Nonequilibrium statistical operator in the generalized molecular hydrodynamics of fluids

We discuss the important role of the Zubarev nonequilibrium statistical operator method in the generalized molecular hydrodynamics of fluids. Using this method allows developing a consistent approach of generalized collective excitations for simple, ion, polar, magnetic, and some other fluids. We construct a nonequilibrium statistical operator and derive the corresponding transport equations for a system that relaxes and passes into the state of molecular hydrodynamics. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 1, pp. 91–101, January, 2008.     

Stationary solutions of the kinetic Broadwell model

We consider a stationary discrete model of the Boltzmann equation for four velocities (the Broadwell model). We obtain new exact automodel solutions of the model corresponding to an incompressible and a compressible gas. We show that one class of solutions satisfies the problem of gas evaporation and condensation on the boundary of a disk and external space. The system turns out to be strongly nonequilibrium, and continuous medium equations are not applicable to it.     

Statistical hydrodynamics of magnetic fluids. I. the nonequilibrium statistical operator method

The Zubarev nonequilibrium statistical operator is used to describe the generalized hydrodynamic state of a magnetic fluid in an external magnetic field. The magnetic fluid is modeled with “liquid-state” and “magnetic” subsystems described using the classical and quantum statistics methods respectively. Equations of the generalized statistical hydrodynamics for a magnetic fluid in a nonhomogeneous external magnetic field with the Heisenberg spin interaction are derived for “liquid-state” and “magnetic” subsystems characterized by different nonequilibrium temperatures. These equations can be used to describe both the weakly and strongly nonequilibrium states. Some limiting cases are analyzed in which the variables of one of the subsystems can be formally neglected. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 115, No. 1, pp. 132–153, April, 1998.     

Dependence of equilibrium properties of channeled particles on the transverse quasitemperature

We use methods of nonequilibrium thermodynamics to investigate the quasiequilibrium and kinetic characteristics of channeled particles regarded as a separate thermodynamic subsystem. For the channeled particles, we derive the energy—momentum balance equation in the moving coordinate system and show that the solution of the balance equation provides an expression for the main thermodynamic parameter, the transverse quasitemperature of the channeled-particle subsystem. We study the quasiequilibrium angular distribution of particles after their passage through a thin single crystal, the quasiequilibrium distribution over the particle exit angles under backscattering conditions, and also the rate constant for the nonequilibrium (dechanneling) process at large deviations of the system as a whole from the thermodynamic equilibrium. We discuss a measurement method for the particle beam transverse temperature over the peak height of the angular particle distribution found in the framework of a “shoot-through” experiment. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 3, pp. 509–524, March, 2006.     

Equation for the Complex Energy of a Bound Particle in a Laser Radiation Field in the Presence of Strong Constant Electromagnetic Fields

We construct exact solutions of the Schrodinger and Pauli equations for charged particles in the external field of the Redmond generalized configuration. We calculate the Green's functions of scalar and spinning particles in this field. Using them, we calculate the equations for the complex quasienergy of a bound particle (bound by a short-range potential) two different ways. In the example of an external constant electric field, we discuss the applicability domain of the obtained equations and the differences between their solutions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 2, pp. 198–211, November, 2005.     

Master equation of the reduced statistical operator of an atom in a plasma

We use the nonequilibrium Liouville equation to derive the master equation for the reduced statistical operator in a heat bath represented by a many-particle environment. Focusing on the case of a weak system-bath coupling, we consider the Born-Markov approximation of the master equation and compare the result to different approaches. The master equation is elaborated for the special case of an atom as a reduced system in a plasma background. We find that the dynamical structure factor determines the effect of the plasma on the reduced system. We consider the operator equation in the atomic eigenstate and in the phase-space representation, which yields two limiting cases: quantum mechanical behavior similar to the isolated atom for the lower strongly bound levels and a semiclassical one for highly excited Rydberg levels. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 1, pp. 31–62, January, 2008.     

Analytic solution to the problem of the behavior of a collisional plasma in a half-space in an external alternating electric field

An exact solution to the linearized problem of the behavior of a collisional plasma in a half-space in an external alternating electric field is obtained. Mirror boundary conditions are used. The eigenvectors of the corresponding characteristic system are found in the space of generalized functions, and the eigenvalue spectrum is investigated. A theorem on the expansion of the solution of the investigated boundary-value problem with respect to eigenvectors is proved. An expression for calculating a discrete mode is found explicitly.Pedagogical University, Moscow. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 103, No. 2, pp. 299–311, May, 1995.     

Integral Equation Method Estimating the Effect of the Electromagnetic Field on Extended Conductors

A simulation procedure is designed for the electromagnetic field on infinitely extended hollow cylindrical conductors enclosed in a thin dielectric layer and embedded in a conducting ground. The surrounding medium consists of two half-spaces representing the air and the ground. The source is an external transverse alternating linear current. The electromagnetic field is computed by solving two one-dimensional integral equations of the second kind for which an efficient algorithm is available.     

