Boltzmann Kinetic Theory Research Articles

The dielectric function of weakly ionized dusty plasmas

Using classical Boltzmann kinetic theory, the dielectric function of weakly ionized unmagnetized dusty plasma is derived. The elastic Coulomb collision and inelastic charging collision of electrons with charged dust particle as well as charge variation on dust surface are taken into account. The theoretical result is applied to analyze the propagation of electromagnetic wave in a dusty plasma. It is demonstrated that the additional collision mechanism provided by charged dust particle can significantly increase the absorbed power of electromagnetic wave. These increases are mainly determined by the dust radius, density, and the charge numbers on the dust surface. The obtained results will support an enhanced understanding of the wave propagation processes in space and laboratory dusty plasmas.

High-order kinetic relaxation schemes as high-accuracy Poisson solvers

We present a new approach to find accurate solutions to the Poisson equation, as obtained from the steady-state limit of a diffusion equation with strong source terms. For this purpose, we start from Boltzmann's kinetic theory and investigate the influence of higher-order terms on the resulting macroscopic equations. By performing an appropriate expansion of the equilibrium distribution, we provide a method to remove the unnecessary terms up to a desired order and show that it is possible to find, with high level of accuracy, the steady-state solution of the diffusion equation for sizeable Knudsen numbers. In order to test our kinetic approach, we discretize the Boltzmann equation and solve the Poisson equation, spending up to six order of magnitude less computational time for a given precision than standard lattice Boltzmann methods (LBMs).

Limite de diffusion linéaire pour un système déterministe de sphères dures

We provide a rigorous derivation of the Brownian motion as the hydrodynamic limit of a deterministic system of hard spheres as the number of particles N goes to infinity and their diameter ε simultaneously goes to 0, in the fast relaxation limit Nεd−1→∞ (with a suitable scaling of the observation time and length). As suggested by Hilbert in his sixth problem, we use Boltzmann's kinetic theory as an intermediate level of description for the gas close to global equilibrium. Our proof relies on the fundamental ideas of Lanford. The main novelty is the detailed study of the branching process, leading to explicit estimates of pathological collision trees.

First-principles study on the electronic structure and transport properties of Mn3Cu4Bi4

Density functional calculations were used to elucidate the electronic structure and some transport properties of Mn3Cu4Bi4. The calculations results indicate that Mn3Cu4Bi4 compound possesses metallic character and has high carrier concentration. Moreover, Mn3Cu4Bi4 compound is a ferromagnetic material due to the exchange interaction of Mn-3d electrons. The Boltzmann kinetic transport theory was used to calculate the Seebeck coefficient with the constant relaxation time approximation. The result shows that the thermoelectric properties of Mn3Cu4Bi4 is isotropic and the Seebeck coefficient reaches 30.9μV/K at 490K when the chemical potential μ is −0.043eV, which is higher than that of many metals. It is the first time to report the Seebeck coefficient of Mn3Cu4Bi4.

The nineteenth century conflict between mechanism and irreversibility

The reversibility problem (better known as the reversibility objection) is usually taken to be an internal problem in the kinetic theory of gases, namely the problem of how to account for the second law of thermodynamics within this theory. Historically, it is seen as an objection that was raised against Boltzmann's kinetic theory of gases, which led Boltzmann to a statistical approach to the kinetic theory, culminating in the development of statistical mechanics. In this paper, I show that in the late nineteenth century, the reversibility problem had a much broader significance—it was widely discussed and certainly not only as an objection to Boltzmann's kinetic theory of gases. In this period, there was a conflict between mechanism and irreversibility in physics which was tied up with central issues in philosophy of science such as materialism, empiricism and the need for mechanistic foundations of physical theories, as well as with concerns about the heat death of the universe. I discuss how this conflict was handled by the major physicists of the period, such as Maxwell, Kelvin, Duhem, Poincaré, Mach and Planck, as well as by a number of lesser-known authors.

