Transport Coefficients for Inelastic Maxwell Mixtures

The Boltzmann equation for inelastic Maxwell models (IMM) is used to determine the Navier–Stokes transport coefficients of a granular binary mixture in d-dimensions. The Chapman–Enskog method is applied to solve the Boltzmann equation for states near the (local) homogeneous cooling state. The mass, heat, and momentum fluxes are obtained to first order in the spatial gradients of the hydrodynamic fields, and the corresponding transport coefficients are identified. There are seven relevant transport coefficients: the mutual diffusion, the pressure diffusion, the thermal diffusion, the shear viscosity, the Dufour coefficient, the pressure energy coefficient, and the thermal conductivity. All these coefficients are exactly obtained in terms of the coefficients of restitution and the ratios of mass, concentration, and particle sizes. The results are compared with known transport coefficients of inelastic hard spheres (IHS) obtained analytically in the leading Sonine approximation and by means of Monte Carlo simulations. The comparison shows a reasonably good agreement between both interaction models for not too strong dissipation, especially in the case of the transport coefficients associated with the mass flux     

Navier–Stokes Transport Coefficients of <Emphasis Type="Italic">d</Emphasis>-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 (Garzó, V., Dufty, J.W. in Phys. Fluids 14:1476–1490, 2002) to an arbitrary number of dimensions and provide explicit expressions for the seven Navier–Stokes transport coefficients in terms of the coefficients of restitution and the masses, composition, and sizes of the constituents of the mixture. In addition, to check the accuracy of our theory, 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).     

Losses and nonlinear steady-state particle distribution functions for fully ionized tokamak-plasmas in the collisional transport regimes

Fully ionized L-mode tokamak plasmas in the fully collisional (Pfirsch-Schlüter) and in the low-collisional (banana) nonlinear transport regimes are analyzed. We derive the expressions for particles and heat losses together with the steady-state particle distribution functions in the several collisional transport regimes. The validity of the nonlinear closure equations, previously derived, has been indirectly tested by checking that the obtained particle distribution functions are indeed solutions of the nonlinear, steady-state, Vlasov-Landau gyro-kinetic equations. A quite encouraging result is the fact that, for L-mode tokamak plasmas a dissymmetry appears between the ion and electron transport coefficients: the latter submits to a nonlinear correction, which makes the radial electron coefficients much larger than the former. In particular we show that when the L-mode JET plasma is out of the linear region, the Pfirsch-Schlüter electron transport coefficients are corrected by an amplification factor, which may reach values of order 10². Such a correction is absent for ions. On the contrary, in the banana regime, the ion transport coefficients are increased by a factor 2 and the nonlinear corrections for electrons are negligible. These results are in line with experiments.     

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用密度调制的方法研究了等离子体中粒子输运问题。采用了注入脉冲式补充送气和超声分子束两种不同的密度调制方法。在HL-2A装置常规欧姆放电的情况下,运用有限差分法和Nagashima矩阵技术,求解了粒子平衡方程。计算出了粒子的输运系数(对流速度v和扩散系数D)。研究了粒子输运系数与等离子体线平均密度之间的关系。实验结果表明,在欧姆放电的情况下,等离子体芯部的粒子对流速度方向始终是向内的,并且密度低时,粒子输运系数(粒子扩散系数D和对流速度v)较大;密度高时,粒子输运系数较小。     

Note on the Kaplan–Yorke Dimension and Linear Transport Coefficients

A number of new relations between the Kaplan–Yorke dimension, phase space contraction, transport coefficients and the maximal Lyapunov exponents are given for dissipative thermostatted systems, subject to a small but non-zero external field in a nonequilibrium stationary state. A condition for the extensivity of phase space dimension reduction is given. A new expression for the linear transport coefficients in terms of the Kaplan–Yorke dimension is derived. Alternatively, the Kaplan–Yorke dimension for a dissipative macroscopic system can be expressed in terms of the linear transport coefficients of the system. The agreement with computer simulations for an atomic fluid at small shear rates is very good.     

Chapman–Enskog solutions to arbitrary order in Sonine polynomials I: Simple,rigid-sphere gas

The Chapman–Enskog solutions of the Boltzmann equations provide a basis for the computation of important transport coefficients for both simple gases and gas mixtures. These coefficients include the viscosities, the thermal conductivities, and diffusion coefficients. The Chapman–Enskog solutions are also useful for computation of the associated slip and jump coefficients near surfaces. Generally, these solutions are expressed in terms of Sonine polynomial expansions. While it has been found that relatively, low-order expansions (of order 4) can provide reasonable precision in the computation of the transport coefficients (to about 1 part in 1000), the adequacy of the low-order expansions for computation of the slip and jump coefficients still needs to be explored. Also of importance is the fact that such low-order expansions do not provide good convergence (in velocity space) for the actual Chapman–Enskog solutions even though the transport coefficients derived from these solutions appear to be reasonable. Thus, it is of some interest to explore Sonine polynomial expansions to higher orders. It is our purpose in this paper to report the results of our investigation of high-order, standard, Sonine polynomial expansions for the viscosity and the thermal conductivity related Chapman–Enskog solutions for a simple, rigid-sphere gas where we have carried out our calculations using expansions to order 150 and where our reported values for the transport coefficients have been demonstrated to converge to at least 25 significant digits. We note that, for a rigid-sphere gas, all of the relevant integrals needed for these solutions are evaluated analytically as pure fractions and, thus, results to any desired precision may be obtained. This work also indicates how results may be obtained in a similar fashion for realistic intermolecular potential models, and how gas-mixture problems may also be addressed with some additional effort.     

