The calculation involves a novel effect in plasma kinetic theory: an enhanced diffusion coefficient arising from multiple binary collisions between the same pair of particles. The multiple collisions are caused by velocity diffusion of the particle trajectories. Before we can explain this new effect, we must first explain why previous theoretical calculations of the test particle diffusion coefficient fail to reproduce the experimental data. The well-known classical diffusion coefficient [2],

Thermal and isothermal slip in a new Liu model transport equation

The aim of this study was to investigate the suitability of the Liu model to solve boundary-value problems in kinetic theory. The problem of isothermal and thermal slip, which is of independent interest, is solved. A constraint on the determining parameter of the model is obtained.

Smoothed particle hydrodynamics techniques for the solution of kinetic theory problems Part 1: Method

The smoothed particle hydrodynamics (SPH) technique has been applied to a problem in kinetic theory, namely, the dynamics of liquid crystalline polymers (LCPs). It is a Lagrangian solution method developed for fluid flow calculations; its adaption to kinetic theory is outlined. The Lagrangian formulation of the Doi theory for LCPs is first described, and the problem is presented in the general framework of nonparametric density estimation. The implementation of the SPH technique in this specific problem is given, highlighting particular aspects of our implementation of SPH, including the form of the kernel function and use of an adaptive kernel. We then present results which demonstrate convergence and other details of the solution method, and also make comparisons with other solution techniques and discuss other potential applications.

An inverse model problem in kinetic theory

This paper deals with an inverse model problem in linear kinetic theory: the identification of a density profile of a scattering medium in a slab geometry from measurements of the reflected portion of a particle flux entering the medium. We prove well-posedness of the problem and present a robust algorithm for the identification.

Existence and uniqueness for nonlinear boundary value problems in kinetic theory

A boundary value problem for the stationary nonlinear Boltzmann equation in a slab has been examined in a weightedL ∞ space. It has been proved that the problem possesses a unique solution for boundary data small enough. The proof is based on the implicit function theorem. It has also been shown that for the linearized problem the Fredholm alternative applies.