Numerical solution of the linearized Boltzmann equation for an arbitrary intermolecular potential

A numerical procedure to solve the linearized Boltzmann equation with an arbitrary intermolecular potential by the discrete velocity method is elaborated. The equation is written in terms of the kernel, which contains the differential cross section and represents a singularity. As an example, the Lennard-Jones potential is used and the corresponding differential cross section is calculated and tabulated. Then, the kernel is calculated so that to overcome its singularity. Once, the kernel is known and stored it can be used for many kinds of gas flows. In order to test the method, the transport coefficients, i.e. thermal conductivity and viscosity for all noble gases, are calculated and compared with those obtained by the variational method using the Sonine polynomials expansion. The fine agreement between the results obtained by the two different methods shows the feasibility of application of the proposed technique to calculate rarefied gas flows over the whole range of the Knudsen number.     

Coarse-graining scale and effectiveness of hydrodynamic modeling

We review our work on the application of the renormalization-group method to obtain first- and second-order relativistic hydrodynamics from the relativistic Boltzmann equation (RBE) as a dynamical system, with some corrections and new unpublished results. For the first-order equation, we explicitly obtain the distribution function in the asymptotic regime as the invariant manifold of the dynamical system, which turns out to be nothing but the matching condition defining the energy frame, i.e., the Landau-Lifshitz one. It is argued that the frame on which the flow of the relativistic hydrodynamic equation is defined must be the energy frame, if the dynamics should be consistent with the underlying RBE. A sketch is also given for derivation of the second-order hydrodynamic equation, i.e., extended thermodynamics, which is accomplished by extending the invariant manifold so that it is spanned by excited modes as well as the zero modes (hydrodynamic modes) of the linearized collision operator. On the basis of thus constructed resummed distribution function, we propose a novel ansatz for the functional form to be used in Grad moment method; it is shown that our theory gives the same expressions for the transport coefficients as those given in the Chapman-Enskog theory as well as the novel expressions for the relaxation times and lengths allowing natural interpretation.     

Mathematical aspects of relativistic kinetic theory

The linearized relativistic Boltzmann equation is studied with respect to its position in the theory of integral equations. A sufficient condition is given for the convergence of the series which are usually employed in the calculation of the transport coefficients pertaining to systems described by this equation. As an illustration, the theory is applied to the neutrino gas.     

Nonlinear transport in the Boltzmann limit

Formal expressions for the irreversible fluxes of a simple fluid are obtained as functionals of the thermodynamic forces and local equilibrium time correlation functions. The Boltzmann limit of the correlation functions is shown to yield expressions for the irreversible fluxes equivalent to those obtained from the nonlinear Boltzmann kinetic equation. Specifically, for states near equilibrium, the fluxes may be formally expanded in powers of the thermodynamic gradients and the associated transport coefficients identified as integrals of time correlation functions. It is proved explicitly through nonlinear Burnett order that the time correlation function expressions for these transport coefficients agree with those of the Chapman-Enskog expansion of the nonlinear Boltzmann equation. For states far from equilibrium the local equilibrium time correlation functions are determined in the Boltzmann limit and a similar equivalence to the Boltzmann equation solution is established. Other formal representations of the fluxes are indicated; in particular, a projection operator form and its Boltzmann limit are discussed. As an example, the nonequilibrium correlation functions for steady shear flow are calculated exactly in the Boltzmann limit for Maxwell molecules.Research supported in part by NSF grant PHY 76-21453.     

Relativistic Boltzmann theory for a plasma: VI. The relativistic Landau equation

The relativistic Landau equation is obtained as the Fokker-Planck approximation to the relativistic Boltzmann equation. It is shown that, as far as the transport coefficients are concerned, both equations yield identical results in the dominant term approximation. Special attention is given to Møller-, Bhabha- and Mott-scattering processes. In the case of relativistic electron-electron collisions a detailed determination of the Coulomb logarithm is given.     

