Equations Of Kinetic Theory Research Articles

Well-posedness of the spatially homogeneous Landau equation for soft potentials

We consider the spatially homogeneous Landau equation of kinetic theory, and provide a differential inequality for the Wasserstein distance with quadratic cost between two solutions. We deduce some well-posedness results. The main difficulty is that this equation presents a singularity for small relative velocities. Our uniqueness result is the first one in the important case of soft potentials. Furthermore, it is almost optimal for a class of moderately soft potentials, that is for a moderate singularity. Indeed, in such a case, our result applies for initial conditions with finite mass, energy, and entropy. For the other moderately soft potentials, we assume additionally some moment conditions on the initial data. For very soft potentials, we obtain only a local (in time) well-posedness result, under some integrability conditions. Our proof is probabilistic, and uses a stochastic version of the Landau equation, in the spirit of Tanaka [H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules, Z. Wahrsch. Verw. Geb. 46 (1) (1978–1979) 67–105].

What Does Kinetic Theory Have to Do with Hydraulics?

This forum article introduces the fundamental equation of kinetic theory the Boltzmann equation and highlights some of its potential usefulness in hydraulics and fluid mechanics. The potential of using kinetic theory to construct computational models for the shallow water equations as well as the Navier-Stokes equations is summarized and illustrated. In addition, the usage of kinetic theory to formulate turbulence models is briefly described. Kinetic theory 1 recognizes that matter is made up of an enormous number of tiny units, called molecules and atoms, that bounce around and interact with one another and 2 provides the link between the gross properties of matter and the motion of its parts. The basic concepts of kinetic theory are quite familiar to the hydraulic engineer. Indeed, the basic concepts of kinetic theory are called upon whenever one explains macroscopic behavior in terms of molecular motion.

Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier–Stokes asymptotics

In this paper, we develop a numerical method to solve Boltzmann like equations of kinetic theory which is able to capture the compressible Navier–Stokes dynamics at small Knudsen numbers. Our approach is based on the micro/macro decomposition technique, which applies to general collision operators. This decomposition is performed in all the phase space and leads to an equivalent formulation of the Boltzmann (or BGK) equation that couples a kinetic equation with macroscopic ones. This new formulation is then discretized with a semi-implicit time scheme combined with a staggered grid space discretization. Finally, several numerical tests are presented in order to illustrate the efficiency of our approach. Incidentally, we also introduce in this paper a modification of a standard splitting method that allows to preserve the compressible Navier–Stokes asymptotics in the case of the simplified BGK model. Up to our knowledge, this property is not known for general collision operators.

Kinetic lattice Boltzmann method for microscale gas flows: Issues on boundary condition, relaxation time, and regularization

It is well known that the Navier-Stokes equations cannot adequately describe gas flows in the transition and free-molecular regimes. In these regimes, the Boltzmann equation (BE) of kinetic theory is invoked to govern the flows. However, this equation cannot be solved easily, either by analytical techniques or by numerical methods. Hence, in order to efficiently maneuver around this equation for modeling microscale gas flows, a kinetic lattice Boltzmann method (LBM) has been introduced in recent years. This method is regarded as a numerical approach for solving the BE in discrete velocity space with Gauss-Hermite quadrature. In this paper, a systematic description of the kinetic LBM, including the lattice Boltzmann equation, the diffuse-scattering boundary condition for gas-surface interactions, and definition of the relaxation time, is provided. To capture the nonlinear effects due to the high-order moments and wall boundaries, an effective relaxation time and a modified regularization procedure of the nonequilibrium part of the distribution function are further presented based on previous work [Guo et al., J. Appl. Phys. 99, 074903 (2006); Shan et al., J. Fluid Mech. 550, 413 (2006)]. The capability of the kinetic LBM of simulating microscale gas flows is illustrated based on the numerical investigations of micro Couette and force-driven Poiseuille flows.

Global Existence for Quadratic Systems of Reaction-Diffusion

Abstract We prove global existence in time of weak solutions to a class of quadratic reaction-diffusion systems for which a Lyapounov structure of L log L-entropy type holds. The approach relies on an a priori dimension-independent L2-estimate, valid for a wider class of systems including also some classical Lotka-Volterra systems, and which provides an L1-bound on the nonlinearities, at least for not too degenerate diffusions. In the more degenerate case, some global existence may be stated with the use of a weaker notion of renormalized solution with defect measure, arising in the theory of kinetic equations.