Electronic transport of Lorentz plasma with collision and magnetic field effects

The electronic transverse transport of Lorentz plasma with collision and magnetic field effects is studied by solving the Boltzmann equation for different electron density distributions. For the Maxwellian distribution, it is shown that transport coefficients decrease as ? increases, ? is the ratio of an electron's magneto-cyclotron frequency to plasma collision frequency. It means that the electrons are possible to be highly collimated by a strong magnetic field. For the quasimonoenergetic distribution with different widths, it is found that the transport coefficients decrease greatly as εˉ decreases.In particular when the width approaches to zero the transverse transport coefficients are hardly affected by the magnetic field and the minimal one is obtained. Results imply that the strong magnetic field and quasi-monoenergetic distribution are both beneficial to reduce the electronic transverse transport. This study is also helpful to understand the relevant problems of plasma transport in the background of the inertial confinement fusion.     

Experimental determination of the transport coefficients of bismuth in the range 1·5–50 K

A method has been developed for the simultaneous measurement of several different transport coefficients on a sample. Experiments were performed on a Bi single crystal at temperatures 1·5–50 K. Six transport properties were measured, including the electrical resistivity, thermal conductivity and thermopower. A set of quantities related to transport integrals of the phenomenological transport theory has been determined from experimental data. All the possible transport coefficients can be calculated from them. The possibilities of the determination from experiments of the full tensorial transport integrals for an anisotropic material like Bi are discussed.     

流注放电低温等离子体中电子输运系数计算的蒙特卡罗模型

流体或者粒子-流体混合数值仿真是研究流注放电基本物理机制的常用手段,而精确的电子输运系数是保证其仿真正确性的必要前提.鉴于现有电子输运系数求解工具存在一定缺陷,本文开发了采用蒙特卡罗方法求解低温等离子体中电子输运系数的仿真工具,测试表明其准确性和精确度均较高.研究了氮氧气体混合比及大气压下三体碰撞吸附对电子输运系数的影响.氮气中流注放电仿真表明,流体仿真中采用本模型改进后的电子输运系数可显著改善流注通道内部的等离子体参数分布.     

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用高密度等离子体模型可以计算出一整套输运参数,并且在很宽的等离子体温度和密度范围内有合理的精度,可广泛应用于Z箍缩等离子体、激光聚变和磁约束聚变等领域,并将这个模型计算出的各种输运参数拟合成了实用的公式。     

Transport properties of high temperature air in local thermodynamic equilibrium

In the paper new calculated transport coefficients of air in the temperature range 50-100 000 K are presented. The results have been obtained by means of the perturbative Chapman-Enskog method, assuming that the plasma is in local thermodynamic equilibrium (LTE). The calculations include viscosity, thermal conductivity, electric conductivity and multicomponent diffusion coefficients. For the calculation, a recent compilation of collision integrals obtained by Capitelli et al. [1] has been utilized. Analytical expression for all transport coefficients and thermodynamic parameters of the air plasma are also reported. Received 17 November 1999     

Self-consistent transport equations for the electron-impurity system in a magnetic field

General transport equations for a system of independent electrons in a random array of impurities of arbitrary range and in a magnetic field of arbitrary strength are derived on the basis of conventional perturbation theory, and the general structure of the transverse transport coefficients is discussed. Contrary to kinetic theory, within this framework it is easy to formulate “conserving” approximations, which include collision-broadening effects self-consistently. It is demonstrated, how for high magnetic field explicit results for the transverse transport coefficients can be obtained, which are free of the divergencies appearing in the kinetic theory.

     

Kubo formulas for relativistic fluids in strong magnetic fields

Magnetohydrodynamics of strongly magnetized relativistic fluids is derived in the ideal and dissipative cases, taking into account the breaking of spatial symmetries by a quantizing magnetic field. A complete set of transport coefficients, consistent with the Curie and Onsager principles, is derived for thermal conduction, as well as shear and bulk viscosities. It is shown that in the most general case the dissipative function contains five shear viscosities, two bulk viscosities, and three thermal conductivity coefficients. We use Zubarev’s non-equilibrium statistical operator method to relate these transport coefficients to correlation functions of the equilibrium theory. The desired relations emerge at linear order in the expansion of the non-equilibrium statistical operator with respect to the gradients of relevant statistical parameters (temperature, chemical potential, and velocity.) The transport coefficients are cast in a form that can be conveniently computed using equilibrium (imaginary-time) infrared Green’s functions defined with respect to the equilibrium statistical operator.     