On the linearized relativistic Boltzmann equation

The linearized relativistic Boltzmann equation inL ²(r,p) is investigated. The detailed analysis of the collision operatorL is carried out for a wide class of scattering cross sections.L is proved to have a form of the multiplication operatorv(p) plus the compact inL ²(p) perturbationK. The collisional frequencyv(p) is analysed to discriminate between relativistic soft and hard interactions. Finally, the existence and uniqueness of the solution to the linearized relativistic Boltzmann equation is proved.     

On the linearized relativistic Boltzmann equation. II. Existence of hydrodynamics

Solutions are analyzed of the linearized relativistic Boltzmann equation for initial data fromL ²(r, p) in long-time and/or small-mean-free-path limits. In both limits solutions of this equation converge to approximate ones constructed with solutions of the set of differential equations called the equations of relativistic hydrodynamics.     

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).     

Single-parameter expression for ions reflection from surfaces

Based on a detailed analysis of the Boltzmann equation for ion transport in solids, it has been shown that for low-energy ions incident on a heavy element target the distribution function of the ions can be determined by one single parameter, called the scaled transport cross-section, which was defined earlier [1]. This means that the transport quantities of different ion-target-energy combinations should be similar only when their scaled transport cross-section is the same. To test this significant conclusion, we undertook a set of systematic and extensive calculations of reflection coefficients using the improved bipartition model of ion transport. The systematic calculations include 3410 ion-target-energy combinations, namely H, D, T, He, Li, B, C, N, O, Ne ions with energy ranges from 10 eV to 1 MeV normally incident to C, Al, Cu, Mo, Ag, W, Au, Pb, U targets. The only restrictions isM ₁/M ₂<1/6. The calculations verify that particle and energy reflection coefficients present an excellent one-to-one correspondence to the scaled transport cross-section. Furthermore, based on the calculations, universal expressions for both particle reflection coefficients and energy reflection coefficients for normal ion incidence have been obtained by fitting the numerical data. By comparing the results calculated by the universal expressions with experimental and Monte Carlo data, it is shown that the expression can describe reflection coefficients well.     

Solution of a linearized kinetic model for an ultrarelativistic gas

A linearized model of the Boltzmann equation for a relativistic gas is shown to be reducible, in the ultrarelativistic limit and for (1 + 1)-dimensional problems, to a system of three uncoupled transport equations, one of which is well known. A general method for solving these equations is recalled, with a few new details, and applied to the solution of two boundary value problems. The first of these describes the propagation of an impulsive change in a half space and is shown to give an explicit example of the recently proved result that no signal can propagate with speed larger than the speed of light, according to the relativistic Boltzmann equation. The second problem deals with steady oscillations in a half space and illustrates the meaning of certain recent results concerning the dispersion relation for linear waves in relativistic gas.     

Kinetic perturbation theory for dilute gases

We prove, for two classes of smooth, repulsive interparticle potentials ø(r) = ø₀(r) + δø₁(r), that the collision integ rals of the linearized Boltzmann equation are analytic functions of λ in the neighborhood of λ = 0. It then follows, for example, that the first Enskog approximation for the transport coefficients can be represented by a power series in λ.     

An efficient method for computing genus expansions and counting numbers in the Hermitian matrix model

We present a method to compute the genus expansion of the free energy of Hermitian matrix models from the large N expansion of the recurrence coefficients of the associated family of orthogonal polynomials. The method is based on the Bleher–Its deformation of the model, on its associated integral representation of the free energy, and on a method for solving the string equation which uses the resolvent of the Lax operator of the underlying Toda hierarchy. As a byproduct we obtain an efficient algorithm to compute generating functions for the enumeration of labeled k-maps which does not require the explicit expressions of the coefficients of the topological expansion. Finally we discuss the regularization of singular one-cut models within this approach.     