Kinetic equations for exactly solvable models of geminate reactions in the presence of scavengers

The kinetic equations for an exactly solvable scavenger problem were presented in the form of the equation of general kinetic theory. Presenting the memory function as the sum of Markovian and non-Markovian parts yielded a binary approximation corresponding to low scavenger concentrations. It was demonstrated that the bulk channel of the formation of product D can be separated into two subchannels: the reaction of a scavenger C with the reactant B that left a geminate pair AB for the bulk solution (bulk subchannel) and with the reactant B that retains a spatial correlation with the reactant A (geminate subchannel). Note that the geminate subchannel of the D-channel and the geminate subchannel of the formation of product P are of three-particle nature, and, therefore, their rates cannot be expressed through the relative mobilities of the reactants in pairs BC and AB. The obtained equations of general kinetic theory were transformed so as to impart them the regular form of the rate equations accepted in chemistry, with a bulk reaction rate constant and nonuniform terms (sources) associated with the initial spatial correlations between the reactants. Nevertheless, these equations differ substantially from those reported in the literature.

Scaling and Expansion of Moment Equations in Kinetic Theory

The set of generalized 13 moment equations for molecules interacting with power law potentials [Struchtrup, Multiscale Model. Simul. 3:211 (2004)] forms the base for an investigation of expansion methods in the Knudsen number and other scaling parameters. The scaling parameters appear in the equations by introducing dimensionless quantities for all variables and their gradients. Only some of the scaling coefficients can be chosen independently, while others depend on these chosen scales–their size can be deduced from a Chapman–Enskog expansion, or from the principle that a single term in an equation cannot be larger in size by one or several orders of magnitude than all other terms.It is shown that for the least restrictive scaling the new order of magnitude expansion method [Struchtrup, Phys. Fluids 16(11):3921 (2004)] reproduces the original equations after only two expansion steps, while the classical Chapman–Enskog expansion would require an infinite number of steps. Both methods yield the Euler and Navier–Stokes–Fourier equations to zeroth and first order. More restrictive scaling choices, which assume slower time scales, small velocities, or small gradients of temperature, are considered as well.

Numerical methods for solving moment equations in kinetic theory of neuronal network dynamics

Recently developed kinetic theory and related closures for neuronal network dynamics have been demonstrated to be a powerful theoretical framework for investigating coarse-grained dynamical properties of neuronal networks. The moment equations arising from the kinetic theory are a system of (1 + 1)-dimensional nonlinear partial differential equations (PDE) on a bounded domain with nonlinear boundary conditions. The PDEs themselves are self-consistently specified by parameters which are functions of the boundary values of the solution. The moment equations can be stiff in space and time. Numerical methods are presented here for efficiently and accurately solving these moment equations. The essential ingredients in our numerical methods include: (i) the system is discretized in time with an implicit Euler method within a spectral deferred correction framework, therefore, the PDEs of the kinetic theory are reduced to a sequence, in time, of boundary value problems (BVPs) with nonlinear boundary conditions; (ii) a set of auxiliary parameters is introduced to recast the original BVP with nonlinear boundary conditions as BVPs with linear boundary conditions – with additional algebraic constraints on the auxiliary parameters; (iii) a careful combination of two Newton’s iterates for the nonlinear BVP with linear boundary condition, interlaced with a Newton’s iterate for solving the associated algebraic constraints is constructed to achieve quadratic convergence for obtaining the solutions with self-consistent parameters. It is shown that a simple fixed-point iteration can only achieve a linear convergence for the self-consistent parameters. The practicability and efficiency of our numerical methods for solving the moment equations of the kinetic theory are illustrated with numerical examples. It is further demonstrated that the moment equations derived from the kinetic theory of neuronal network dynamics can very well capture the coarse-grained dynamical properties of integrate-and-fire neuronal networks.

Quantum kinetic theory of a Bose-Einstein gas confined in a lattice

We continue our earlier work [Ana Maria Rey, B. L. Hu, Esteban Calzetta, Albert Roura and Charles W. Clark, Phys. Rev. A 69, 033610 (2004)] on the nonequilibrium dynamics of a Bose Einstein condensate (BEC) selectively loaded into every third site of a one-dimensional optical lattice. From the two-particle irreducible (2PI) closed-time-path (CTP) effective action for the Bose- Hubbard Hamiltonian, we show how to obtain the Kadanoff-Baym equations of quantum kinetic theory. Using the quasiparticle approximation, we show that the local equilibrium solutions of these equations reproduce the second- order corrections to the self-energy originally derived by Beliaev. This work paves the way for the use of effective action methods in the derivation of quantum kinetic theory of many atom systems.

