Kinetic Models for Granular Flow

The generalization of the Boltzmann and Enskog kinetic equations to allow inelastic collisions provides a basis for studies of granular media at a fundamental level. For elastic collisions the significant technical challenges presented in solving these equations have been circumvented by the use of corresponding model kinetic equations. The objective here is to discuss the formulation of model kinetic equations for the case of inelastic collisions. To illustrate the qualitative changes resulting from inelastic collisions the dynamics of a heavy particle in a gas of much lighter particles is considered first. The Boltzmann–Lorentz equation is reduced to a Fokker–Planck equation and its exact solution is obtained. Qualitative differences from the elastic case arise primarily from the cooling of the surrounding gas. The excitations, or physical spectrum, are no longer determined simply from the Fokker–Planck operator, but rather from a related operator incorporating the cooling effects. Nevertheless, it is shown that a diffusion mode dominates for long times just as in the elastic case. From the spectral analysis of the Fokker–Planck equation an associated kinetic model is obtained. In appropriate dimensionless variables it has the same form as the BGK kinetic model for elastic collisions, known to be an accurate representation of the Fokker–Planck equation. On the basis of these considerations, a kinetic model for the Boltzmann equation is derived. The exact solution for states near the homogeneous cooling state is obtained and the transport properties are discussed, including the relaxation toward hydrodynamics. As a second application of this model, it is shown that the exact solution for uniform shear flow arbitrarily far from equilibrium can be obtained from the corresponding known solution for elastic collisions. Finally, the kinetic model for the dense fluid Enskog equation is described.     

Resonant generation of plasma oscillations by a plane gravitational wave

An exact solution of the universal Maxwell equations linearized in the amplitude of the induced field is found with the help of a previously found exact solution of the kinetic equation with a model collision integral. Two oscillation modes arise under the action of a gravitational wave in a plasmalike medium: an undamped wave with the frequency of the gravitional wave and a damped one with the plasma frequency. When these frequencies coincide, a resonance arises, as a result of which the amplitude of the electric oscillations increases sharply.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 74–77, January, 1985.     

Existence theory of the linear equations appearing in the Chapman-Enskog solutions to two kinetic equations for liquids

The linear operators appearing in the Chapman-Enskog solutions to Kirkwood's Fokker-Planck kinetic equation and to Rice and Allnatt's kinetic equation are studied in this article. Existence proofs are given for the linearized Chapman-Enskog equations involving either the Fokker-Planck or the Rice-Allnatt operators. It is shown that the Fokker-Planck and Rice-Allnatt operators, defined in the domain appropriate to kinetic theory, are essentially self-adjoint. It is also shown that the spectrum of either of these operators coincides with the spectrum of the self-adjoint extension of the corresponding operator.Sloan Foundation Fellow 1968–70. Guggenheim Fellow 1969–70.     

Relativistic kinetics of an anisotropic plasmalike medium with damping in a field of gravitational radiation

An exact solution is obtained for the general relativistic kinetic equation with model collision integral that describes the motion of an initially homogeneous anisotropic plasma in a field of plane gravitational waves (GW) of arbitrary amplitude and polarization. The amplitude of the induced current is calculated.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 70–74, December, 1984.     

Kinetic theory of the generation of magnetic fields in the radiation-dominated stage of expansion of the universe

The system of linearized Einstein and Maxwell equations and a kinetic equation with model collision integral for the cosmological plasma are used to calculate the magnetic field generated by solenoidal perturbations in the radiation-dominated stage of expansion of the universe. The magnetic field is generated by two effects — the Harrison effect and a new effect due to kinetic processes. The second effect contributes to the magnetic field if solenoidal gravitational perturbations exist from the very beginning of the radiation-dominated stage.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 122–126, January, 1992.     

Coordinate-dependent 3+1 formulation of the general relativity equations for a perfect fluid

This paper defines mass, momentum, and energy densities for a perfect fluid, and derives a coordinate-dependent 3+1 decomposition of the equation of motion in terms of a scalar potential c ² [(–g _g44) 1/2 –1] and a vector potentialA _i cg _4i /(–g ₄₄)^1/2. The momentum equation has the form of the Euler equation except there is an additional force proportional to the vector potential and the rate of change of kinetic energy per unit volume. The momentum and energy equations are integrated to obtain the equations previously derived for a particle. The momentum equation is solved for the total acceleration of a fluid element. The equations are exact and do not depend on the choice of coordinate system.     

