Showing papers on "Kinetic theory of gases published in 2002"

Kinetic Theory and Fluid Dynamics

Abstract: In this series of talks, I will discuss the fluid-dynamic-type equations that is derived from the Boltzmann equation as its the asymptotic behavior for small mean free path. The study of the relation of the two systems describing the behavior of a gas, the fluid-dynamic system and the Boltzmann system, has a long history and many works have been done. The Hilbert expansion and the Chapman–Enskog expansion are well-known among them. The behavior of a gas in the continuum limit, however, is not so simple as is widely discussed by superficial understanding of these solutions. The correct behavior has to be investigated by classifying the physical situations. The results are largely different depending on the situations. There is an important class of problems for which neither the Euler equations nor the Navier–Stokes give the correct answer. In these two expansions themselves, an initialor boundaryvalue problem is not taken into account. We will discuss the fluid-dynamic-type equations together with the boundary conditions that describe the behavior of the gas in the continuum limit by appropriately classifying the physical situations and taking the boundary condition into account. Here the result for the time-independent case is summarized. The time-dependent case will also be mentioned in the talk. The velocity distribution function approaches a Maxwellian fe, whose parameters depend on the position in the gas, in the continuum limit. The fluid-dynamictype equations that determine the macroscopic variables in the limit differ considerably depending on the character of the Maxwellian. The systems are classified by the size of |fe− fe0|/fe0, where fe0 is the stationary Maxwellian with the representative density and temperature in the gas. (1) |fe − fe0|/fe0 = O(Kn) (Kn : Knudsen number, i.e., Kn = `/L; ` : the reference mean free path. L : the reference length of the system) : S system (the incompressible Navier–Stokes set with the energy equation modified). (1a) |fe − fe0|/fe0 = o(Kn) : Linear system (the Stokes set). (2) |fe − fe0|/fe0 = O(1) with | ∫ ξifedξ|/ ∫ |ξi|fedξ = O(Kn) (ξi : the molecular velocity) : SB system [the temperature T and density ρ in the continuum limit are determined together with the flow velocity vi of the first order of Kn amplified by 1/Kn (the ghost effect), and the thermal stress of the order of (Kn) must be retained in the equations (non-Navier–Stokes effect). The thermal creep[1] in the boundary condition must be taken into account. (3) |fe − fe0|/fe0 = O(1) with | ∫ ξifedξ|/ ∫ |ξi|fedξ = O(1) : E+VB system (the Euler and viscous boundary-layer sets). E system (Euler set) in the case where the boundary is an interface of the gas and its condensed phase. The fluid-dynamic systems are classified in terms of the macroscopic parameters that appear in the boundary condition. Let Tw and δTw be, respectively, the characteristic values of the temperature and its variation of the boundary. Then, the fluid-dynamic systems mentioned above are classified with the nondimensional temperature variation δTw/Tw and Reynolds number Re as shown in Fig. 1. In the region SB, the classical gas dynamics is inapplicable, that is, neither the Euler

Kinetic boundary conditions in the lattice Boltzmann method.

Abstract: Derivation of the lattice Boltzmann method from the continuous kinetic theory [X. He and L. S. Luo, Phys. Rev. E 55, R6333 (1997); X. Shan and X. He, Phys. Rev. Lett. 80, 65 (1998)] is extended in order to obtain boundary conditions for the method. For the model of a diffusively reflecting moving solid wall, the boundary condition for the discrete set of velocities is derived, and the error of the discretization is estimated. Numerical results are presented which demonstrate convergence to the hydrodynamic limit. In particular, the Knudsen layer in the Kramers' problem is reproduced correctly for small Knudsen numbers.