Anomaly in the nuclear curvature energy

By use of semiclassical mean-field methods, we study the dependence of the curvature-energy coefficient and other surface properties of nuclear matter upon the energy-density functional. This is done both by solving the Euler-Lagrange equation with a simplified phenomenological functional and by obtaining a variational solution with a fourth-order extended Thomas-Fermi functional. The calculated curvature-energy coefficient a_c decreases with increasing value of the bulk nuclear incompressibility coefficient K for physically relevant values of K, but always remains larger than 3 MeV in either approach when the volume-energy coefficient, saturation density, surface-energy coefficient and surface diffuseness are constrained to their experimental values. The calculated values of the curvature-energy coefficient a_c are significantly larger than experimental values obtained from analyses of fission-barrier heights and ground-state masses of nuclei throughout the periodic table. Among possible resolutions of this anomaly, we suggest that relativistic effects or correlations may play a significant role in the nuclear surface, or that the leptodermous expansion may break down in regions of large curvature, such as occurs for highly deformed shapes and for light nuclei.     

无限长黑圆柱情形下密恩(Milne)问题的近似解(球谐函数展开法)

本文探讨当一无限长黑圆柱放在一满足密恩问题中诸条件的无限介质中时,介质中的中子分布。计算采用球谐函数展开法,把中子分布函数对球谐函数展开,保留展开式的起首若干项,从而求得近似解。具体计算作到P₅近似为止。表2及附图示各次近似中对於圆柱半径α的不同值求出的外推长度λ之值。作为长度单位的是中子在介质中的平均自由路程l。为比较起见,我们在图中也画出了达维逊(Davison)给出的曲线(曲线D)。他的曲线是根据α《1及α》1二极限情形下派耳斯(Peierls)积分方程的近似解,中间参照P₃近似的结果画出的。由图可见,α大时P₅近似的结果已很接近於曲线D,而在α=1附近,则曲线D似乎远应该略低一些,才更符合曲线P₅的趋势(例如,像图中虚线所表示的那样)。     

Dissipative relativistic fluid dynamics: a new way to derive the equations of motion from kinetic theory

We rederive the equations of motion of dissipative relativistic fluid dynamics from kinetic theory. In contrast with the derivation of Israel and Stewart, which considered the second moment of the Boltzmann equation to obtain equations of motion for the dissipative currents, we directly use the latter's definition. Although the equations of motion obtained via the two approaches are formally identical, the coefficients are different. We show that, for the one-dimensional scaling expansion, our method is in better agreement with the solution obtained from the Boltzmann equation.     

Transport coefficients of quark-gluon plasma

Transport coefficients of quark-gluon plasma are discussed in the framework of relativistic kinetic theory with the relaxation time approximation of Boltzmann transport equation. The expressions for the coefficients of shear and volume viscosities and heat conductivity are derived assuming quark-gluon plasma to be a non-reactive mixture of quarks, anti-quarks and gluons. A lowest order in deviations from local thermal equilibrium and in plasma phase, lowest order in coupling constant are assumed. Entropy production due to irreversible processes is discussed.     

Velocity of sound in curved space-time

The small-perturbation method of E. M. Lifshits is used to derive expressions for the phase and group velocities of sound in matter described by the equation of state ε=3p in the curved 4-space of Fridman's cosmological model. It is inferred that the curvature of 4-space significantly influences the velocity of sound. Thus, the maximum value of the group velocity of sound in the given model exceeds the maximum value of the velocity of sound in flat 4-space for the same equation of state of matter; for matter of infinite density the velocity of sound is equal to zero at the initial instant of expansion of the universe. These results indicate that when the condition v_so ≤c is taken as the criterion for the choice of equation of state of superdense matter, as for example in relativistic astrophysics, it is necessary to take into account the influence of the curvature of 4-space on the velocity of sound.     

Large energy behavior of the velocity distribution for the hard-sphere gas

The initial-value problem for the Boltzmann-Lorentz equation for hard spheres at zero temperature is shown to be ill defined, the general solution depending on an arbitrary function. The uniqueness of the solution can be obtained by imposing the conservation of the number of particles (Carleman's type of condition does not suffice). The linearized Boltzmann equation for hard spheres is then analyzed, as it occurs in Enskog's method for calculating transport coefficients. It is demonstrated that in the case of viscosity and diffusion it is necessary to add supplementary conditions to obtain the uniqueness of the solution. The nonuniform character of Enskog's expansion and violation of positivity in the large velocity region are exhibited.