Thermal segregation of intruders in the Fourier state of a granular gas

A low density binary mixture of granular gases is considered within the Boltzmann kinetic theory. One component, the intruders, is taken to be dilute with respect to the other, and thermal segregation of the two species is described for a special solution to the Boltzmann equation. This solution has a macroscopic hydrodynamic representation with a constant temperature gradient and is referred to as the Fourier state. The thermal diffusion factor characterizing conditions for segregation is calculated without the usual restriction to Navier-Stokes hydrodynamics. Integral equations for the coefficients in this hydrodynamic description are calculated approximately within a Sonine polynomial expansion. Molecular dynamics simulations are reported, confirming the existence of this idealized Fourier state. Good agreement is found for the predicted and simulated thermal diffusion coefficient, while only qualitative agreement is found for the temperature ratio.

Classical relativistic ideal gas in thermodynamic equilibrium in a uniformly accelerated reference frame

A classical (non-quantum-mechanical) relativistic ideal gas in thermodynamic equilibrium in a uniformly accelerated frame of reference is studied using Gibbs's microcanonical and grand canonical formulations of statistical mechanics. Using these methods explicit expressions for the particle, energy and entropy density distributions are obtained, which are found to be in agreement with the well-known results of the relativistic formulation of Boltzmann's kinetic theory. Explicit expressions for the total entropy, total energy and rest mass of the gas are obtained. The position of the center of mass of the gas in equilibrium is found. The non-relativistic and ultrarelativistic approximations are also considered. The phase space volume of the system is calculated explicitly in the ultrarelativistic approximation.

Gas-Kinetic Scheme for Continuum and Near-Continuum Hypersonic Flows

We present a unified approach for both continuum and near-continuum flows based on the Boltzmann equation and kinetic theory. We use the gas-kinetic scheme (GKS) developed from the linearized Boltzmann equation for the continuum flows and a modified GKS with a variable relaxation time for the near-continuum flows. The gas-kinetic schemes are validated with simulations of the hypersonic flow past a hollow flare at Mach number 9.91 and the shock structures of Argon gas at Mach 8.0 and 9.0. Our results show that the gas-kinetic schemes can indeed simulate both the continuum and near-continuum flows.

Navier–Stokes Transport Coefficients of d-Dimensional Granular Binary Mixtures at Low Density

The Navier-Stokes transport coefficients for binary mixtures of smooth inelastic hard disks or spheres under gravity are determined from the Boltzmann kinetic theory by application of the Chapman-Enskog method for states near the local homogeneous cooling state. It is shown that the Navier-Stokes transport coefficients are not affected by the presence of gravity. As in the elastic case, the transport coefficients of the mixture verify a set of coupled linear integral equations that are approximately solved by using the leading terms in a Sonine polynomial expansion. The results reported here extend previous calculations [V. Garz\'o and J. W. Dufty, Phys. Fluids {\bf 14}, 1476 (2002)] to an arbitrary number of dimensions. To check the accuracy of the Chapman-Enskog results, the inelastic Boltzmann equation is also numerically solved by means of the direct simulation Monte Carlo method to evaluate the diffusion and shear viscosity coefficients for hard disks. The comparison shows a good agreement over a wide range of values of the coefficients of restitution and the parameters of the mixture (masses and sizes).

Thermoelectric Properties of Na x CoO2 and Prospects for Other Oxide Thermoelectrics

We discuss the thermoelectric properties of Na x CoO2 using the electronic structure, as determined in first principles calculations, and Boltzmann kinetic transport theory. The Fermi energy lies near the top of a manifold of Co t 2g bands. These t 2g bands are separated by a large gap from the higher-lying e g states. Although the large crystal-field splitting implies substantial Co–O hybridization, the bands are narrow. Application of standard Boltzmann transport theory to such a narrow band structure yields high thermopowers in accord with experimental observations, even for high metallic carrier densities. The high thermopowers observed for Na x CoO2 can therefore be explained by standard band theory and do not rely on low dimensionality or correlation effects specific to Co. We also present results for the cubic spinel structure ZnRh2O4. Like Na x CoO2, this compound has very narrow valence bands. We find that if it could be doped with mobile carriers, it would also have a high thermopower, comparable with that of Na x CoO2.