The Einstein-Vlasov System/Kinetic Theory

Abstract: The main purpose of this article is to provide a guide to theorems on global properties of solutions to the Einstein-Vlasov system. This system couples Einstein’s equations to a kinetic matter model. Kinetic theory has been an important field of research during several decades in which the main focus has been on nonrelativistic and special relativistic physics, i.e. to model the dynamics of neutral gases, plasmas, and Newtonian self-gravitating systems. In 1990, Rendall and Rein initiated a mathematical study of the Einstein-Vlasov system. Since then many theorems on global properties of solutions to this system have been established. The Vlasov equation describes matter phenomenologically, and it should be stressed that most of the theorems presented in this article are not presently known for other such matter models (i.e. fluid models). This paper gives introductions to kinetic theory in non-curved spacetimes and then the Einstein-Vlasov system is introduced. We believe that a good understanding of kinetic theory in non-curved spacetimes is fundamental to good comprehension of kinetic theory in general relativity.

Nonhydrodynamic modes and a priori construction of shallow water lattice Boltzmann equations

Abstract: Lattice Boltzmann equations for the isothermal Navier-Stokes equations have been constructed systematically using a truncated moment expansion of the equilibrium distribution function from continuum kinetic theory. Applied to the shallow water equations, with its different equation of state, the same approach yields discrete equilibria that are subject to a grid scale computational instability. Different and stable equilibria were previously constructed by Salmon [J. Marine Res. 57, 503 (1999)]. The two sets of equilibria differ through a nonhydrodynamic or ``ghost'' mode that has no direct effect on the hydrodynamic behavior derived in the slowly varying limit. However, Salmon's equilibria eliminate a coupling between hydrodynamic and ghost modes, one that leads to instability with a growth rate increasing with wave number. Previous work has usually assumed that truncated moment expansions lead to stable schemes. Such instabilities have implications for lattice Boltzmann equations that simulate other nonideal equations of state, or that simulate fully compressible, nonisothermal fluids using additional particles.

Hydrodynamics for a granular binary mixture at low density

Abstract: Hydrodynamic equations for a binary mixture of inelastic hard spheres are derived from the Boltzmann kinetic theory. A normal solution is obtained via the Chapman–Enskog method for states near the local homogeneous cooling state. The mass, heat, and momentum fluxes are determined to first order in the spatial gradients of the hydrodynamic fields, and the associated transport coefficients are identified. In the same way as for binary mixtures with elastic collisions, these coefficients are determined from a set of coupled linear integral equations. Practical evaluation is possible using a Sonine polynomial approximation, and is illustrated here by explicit calculation of the relevant transport coefficients: the mutual diffusion, the pressure diffusion, the thermal diffusion, the shear viscosity, the Dufour coefficient, the thermal conductivity, and the pressure energy coefficient. All these coefficients are given in terms of the restitution coefficients and the ratios of mass, concentration, and particle s...

Gaseous mixture flow through a long tube at arbitrary Knudsen numbers

Abstract: The mass flow, heat flux, and diffusion flux of rarefied gas mixture through a tube caused by gradients of pressure, temperature, and concentration were calculated over a wide range of the Knudsen number on the basis of the kinetic equation. The thermodynamic fluxes are presented in the form that allows us to prove the Onsager relations and then to reduce the number of kinetic coefficients determining the solution down to six. The numerical values of the kinetic coefficients are tabulated and the velocity profiles are given in figures.

High-energy tails for inelastic Maxwell models

Abstract: Monte Carlo simulations of the spatially homogeneous Boltzmann equation for inelastic Maxwell molecules, performed by Baldassarri et al. (cond-mat/0111066), have shown that general classes of initial distributions evolve for large times into a singular nonlinear scaling solution with a power law tail. By applying an asymptotic analysis we derive these results from the nonlinear Boltzmann equation, and obtain a transcendental equation from which the exponents, appearing in the power law tails, can be calculated. The dynamics of this model describes a dissipative flow in v-space, which drives the system to an attractor, the nonlinear scaling solution, with a constant negative rate of irreversible entropy production, given by − ¼(1 − α2), where α is the coefficient of restitution.