Transport phenomena in a partially lonized Lorentzian magnetoplasma

Some transport coefficients are evaluated for a homogeneous, partially ionized Lorentzian plasma in the presence of a uniform external magnetic field. The electronion collisions are taken into account by means of the modified Fokker-Planck equation and the electron-neutral collision frequency, ν_en is taken as velocity (ω) dependent or velocity independent, depending on the energy range under consideration. The variation of the transport coefficients with magnetic field is determined for ν_en ∝ ω^s (s being a positive or negative integer) and one finds that qualitatively this behavior does not change by changing either the collision frequency or the velocity dependence of the collision frequency; however for weak magnetic fields the magnitudes of these transport coefficients increase with the decrease in ν_en or s, whereas for strong magnetic fields the transverse components of the transport coefficients decrease and the Hall components tend to saturate with the decrease of either the collision frequency or s.     

双温度氦等离子体输运性质计算

获得覆盖较宽温度和压力范围内的等离子体输运性质是进行等离子体传热和流动过程数值模拟的必要条件.本文采用Saha方程计算等离子体组分, 采用基于将Chapman-Enskog方法扩展到高阶近似的方法, 计算获得了电子温度(T_e)不等于重粒子温度(T_h)的情形下, 在300 K到40000 K的温度范围内氦等离子体的黏性、热导率和电导率. 研究结果表明压力和热力学非平衡参数(θ =T_e/T_h)对氦等离子体的输运性质有较大的影响. 在局域热力学平衡条件下,计算获得的氦等离子体输运性质和文献报道的数据符合良好.     

Transport Coefficients in Some Stochastic Models of the Revised Enskog Equation

A stochastic model of the revised Enskog equation is considered. A choice of the smearing function suggested by the work of Leegwater is used to apply the model to the repulsive part of the Lennard-Jones potential and the inverse-power soft-sphere potential. The virial coefficients obtained from the equilibrium properties of the models are in excellent agreement with the known exact coefficients for these models. The transport coefficients for the repulsive Lennard-Jones (RLP) model are also computed and appear to be of comparable accuracy to the Enskog-theory coefficients applied directly to a hard-sphere system, although exact results for the RLP with which to make an extensive comparison are not yet available. The pressure and the transport coefficients obtained from the model (shear viscosity, thermal conductivity, and self-diffusion) are compared with the pressure and the corresponding transport coefficients predicted by the Enskog and square-well kinetic theories.     

Thermoelectric transport coefficients of n-doped CaTiO3, SrTiO3 and BaTiO3: A theoretical study

Boltzmann transport equations and density functional theory calculations were employed to calculate the thermoelectric transport coefficients of CaTiO₃, SrTiO₃ and BaTiO₃. It was found that BaTiO₃ has the largest Seebeck coefficient and power factor. Then the transport coefficients were analyzed using the ‘Tight Binding Model’. The band narrowing, caused by the increasing lattice constants from CaTiO₃ to BaTiO₃, was the main reason for the increasing Seebeck coefficients and the decreasing electrical conductivity. The calculated electrical conductivity and electronic thermal conductivity were in line with the Wiedemann-Franz law and the Lorenz factor was determined to be 2.45 for these oxides as degenerate semiconductors. Our theoretical results are helpful for seeking high performance thermoelectric oxides.     

Relativistische kinetische Theorie von Transporterscheinungen

The transport coefficients and the absorption and dispersion of sound waves are calculated for systems ofmassive particles- with arbitrary interaction- and forneutrino gases on the basis of a covariant transport equation.     

Microscopic transport theory of heavy-ion collisions

Drift and diffusion coefficients for the variables of mass fragmentation and excitation energy are studied for deeply inelastic collisions. The transport coefficients are obtained in closed form as functions of the parameters of the interaction matrix elements between nucleonic states and as functions of the binding energy of the intermediate rotating quasimolecular configuration. Drift and diffusion coefficients for excitation and mass transfer are related (dissipation fluctuation theorem). In good approximation these relations take the simple form of Einstein's relation between the mobility and the diffusion coefficient of a particle in a medium. For the total mass numbers 100, 250 and 500 the results are discussed in detail. The transport coefficients are compared with experimental results. Within the uncertainties of determination from experimental data, the drift and diffusion coefficients are well described by two previously adjusted parameters of the mean interaction matrix elements. Consequences for the production of superheavy elements in deeply inelastic collisions of U on U or Cf are discussed.     

等离子体中的迁移系数

从等离子体动力学方程出发,采用玻尔兹曼碰撞项,在粒子分布函数偏离麦克斯韦分布甚小的情况下,作三级矩近似得到了等离子体迁移方程组。迁移方程组中忽略时间微分项后,得出了电子-离子等离子体的张量形式迁移系数,以及与磁场方向成平行(∥)、垂直(⊥)和交叉方向的迁移系数。导出的结果和文献[1]采用朗道碰撞项得出的结果进行了比较。结果表明,迁移系数和等离子体参数的依赖关系是一致的,但对磁场B和离子电荷Z_i值取定时,个别迁移系数要差一倍。此外,导出的迁移系数形式比文献[1]的要简单些。