The analysis of the derivation principles of kinetic equations based on exactly solvable models of the bulk reaction A + B → Product

We have considered two many-particle models of the irreversible reaction A + B → Product for which closed kinetic equations for the mean concentration N A ( t) of A species can be exactly obtained. These equations are identically recast into a unified form of integro-differential equation of general kinetic theory. It is shown that the memory functions for both models under consideration can be represented as a sum of the Markovian and non-Markovian parts. It is essential that the Markovian part of the Laplace transform of any kernel can be obtained using the Laplace transform of the kernel itself, and is the root of the non-Markovian part of the Laplace transform of the kernel. The properties established allowed us to perform correct approximation of the memory functions at small concentrations [ B] of B species and derive the binary non-Markovian integro-differential equation. Within the binary theory accuracy this equation has been rewritten in a regular frame of a familiar rate equation satisfying general principles of binary kinetic equations. Thus using particular exactly solvable many-particle models, we have reproduced the most essential steps of the known general way for the derivation of the binary kinetic equation avoiding the sophisticated many-particle technique and the corresponding approximations. Besides, the results obtained can serve as an additional evidence of the approximations made in a general many-particle approach to the derivation of the binary kinetic equation.

Kinetic equations for exactly solvable models of a geminate reaction in the presence of scavengers

On the basis of two many-particle models of a geminate reaction A + B → Product in the presence of bulk scavengers C for which the closed kinetic equations for the mean concentration of species can be exactly derived the unified form of integro-differential equation of general kinetic theory is obtained. A clear physical interpretation has been done for every term of the equations. For low concentration [C] of scavengers, the kinetic equations are reduced to the so-called quasi-binary ones that in turn are brought into the equivalent regular form (the rate type equations).

On the Vlasov Limit for Systems of Nonlinearly Coupled Oscillators without Noise

We discuss the application to systems of coupled nonlinear oscillators of methods originally developed for the Vlasov-Poisson equations of plasma kinetic theory. These methods lead to a simple but rigorous derivation of the infinite-oscillator limit for the well-known Kuramoto model, and also to a compact proof of global existence and uniqueness of solutions to the resulting Kuramoto-Sakaguchi equations. Vlasov-limit techniques can also be adapted to prove a central limit theorem for statistical ensembles of systems of oscillators. However, we emphasize that the noiseless Kuramoto model itself is strictly deterministic, and that the Kuramoto-Sakaguchi equations can be formulated without any reference to a stochastic distribution of oscillator frequencies.

Viscosity of meson matter

We report a calculation of the shear viscosity in a relativistic multicomponent meson gas as a function of temperature and chemical potentials. We approximately solve the Uehling-Uhlenbeck transport equation of kinetic theory, appropriate for a boson gas, with relativistic kinematics. Since at low temperatures the gas can be taken as mostly composed of pions, with a fraction of kaons and etas, we explore the region where binary elastic collisions with at least one pion are the dominant scattering processes. Our input meson scattering phase shifts are fits to the experimental data obtained from chiral perturbation theory and the inverse amplitude method. Our results take the correct nonrelativistic limit (viscosity proportional to the square root of the temperature), show a viscosity of the order of the cube of the pion mass up to temperatures somewhat below that mass, and then a large increase due to kaons and etas. Our approximation may break down at even higher temperatures, where the viscosity follows a temperature power law with an exponent near 3.

Kinetic theory and Bose–Einstein condensation

In this article we discuss the role played by kinetic theory in describing the non-equilibrium dynamics of dilute systems of weakly interacting bosons. We illustrate how a simple kinetic equation for the time evolution of the spectral particle density can be derived from the spatially homogeneous Gross–Pitaevskii equation. This kinetic equation agrees with the usual Boltzmann–Nordheim equation of quantum kinetic theory in the long wavelength limit where the occupation numbers are expected to be large. The stationary solutions of the Gross–Pitaevskii kinetic equation are described. These include both thermodynamic equilibrium spectra and finite flux Kolmogorov–Zakharov spectra. These latter spectra are intrinsically nonequilibrium states and are expected to be relevant in the transfer of particles to low momenta in the initial stage of the condensation process. This is illustrated by some computations of a solution of the kinetic equation beginning with initial conditions far from equilibrium. The solution generates a flux of particles from large to small momenta which results in a singularity at zero momentum within finite time. We interpret this singularity as incipient condensate formation. We then present some numerical results on the post-singularity dynamics and the approach to equilibrium. Contrary to our original expectations we do not observe the Kolmogorov–Zakharov spectrum during the period of condensate growth. In the closing sections we address the issue of the connection between the Gross–Pitaevskii and Boltzmann–Nordheim kinetic equations. We argue that the two equations have differing regimes of applicability in momentum space, matching in an intermediate range. We make some suggestions of how this matching can be modeled in practice. To cite this article: C. Connaughton, Y. Pomeau, C. R. Physique 5 (2004).

Homogeneous Boltzmann equation in quantum relativistic kinetic theory

We consider some mathematical questions about Boltzmann equations for quantum particles, relativistic or non relativistic. Relevant particular cases such as Bose, Bose-Fermi, and photon-electron gases are studied. We also consider some simplifications such as the isotropy of the distribution functions and the asymptotic limits (systems where one of the species is at equilibrium). This gives rise to interesting mathematical questions from a physical point of view. New results are presented about the existence and long time behaviour of the solutions to some of these problems.