Transport coefficients of a heated granular gas

Abstract: The Navier–Stokes transport coefficients of a granular gas are obtained from the Chapman–Enskog solution to the Boltzmann equation. The granular gas is heated by the action of an external driving force (thermostat) which does work to compensate for the collisional loss of energy. Two types of thermostats are considered: (a) a deterministic force proportional to the particle velocity (Gaussian thermostat), and (b) a random external force (stochastic thermostat). As happens in the free cooling case, the transport coefficients are determined from linear integral equations which can be approximately solved by means of a Sonine polynomial expansion. In the leading order, we get those coefficients as explicit functions of the restitution coefficient α. The results are compared with those obtained in the free cooling case, indicating that the above thermostat forces do not play a neutral role in the transport. The kinetic theory results are also compared with those obtained from Monte Carlo simulations of the Boltzmann equation for the shear viscosity. The comparison shows an excellent agreement between theory and simulation over a wide range of values of the restitution coefficient. Finally, the expressions of the transport coefficients for a gas of inelastic hard spheres are extended to the revised Enskog theory for a description at higher densities.

Relativistic Boltzmann Equation

Abstract: The purpose of this chapter is to introduce the basic concepts of relativistic kinetic theory and the relativistic Boltzmann equation, which rules the time evolution of the distribution function.

Hydrodynamics from Grad's equations: What can we learn from exact solutions?

Abstract: A detailed treatment of the classical Chapman-Enskog derivation of hydrodynamics is given in the framework of Grad's moment equations. Grad's systems are considered as the minimal kinetic models where the Chapman-Enskog method can be studied exactly, thereby providing the basis to compare various approximations in extending the hydrodynamic description beyond the Navier-Stokes approximation. Various techniques, such as the method of partial summation, Pade approximants, and invariance principle are compared both in linear and nonlinear situations.

Effective Kinetic Theory for High Temperature Gauge Theories

Abstract: Quasiparticle dynamics in relativistic plasmas associated with hot, weakly-coupled gauge theories (such as QCD at asymptotically high temperature $T$) can be described by an effective kinetic theory, valid on sufficiently large time and distance scales. The appropriate Boltzmann equations depend on effective scattering rates for various types of collisions that can occur in the plasma. The resulting effective kinetic theory may be used to evaluate observables which are dominantly sensitive to the dynamics of typical ultrarelativistic excitations. This includes transport coefficients (viscosities and diffusion constants) and energy loss rates. We show how to formulate effective Boltzmann equations which will be adequate to compute such observables to leading order in the running coupling $g(T)$ of high-temperature gauge theories [and all orders in $1/\log g(T)^{-1}$]. As previously proposed in the literature, a leading-order treatment requires including both $2 2$ particle scattering processes as well as effective ``$1 2$'' collinear splitting processes in the Boltzmann equations. The latter account for nearly collinear bremsstrahlung and pair production/annihilation processes which take place in the presence of fluctuations in the background gauge field. Our effective kinetic theory is applicable not only to near-equilibrium systems (relevant for the calculation of transport coefficients), but also to highly non-equilibrium situations, provided some simple conditions on distribution functions are satisfied.

Dynamics: Models and Kinetic Methods for Non-equilibrium Many Body Systems

Abstract: Preface. 1: Brownian Motion. Dynamics of colloidal systems: beyond the stochastic approach L. Bocquet, J.P. Hansen. Lattice-Boltzmann simulations of hydrodynamically interacting particles A.J.C. Ladd. Orientational relaxation and Brownian motion B.U. Felderhof, R.B. Jones. Viscosity and diffusion of concentrated hard-sphere-like colloidal suspensions R. Verberg, et al. Interacting Brownian particles B. Cichocki. Spinodal decomposition kinetics: the initial and intermediate stages J.K.G. Dhont. First Interlude. Seventeenth century Dutch art: a brief guide to the Mauritshuis, The Hague, and to the Rijksmuseum, Amsterdam J.R. Dorfman. 2: Dynamical Systems. Kinetic Theory of Dynamical Systems R. van Zon, et al. Multifractal phase-space distributions for stationary nonequilibrium systems H.A. Posch, et al. A model of non-equilibrium statistical mechanics J. Piasecki, Y.G. Sinai. Statistical properties of chaotic systems in high dimensions N. Chernov. Dynamical systems and statistical mechanics: Lyapunov exponents and transport coefficients E.G.D. Cohen. Second Interlude. Boltzmann and statistical mechanics E.G.D. Cohen. 3: Granular Flows. Kinetic theory of granular fluids: hard and soft inelastic spheres M.H. Ernst. Inelastic collapse S. McNamara. Stationary states in systems with dissipative interactions J. Piasecki. Motion along a rough inclined surface A. Hansen. Hydraulic theory for a granular heap on an incline J.T. Jenkins. Experimental studies of granular flows J.M. Huntley, R.D. Wildman. Model kinetic equations for rapid granular flows J.J. Brey. Velocity correlations in driven two-dimensional granular media C. Bizon, et al. Third Interlude. G.E. Uhlenbeck on Paul Ehrenfest E.G.D. Cohen. 4: Quantum Kinetic Theory. Quantum kinetic theory: the disordered electron problem T.R. Kirkpatrick, D. Belitz. Quantum phase transitions D. Belitz, T.R. Kirkpatrick. Scattering, transport & Stochasticity in quantum systems P. Gaspard. Some problems of the kinetic theory of mesoscopic systems V.L. Gurevich. Kinetic theory for electron dynamics in semiconductors and plasmas J.W. Dufty. Quantum kinetic theory of trapped atomic gases H.T.C. Stoof. Participants. Index.