Molecular dynamics simulations of ballistic annihilation

Using event-driven molecular dynamics we study one- and two-dimensional ballistic annihilation. We estimate exponents xi and gamma, which describe the long-time decay of the number of particles [n(t) approximately t-xi] and of their typical velocity [v(t) approximately t-gamma]. To a good accuracy our results confirm the scaling relation xi+gamma=1. In the two-dimensional case our results are in good agreement with those obtained from Boltzmann kinetic theory.

Two generalizations of the Boltzmann equation

We connect two different generalizations of Boltzmann's kinetic theory by requiring the same stationary solution. Non-extensive statistics can be produced by either using corresponding collision rates nonlinear in the one-particle densities or equivalently by using nontrivial energy composition rules in the energy conservation constraint. Direct transformation formulas between key functions of the two approaches are given.

Approach to a non-equilibrium steady state

We consider a non-interacting one-dimensional gas accelerated by a constant and uniform external field. The energy absorbed from the field is transferred via elastic collisions to a bath of scattering obstacles. At gas–obstacle encounters the particles of the gas acquire a fixed kinetic energy. The approach to the resulting stationary state is studied within the Boltzmann kinetic theory. It is shown that the long time behavior is governed by the hydrodynamic mode of diffusion superposed on a convective flow. The diffusion coefficient is analytically calculated for any value of the field showing a minimum at intermediate field intensities. It is checked that the properly generalized Green–Kubo formula applies in the non-equilibrium stationary state.

Consistent Lattice Boltzmann Method

Lack of energy conservation in lattice Boltzmann models leads to unrealistically high values of the bulk viscosity. For this reason, the lattice Boltzmann method remains a computational tool rather than a model of a fluid. A novel lattice Boltzmann model with energy conservation is derived from Boltzmann's kinetic theory. Simulations demonstrate that the new lattice Boltzmann model is the valid approximation of the Boltzmann equation for weakly compressible flows and microflows.

Colloquium: Physically based fluid modeling of collisionally dominated low-temperature plasmas

This colloquium examines the theoretical modeling of nonequilibrium low-temperature (tens of thousands of degrees) plasmas, which involves a juxtaposition of three distinct fields: atomic and molecular physics, for the input of scattering cross sections; statistical mechanics, for the kinetic modeling; and electromagnetic theory, for the simultaneous solution of Maxwell's equations. Cross sections come either from single-scattering beam experiments or, at very low energies (<0.5 eV), from multiple-scattering experiments on swarms in gases—the free diffusion or large Debye length limit of a plasma, where they are embedded in transport coefficient data. The same Boltzmann kinetic theory that has been developed to a high level of sophistication over the past 50 years, specifically for the purpose of unfolding these transport data, can be employed for low-temperature plasmas with appropriate modification to allow for self-consistent rather than externally prescribed fields. A full kinetic treatment of low-temperature plasmas is, however, a daunting task and remains at the developmental level. Fortunately, since the accuracy requirements for modeling plasmas are generally much less stringent than for swarms, such a sophisticated phase-space treatment is not always necessary or desirable, and a computationally more efficient but correspondingly less accurate macroscopic theoretical model in configuration space at the fluid level is often considered sufficient. There has been a proliferation of such fluid modeling in recent times and this approach is now routinely used in the design and development of a large variety of plasma technologies, ranging from plasma display panels to plasma etching reactors for microelectronic device fabrication. However, many of these models have been developed empirically with specific applications in mind, and rigor and sophistication vary accordingly. In this colloquium, starting from the governing Boltzmann kinetic equation, a unified, general formulation of fluid equations is given for both ions and electrons in gaseous media with transparent and internally consistent approximations, all benchmarked against established results. Thereby a fluid model is obtained that is appropriate for practical application but at the same time is based on a firmer physical foundation.

Expanded analogy between Boltzmann kinetic theory of fluids and turbulence

We demonstrate that the effects of turbulent fluctuations have a striking resemblance to those of microscale (thermal) fluctuations in laminar flows, even to higher order in the Knudsen number. This suggests that there may be a good basis for understanding turbulence in terms of Boltzmann kinetic theory. If so, turbulence may be better described in terms of ‘mixing times’ rather than the more classical ‘mixing lengths’. Comparisons are made to Reynolds-stress turbulence models.