Propagation of radiation in a plasma situated in a gravitational field

Weak electromagnetic and gravitational fields in a plasma situated in a strong gravitational field, are studied using linearized, general-relativistic, kinetic equations. A tensor operator is constructed for the electrical conductivity of a plasma in a gravitational field, which is a general-relativistic generalization of the electrical conductivity of a homogeneous plasma. Similar tensor operators, which allow one to determine the energy-momentum tensor and the vector current, induced by electromagnetic and gravitational fields in a plasma, are also obtained.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 57–62, September, 1976.     

Hopf equation in random force fields of arbitrary distribution

An equation is obtained in variational derivatives over the external fields for the space-time functional of the velocity field with an arbitrary distribution of the random fields. Equations are obtained for the moments and response functions. The exact operator solution of these equations is obtained.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 77–82, February, 1977.The author would like to express his thanks to L. Ya. Kobelev for helpful discussions.     

Exact solutions of the equations of motions of quarks in a non-Abelian field of a flat color wave

Exact solutions of the equations of motion of quarks in a non-Abelian field of a flat color wave of a spectral configuration are obtained. The gauge field of the wave takes values from the SU(2) group and is an exact solution of the Yang-Mills equations.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 76–78, December, 1986.     

Electronic properties of a large quantum dot at a finite temperature

The physical properties of a two-dimensional parabolic quantum dot composed of large number of interacting electrons are numerically determined by the Thomas–Fermi (TF) method at a finite temperature. Analytical solutions are given for zero temperature for comparative purposes. The exact solution of the TF equation is obtained for the non-interacting system at finite temperatures. The effect of the number of particles and temperature on the properties are investigated both for interacting and non-interacting cases. The results indicate that the effect of e–e interaction on the density profile shows different temperature dependencies above and below a certain temperature T_c.     

Permittivity of plasma and nonstationary theory of nonlocal transport

A regular procedure is proposed for finding the solution to a linearized kinetic equation for charged particles with the Landau collision integral in a plasma with large Z. The expression for longitudinal permittivity of a collisional plasma, which is obtained using this procedure for the entire range of frequencies and wavenumbers, as well as the collision parameter, is transformed to the known expressions in the corresponding asymptotic limits. The nonlocal transport equations for small perturbations are also formulated for arbitrary relations between the characteristic space and time scales of the plasma; these relations considerably extend the limits of applicability for previously developed theories.     

Asymptotic relaxation of a system of a large number of interacting particles

A model of an open relaxing system of scalar bosons with one state and random interaction, described by a system of linking equations for the moments, is considered in a Markov approximation. It is shown that in the self-consistent field limit the equations for the first moments are equations of the characteristics with respect to the equation for the generating functions of the system. They can be understood as the scalar analog of a quasilinear Schrödinger equation with a purely imaginary Hamiltonian describing non-Hamiltonian dynamics of an open system of randomly interacting quantum particles. By virtue of the assumption made about the nature of the interaction, the equations of the characteristics obtained differ from those presented earlier, where the means are solutions of linearized relaxation equations.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 89–92, October, 1988.In conclusion, I am grateful to V. P. Belavkin for formulating the problems and useful discussions.     

The moment method for boundary layer problems in Brownian motion theory

We apply Grad's moment method, with Hermite moments and Marshak-type boundary conditions, to several boundary layer problems for the Klein-Kramers equation, the kinetic equation for noninteracting Brownian particles, and study its convergence properties as the number of moments is increased. The errors in various quantities of physical interest decrease asymptotically as inverse powers of this number; the exponent is roughly three times as large as in an earlier variational method, based on an expansion in the exact boundary layer eigenfunctions. For the case of a fully absorbing wall (the Milne problem) we obtain full agreement with the recent exact solution of Marshall and Watson; the relevant slip coefficient, the Milne length, is reproduced with an accuracy better than 10^–6. We also consider partially absorbing walls, with specular or diffuse reflection of nonabsorbed particles. In the latter case we allow for a temperature difference between the wall and the medium in which the particles move. There is noa priori reason why our method should work only for Brownian dynamics; one may hope to extend it to a broad class of linear transport equations. As a first test, we looked at the Milne problem for the BGK equation. In spite of the completely different analytic structure of the boundary layer eigenfunctions, the agreement with the exact solution is almost as good as for the Klein-Kramers equation.     