Viscoelastic fluid models derived from kinetic equations for polymers

Abstract: Differential relationships between velocity and stress are derived from the kinetic theory of polymeric liquids. The Fokker--Planck equation corresponding to the so-called dumbbell theory for dilute solutions of polymers is considered. AChapman--Enskog expansion for time-dependent, nonhomogeneous flows is proposed when the elastic character of the fluid is small, and the results included in the book of R. Bird, C. Curtiss, R. Armstrong, and O. Hassager [Dynamics of Polymeric Liquids, Vol., Kinetic Theory, John Wiley & Sons, New York, 1987] are recovered. A comparison with Oldroyd-B, FENE, and FENE-P fluids is presented in the frame of the plane Couette flow.

Conductivity of the classical two-dimensional electron gas

Abstract: We discuss the applicability of the Boltzmann equation to the classical two-dimensional electron gas. We show that in the presence of both electron-impurity and electron-electron scattering an approach which treats the electron-impurity scattering through the scattering integral in the Boltzmann equation can be inapplicable. The correct result for conductivity can be different from the one obtained from the kinetic equation by a logarithmically large factor.

Heat transfer in a gas mixture between two parallel plates: Finite-difference analysis of the Boltzmann equation

Abstract: The problem of heat transfer and temperature distribution in a binary mixture of rarefied gases between two parallel plates with different temperatures is investigated on the basis of kinetic theory. Under the assumption that the gas molecules are hard spheres and undergo diffuse reflection on the plates, the Boltzmann equation is analyzed numerically by means of an accurate finite-difference method, in which the complicated nonlinear collision integrals are computed efficiently by the deterministic numerical kernel method. As a result, the overall quantities (the heat flow in the mixture, etc.) as well as the profiles of the macroscopic quantities (the molecular number densities of the individual components, the temperature of the total mixture, etc.) are obtained accurately for a wide range of the Knudsen number. At the same time, the behavior of the velocity distribution function is clarified with high accuracy.

A Diagrammatic Formulation of the Kinetic Theory of Fluctuations in Equilibrium Classical Fluids. I. The Fluctuation Basis and the Cluster Properties of Associated Functions

Abstract: This is the first of a series of papers that presents a kinetic theory of fluctuations in equilibrium classical fluids that makes extensive use of diagrammatic techniques in its development and that will facilitate the use of diagrammatic techniques in the derivation of approximate kinetic theories. We develop the theory for atomic liquids, but the results are easily generalizable to molecular liquids. The fundamental fluctuating quantity in the theory is f(R,P), the density of particles (atoms) at a point in single-particle phase space. The time correlation function for fluctuations of this quantity from its average, , is the most basic correlation function of concern in generalized kinetic theories of fluctuations in liquids. In this paper, we investigate the properties of a basis set of vectors for the Hilbert space of classical dynamical variables that was suggested by Gross, Boley, and Lindenfeld. In later papers, this basis set, which we call the "fluctuation basis", will be used to construct a diagrammatic theory for this correlation function, its generalizations, and its memory function.