Numerical modeling of geophysical granular flows: 1. A comprehensive approach to granular rheologies and geophysical multiphase flows

Geophysical granular materials display a wide variety of behaviors and features. Typically, granular flows (1) are multiphase flows, (2) are very dissipative over many different scales, (3) display a wide range of grain concentrations, and (4), as a final result of these previous features, display complex nonlinear, nonuniform, and unsteady rheologies. Therefore the objectives of this manuscript are twofold: (1) setting up a hydrodynamic model which acknowledges the multiphase nature of granular flows and (2) defining a comprehensive rheological model which accounts for all the different forms of viscous dissipations within granular flows at any concentration. Hence three important regimes within granular flows must be acknowledged: kinetic (pure free flights of grain), kinetic‐collisional, and frictional. The momentum and energy transfer will be different according to the granular regimes, i.e., strain rate dependent in the kinetic and kinetic‐collisional cases and strain rate independent in the frictional case. A “universal” granular rheological model requires a comprehensive unified stress tensor able to adequately describe viscous stress within the flow for any of these regimes, and without imposing a priori what regime will dominate over the others. The kinetic‐collisional viscous regime is defined from a modified Boltzmann's kinetic theory of dense gas. The frictional viscous regime is defined from the plastic potential and the critical state theories which account for compressibility of granular matter (e.g., dilatancy, consolidation, and critical state). In the companion paper [Dartevelle et al., 2004] we will introduce a multiphase computer code, (G)MFIX, which accounts for all the granular regimes and rheology and present typical simulations of diluted (e.g., plinian clouds) and concentrated geophysical granular flows (i.e., pyroclastic flows and surges).

Effect of energy nonequipartition on the transport properties in a granular mixture

The Boltzmann kinetic theory is used to analyze the effect of energy nonequipartition on the pressure and the shear viscosity of a granular binary mixture under simple shear flow. Theory and Monte Carlo simulations show that both quantities exhibit a non−monotonic behaviour with the mass ratio in contrast to the predictions made from previous theories based on the equipartition assumption. Our results agree qualitatively well with recent molecular dynamics simulations performed by Alam and Luding [Granular Matter 4, 139 (2002)].

Energy Nonequipartition in a Sheared Granular Mixture

The kinetic granular temperatures of a binary granular mixture in simple shear flow are calculated by means of the Direct Simulation Monte Carlo method. The results show that the temperature ratio is clearly different from unity (as may be expected since the system is out of equilibrium) and strongly depends on the restitution coefficients as well as on the parameters of the mixture. The influence of the temperature differences on the rheological properties is also discussed. The results are compared with the theoretical predictions obtained from the Boltzmann kinetic theory by using a Sonine polynomial expansion. The comparison shows an excellent agreement over the range of parameters investigated.

Shear viscosity for a heated granular binary mixture at low density.

The shear viscosity for a heated granular binary mixture of smooth hard spheres at low density is analyzed. The mixture is heated by the action of an external driving force (Gaussian thermostat) that exactly compensates for cooling effects associated with the dissipation of collisions. The study is made from the Boltzmann kinetic theory, which is solved by using two complementary approaches. First, a normal solution of the Boltzmann equation via the Chapman-Enskog method is obtained up to first order in the spatial gradients. The mass, heat, and momentum fluxes are determined and the corresponding transport coefficients identified. As in the free cooling case [V. Garzó and J. W. Dufty, Phys. Fluids 14, 1476 (2002)], practical evaluation requires a Sonine polynomial approximation, and here it is mainly illustrated in the case of the shear viscosity. Second, to check the accuracy of the Chapman-Enskog results, the Boltzmann equation is numerically solved by means of the direct simulation Monte Carlo method. The simulation is performed for a system under uniform shear flow, using the Gaussian thermostat to control inelastic cooling. The comparison shows an excellent agreement between theory and simulation over a wide range of values of the restitution coefficients and the parameters of the mixture (masses, concentrations, and sizes).