Constructing Phantom With a Nonminimally Coupled Complex Scalar Field

In view of the recent observation data indicating that the equation of state of the dark energy might be smaller than −1, this leads to introduction of phantom models featured by their negative kinetic energy to account for the regime of equation of state w < −1. In this paper, we discuss the possibility of using a nonminimally coupled complex scalar field as phantom to realize the equation-of-state parameter w < −1. The main equations which govern the evolution of the universe are obtained. The relations between the potential of the field and red-shift, namely, the reconstruction equations are derived. PACS 04.40.-b, 98.80.Cq, 98.80.Hw     

Problems of classical and quantum gravitational collapse

An exact solution of Einstein's equations is obtained in the case of a spherically symmetric distribution of a perfect fluid with maximally hard equation of state. A sufficient condition of halting of the collapse of charged dust is obtained. The dynamics and statics of a vector massive configuration with massless source are considered.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 7–12, September, 1977.We are very grateful to N. V. Pavlov and K. A. Bronnikov for valuable discussions and helpful comments.     

Decrease in durability of brittle solids in cyclic tests

In this work a phenomenological model Is proposed for fatigue fracture, based on analysis of basic experimental results on cyclic testing taking into account the basic physical assumptions on which the mechanism for the growth of a fracture crack is based. A number of theoretical relations are obtained that reveal qualitatively and quantitatively the basic concepts concerning the mechanism of fatigue fracture: the magnitude of the local heating at the vertex of the fracture crack, equations for its rate of growth, decrease in the potential barrier, safe stress amplitude, and the kinetic equation for cyclic durability.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 3–8, June 1981.     

An “instantaneous” distribution of the ionization-passive electrons and holes under irradiation of a dielectric by intense electron or laser beams

A method is worked out for calculation of an “instantaneous” energy distribution of the ionization-passive electrons and holes resulting from the electron-electron collisions before the onset of electron-phonon relaxation under 10⁻¹⁵–10⁻¹⁴ s irradiation of a dielectric by an intense electron or laser beam. The method is based on the solution of a system of integral-differential kinetic equations of general form. The Auger and impact ionization as well as hole recoil due to the momentum conservation law are taken into account in calculations. The “instantaneous” distribution is calculated in NaCl under irradiation of the sample by a high-density electron beam. The “instantaneous” distribution of ionization-passive electrons and holes is the initial one in solutions of all kinetic equations describing further relaxation of electron excitations in irradiated materials.__________Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 15–22, November, 2004.     

Traffic Flow Models Considering an Internal Degree of Freedom

We solve numerically the integrodifferential equation for the equilibrium case of Paveri–Fontana's Boltzmann-like traffic equation. Beside space and actual velocity, Paveri–Fontana used an additional phase space variable, the desired velocity, to distinguish between the various driver characters. We refine his kinetic equation by introducing a modified cross section in order to incorporate finite vehicle length. We then calculate from the equilibrium solution the mean-velocity–density relation and investigate its dependence on the imposed desired velocity distribution. A further modification is made by modeling the interaction as an imperfect showing-down process. We find that the velocity cumulants of the stationary homogeneous solution essentially only depend on the first two cumulants, but not on the exact shape of the imposed desired velocity distribution. The equilibrium solution can therefore be approximated by a bivariate Gaussian distribution which is in agreement with empirical velocity distributions. From the improved kinetic equation we then derive a macroscopic model by neglecting third and higher order cumulants. The equilibrium solution of the macroscopic model is compared with the cumulants of the kinetic equilibrium solution and shows good agreement, thus justifying the closure assumption.     

Nonequilibrium statistical theory of a cascade of displacements in a solid. Dynamic stage

On the basis of analytical solution of the kinetic equation, a single-particle distribution describing the time development of the low-energy cascade at the dynamic stage of the process is obtained. A generalized cascade function is obtained, and its dependence on the basic parameters of the theory is analyzed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 45–50, June, 1990.     

Nonlinear gravitational waves in plasma

An exact solution is obtained for the relativistic collisionless kinetic equations describing a test plasma in the field of a strong plane gravitational wave. The gravitational wave induces in the plasma a longitudinal electric current whose amplitude is maximum at temperatures Te ip m_ec². The interaction of gravitational waves with a system consisting of Boltzmann ions and degenerate electrons is also considered.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 20–24, July 1981.The authors thank G. G. Ivanov for a number of valuable comments.