Generalized kinetic theory of electrons and phonons

Abstract: A generalized kinetic theory [1] was proposed in order to have the possibility to treat particles which obey a very general statistics. By adopting the same approach, we generalize here the kinetic theory of electrons and phonons. Equilibrium solutions and their stability are investigated.

Study of a shock wave structure in gas mixtures on the basis of the Boltzmann equation

Abstract: The structure of a shock wave for a binary gas mixture is studied on the basis of numerical solution of the complete kinetic Boltzmann equation for the model of hard sphere molecules. For the evaluation of the collision integral we apply a generalization of the conservative discrete ordinate method (the kernel method for a single gas was developed by Tcheremissine) for binary gas mixtures and for the case of cylindrical symmetry. The transition from up-stream to downstream uniform state is presented by macroscopic values and by distribution functions.

Dynamics of ballistic annihilation.

Abstract: The problem of ballistically controlled annihilation is revisited for general initial velocity distributions and an arbitrary dimension. An analytical derivation of the hierarchy equations obeyed by the reduced distributions is given, and a scaling analysis of the corresponding spatially homogeneous system is performed. This approach points to the relevance of the nonlinear Boltzmann equation for dimensions larger than 1 and provides expressions for the exponents describing the decay of the particle density n(t) proportional, variant t(-xi) and the root-mean-square velocity v proportional, variant t(-gamma) in terms of a parameter related to the dissipation of kinetic energy. The Boltzmann equation is then solved perturbatively within a systematic expansion in Sonine polynomials. Analytical expressions for the exponents xi and gamma are obtained in arbitrary dimension as a function of the parameter mu characterizing the small velocity behavior of the initial velocity distribution. Moreover, the leading non-Gaussian corrections to the scaled velocity distribution are computed. These expressions for the scaling exponents are in good agreement with the values reported in the literature for continuous velocity distributions in d=1. For the two-dimensional case, we implement Monte Carlo and molecular dynamics simulations that turn out to be in excellent agreement with the analytical predictions.

Stationary state of thermostated inelastic hard spheres

Abstract: We consider a fluid composed of inelastic hard spheres moving in a thermostat modelled by a hard sphere gas. The losses of energy due to inelastic collisions are balanced by the energy transfer via elastic collisions from the thermostat particles. The resulting stationary state is analysed within the Boltzmann kinetic theory. A numerical iterative method permits to study the nature of deviations from the Gaussian state. Some analytic results are obtained for a one-dimensional system.

Higher-order hydrodynamics: Extended Fick’s Law, evolution equation, and Bobylev’s instability

Abstract: A higher-order hydrodynamics for material motion in fluids, under arbitrary nonequilibrium conditions, is constructed. We obtain what is a generalized—to that conditions—Fick-type Law. It includes a representation of Burnett-type contributions of all order, in the form of a continuous-fraction expansion. Also, the equation includes generalized thermodynamic forces, which are characterized and discussed. All kinetic coefficients are given as correlations of microscopic mechanical quantities averaged over the nonequilibrium ensemble, and then are time- and space-dependent as a consequence of accounting for the dissipative processes that are unfolding in the medium. An extended evolution equation for the density of particles is derived, and the conditions when it goes over restricted forms of the type of the telegraphist equation and Fick’s diffusion equation are presented.

Collision group and renormalization of the Boltzmann collision integral.

Abstract: On the basis of a recently discovered collision group [V. L. Saveliev, in Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. Gallis, AIP Conf. Proc. No. 585 (AIP, Melville, NY, 2001), p. 101], the Boltzmann collision integral is exactly rewritten in two parts. The first part describes the scattering of particles with small angles. In this part the infinity due to the infinite cross sections is extracted from the Boltzmann collision integral. Moreover, the Boltzmann collision integral is represented as a divergence of the flow in velocity space. Owing to this, the role of collisions in the kinetic equation can be interpreted in terms of the nonlocal friction force that depends on the distribution function.