Dynamics of neuronal populations: eigenfunction theory; some solvable cases

A novel approach to cortical modelling was introduced by Knight and co-workers in 1996. In their presentation cortical dynamics is formulated in terms of interacting populations of neurons, a perspective that is in part motivated by modern cortical imaging. The approach may be regarded as the application of statistical mechanics to neuronal populations, and the simplest exemplar bears a kinship to the Boltzmann equation of kinetic theory. The disarming simplicity of this linear equation hides the complex behaviour it produces. A purpose of this paper is to investigate and reveal its intricacies by treating a series of solvable special cases. In particular we will focus on issues that relate to the spectral analysis of the underlying operators. A fairly thorough treatment is presented for a simple, but still useful example, that has important consequences for more general situations.

Quantum kinetic model of a driven radiative two-level system

We introduce a system of two-level atoms in one dimension which allow collision-induced radiative transitions between the two states. In addition, we subject the system to a driving force. A linear, but non-local master equation in momentum is solved exactly, illustrating three results: the development of multi-streaming motion, the occurrence of acceleration reversals, and an observed approach to a steady state as the emission of radiation and the driving field balance each other. We relate our results to the theory of kinetic equations with internal degrees of freedom used in rarefied gas dynamics.

DISCRETE VELOCITY MODEL AND IMPLICIT SCHEME FOR THE BGK EQUATION OF RAREFIED GAS DYNAMICS

We present a numerical method for computing transitional flows as described by the BGK equation of gas kinetic theory. Using the minimum entropy principle to define a discrete equilibrium function, a discrete velocity model of this equation is proposed. This model, like the continuous one, ensures positivity of solutions, conservation of moments, and dissipation of entropy. The discrete velocity model is then discretized in space and time by an explicit finite volume scheme which is proved to satisfy the previous properties. A linearized implicit scheme is then derived to efficiently compute steady-states; this method is then verified with several test cases.

The Complete A-Z Physics Handbook, Second Edition

The first edition of this book was published in 1997 and reviewed in Physics Education in March 1988. This second edition is 13 pages longer and contains a significant number of new entries, mostly in particle physics and astrophysics, to match the latest AS/A-level specifications. Some entries on topics such as surface tension that were removed from previous AS/A-level syllabuses have now been deleted. The revised book thus provides an effective alphabetical guide to key terms used in the latest AS/A-level specifications as most AS/A-level core terms are included, apart from terms for some of the new topics such as image processing in the new IOP AS/A-level specification. The style of the second edition is the same as the first, each entry commencing with the key term in bold print followed by an explanation of the term, accompanied by the relevant equation and/or a simple diagram where appropriate. Cross-references are given in italics. For example, the entry under quarks (which is one of the new entries) lists the six quarks and their characteristic properties in bullet point format then follows up with the information about the quark composition of baryons and mesons, giving the composition of the proton, the neutron and the pions as examples. The change of quark composition due to beta decay is mentioned with a cross-reference to beta decay. Other cross-references link to entries on the proton, the neutron and the four fundamental interactions. Lengthy entries such as the photoelectric effect each cover several pages, thus providing opportunities for in-depth revision. Useful theme entries provided in the first edition have been retained and referred to in a new two-page appendix on preparing for synoptic questions. Some proofs are provided such as the proof of the kinetic theory equation. Some experiments are described, each under an appropriate entry. Technically, the entries are reasonably accurate and usually expressed in appropriate and familiar language. The book ends with the same appendices as in the first edition (units, formulae, physical constants, examiners' terms, a suggested revision list and notes on examination preparation) plus the two-page synoptic appendix. The second edition with its additional entries on particle physics and astrophysics should prove useful as a study guide to supplement a textbook or as a revision guide as AS or A2 examinations approach. In addition, the concise nature of many of the entries could be valuable in helping physics students sharpen their written communication skills, thereby securing marks which all too often they lose on account of inadequate answers to descriptive questions.

Kinetic schemes and boundary conditions for moment equations

A numerical scheme for moment equations of kinetic theory, due to LeTallec & Perlat, is considered for the calculation of stationary heat transfer in the Grad 13 moment system and linearized extended thermodynamics of 14 moments. It is shown that the required distance of grid points must be considerably smaller than the mean free path. Thus, the kinetic scheme is useful only in the case of large Knudsen numbers. Results of the numerical calculation for 13 and 14 moments are compared with an analytical solution for heat transfer with 13 moments. The results indicate that the boundary conditions do not guarantee conservation of energy at the walls. In order to overcome this deficiency a modification of the boundary conditions is presented and discussed.