Fluid Dynamic Limits of the Boltzmann Equation

Abstract: In 1866, James Clerk Maxwell (1831–1879) developed a fundamental theoretical basis for the kinetics theory of gases. Maxwell’s theory is based on the idea of Daniel Bernoulli (1738), which gave birth to the kinetic theory of gases, that gases are formed of electric molecules rushing hither and thither at high speeds, colliding and rebounding according to the laws of elementary mechanics (see, Cercignani, Illner, and Pulvirenti 1994, pp. 8–12). In fact, Maxwell developed, first, a theory of transport processes and gave a heuristic derivation of the velocity distribution function that bears his name. Next, he developed a much more accurate model (Maxwell 1867), based on transfer equations, in fact, a model, according to which the molecules interact with a force inversely proportional to the fifth power of the distance between them (now commonly called Maxwellian molecules). With these transfer equations, Maxwell came very close to an evolution equation for the distribution, but this step (1872) must be credited to Ludwig Boltzmann (1844–1906). The equation under consideration is usually called the Boltzmann equation.

Conservative force fields in nonextensive kinetic theory

Abstract: We investigate the nonextensive q-distribution function for a gas in the presence of an external field of force possessing a potential U( r ) . In the case of a dilute gas, we show that the power-law distribution including the potential energy factor term can rigorously be deduced based on kinetic theoretical arguments. This result is significant as a preliminary to the discussion of long-range interactions according to nonextensive thermostatistics and the underlying kinetic theory. As an application, the historical problem of the unbounded isothermal planetary atmospheres is rediscussed. It is found that the maximum height for the equilibrium atmosphere is zmax=KBT/mg(1−q). In the extensive limit, the exponential Boltzmann factor is recovered and the length of the atmosphere becomes infinite.

Kinetic Phenomena in the Diffusion of Gases in Capillaries and Porous Bodies

Abstract: The flow of gases and gaseous mixtures in capillaries and porous bodies were considered. Phenomenological transfer equations were derived using methods of non-equilibrium thermodynamics. Integral and local equations of non-equilibrium thermodynamics were constructed and phenomenological relations of two types were obtained. Different approaches to the calculation of kinetic coefficients were indicated. Methods of kinetic theory based on Boltzmann's equation and the dusty gas model were analyzed separately. Results of the calculations of kinetic coefficients for small and large Knudsen numbers were reported with allowance for incomplete accommodation of molecules on the surface of channel walls. A number of kinetic effects exhibiting during the flow of gases and gaseous mixtures through the capillaries and porous bodies was discussed.

Kinetic Theory of a Dilute Gas System under Steady Heat Conduction

Abstract: The velocity distribution function of the steady-state Boltzmann equation for hard-core molecules in the presence of a temperature gradient has been obtained explicitly to second order in density and the temperature gradient. Some thermodynamical quantities are calculated from the velocity distribution function for hard-core molecules and compared with those for Maxwell molecules and the steady-state Bhatnagar-Gross-Krook(BGK) equation. We have found qualitative differences between hard-core molecules and Maxwell molecules in the thermodynamical quantities, and also confirmed that the steady-state BGK equation belongs to the same universality class as Maxwell molecules.

Memory effects and conservation laws in the quantum kinetic evolution of a dilute Bose gas

Abstract: We derive a non-Markovian generalization to the quantum kinetic theory described by Walser et al. [Phys. Rev. A 59, 3878 (1999)] in the absence of a condensed fraction for temperatures above the Bose-Einstein condensationtemperature i.e., T>T c . Within this framework, quasiparticle damping arises naturally due the finite duration of a binary collision and it leads to a systematic Markov approximation from the non-Markovian Born theory. Such a self-consistent theory conserves the total energy to second order in the interaction strength. By introducing an improved damping function, we demonstrate global energy conservation at the order of the perturbation theory. Finally, we apply this kinetic theory to a simple model of an inhomogeneous Bose gas that is confined in a spherical box. By studying numerically the real-time quantum evolution towards equilibrium, we obtain damping rates and frequencies of the collective modes and illustrate the emergence of differing time scales for correlation and relaxation.