Shear and Flexural Buckling Modes of a Spherical Sandwich Shell in a Centrosymmetric Temperature Field Inhomogeneous across the Thickness

Problems on buckling modes (BMs) are considered for a spherical sandwich shell with thin isotropic external layers and a transversely soft core of arbitrary thickness in a centrosymmetric temperature field inhomogeneous across the shell thickness. For their statement, the two-dimensional equations of the theory of moderate bending of thin Kirchhoff–Love shells are used for the external layers, with regard for their interaction with the core; for the core, maximum simplified geometrically nonlinear equations of thermoelasticity theory, in which a minimum number of nonlinear summands is retained to correctly describe its pure shear BM, are utilized. An exact analytical solution to the problem on initial centrosymmetric deformation of the shell is found, assuming that the temperature increments in the external layers are constant across their thickness. It is shown that the three-dimensional equations for the core, linearized in the neighborhood of the solution, can be integrated along the radial coordinate and reduced to two two-dimensional differential equations, which supplement the six equations that describe the neutral equilibrium of the external layers. It is established that the system of eight differential equations of stability, upon introduction of new unknowns in the form of scalar and vortical potentials, splits into two uncoupled sets of equations. The first of them has two kinds of solutions, by which the pure shear BM is described at an identical value of the parameter of critical temperature. The second system describes a mixed flexural BM, whose realization, at definite combinations of determining parameters of the shell and over wide ranges of their variation, is possible for critical parameters of temperature by orders of magnitude exceeding the similar parameter of shear BM.     

一类单调变分不等式的非精确交替方向法

交替方向法适合于求解大规模问题.该文对于一类变分不等式提出了一种新的交替方向法.在每步迭代计算中,新方法提出了易于计算的子问题,该子问题由强单调的线性变分不等式和良态的非线性方程系统构成.基于子问题的精确求解,该文证明了算法的收敛性.进一步,又提出了一类非精确交替方向法,每步迭代计算只需非精确求解子问题.在一定的非精确条件下,算法的收敛性得以证明.     

Nonequilibrium statistical thermodynamics of positronium atoms moving in solids

We calculate the quasitemperature of positronium atoms moving in a solid in the framework of nonequilibrium statistical thermodynamics. We represent the quasitemperature in the form of the renormalized temperature of the thermostat. The renormalization results from taking the positronium formation rate, the kinetic effects of interaction, and the dynamical effects of collisions into account. We obtain an expression for the thermalization time of positronium atoms.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 1, pp. 112–126, January, 2005.     

Exact sets of test uncharged massive spin 3/2 fields in general relativity

A theorem giving the necessary and sufficient conditions for Penrose's exact sets of spinor fields in curved spacetimes is proved. An algorithm for augmenting a system of fields to an exact set and constructing covariant Taylor expansions for the fields of an exact set is proposed. The general approach is applied to test massive spin 3/2 fields. Two possible forms of exact sets are constructed for them on the basis of modified Dirac-Fierz-Pauli equations. The functional arbitrariness in the solutions of the equations for exact sets is determined. In one of the cases, the obtained exact set can be interpreted as a system of fields of spins 3/2 and 1/2 interacting through the gravitational field.State University, Kiev. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 102, No. 1, pp. 119–133, January, 1995.     

Exact solution to the behavior of the electron plasma in a metal layer in an alternating electric field

Exact expressions are obtained for the first time for the response of the conduction electrons in a metal layer to an external alternating electric field. Case's well-known method of separation of the variables is modified, and the methods of singular integral equations and boundary-value problems of complex analysis are used. The solutions can be used to determined the screened field and also the absorption in a metal layer in a wide range of frequencies.N. K. Krupskaya Moscow Region Pedagogical Institute; Moscow Institute of Chemical Mechanical Engineering. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 2, pp. 179–189, February, 1992.     

Global existence and asymptotic analysis of weak solutions to the equations of Ferrohydrodynamics

In this paper, we consider the three-dimensional equations of Ferrohydrodynamics describing the motion of an incompressible ferrofluid under the action of an external magnetic field. We first consider the system introduced by Rosensweig. We establish the existence of globally defined finite energy weak solutions and also investigate the large-time behavior of such solutions. Then we study the asymptotic behavior of the solutions as some physical parameters in the angular momentum equations tend to zero, we prove that the limit functions will be weak solutions of the equations proposed by Shliomis. This shows rigorously the exact relationship of these two principal systems which describe the motion of ferrofluids.     

Lie group analysis and propagation of weak discontinuity in one-dimensional ideal isentropic magnetogasdynamics

The aim of this paper is to carry out symmetry group analysis to obtain important classes of exact solutions from the given system of nonlinear partial differential equations (PDEs). Lie group analysis is employed to derive some exact solutions of one dimensional unsteady flow of an ideal isentropic, inviscid and perfectly conducting compressible fluid, subject to a transverse magnetic field for the magnetogasdynamics system. By using Lie group theory, the full one-parameter infinitesimal transformations group leaving the equations of motion invariant is derived. The symmetry generators are used for constructing similarity variables which leads the system of PDEs to a reduced system of ordinary differential equations; in some cases, it is possible to solve these equations exactly. Further, using the exact solution, we discuss the evolutionary behavior of weak discontinuity.     

Global analysis for strong solutions to the equations of a ferrofluid flow model

In this paper we are concerned with the differential system proposed by Shliomis to describe the motion of an incompressible ferrofluid submitted to an external magnetic field. The system consists of the Navier-Stokes equations, the magnetization equations and the magnetostatic equations. No regularizing term is added to the magnetization equations. We prove the local existence of unique strong solution for the Cauchy problem and establish a finite time blow-up criterion of strong solutions. Under the smallness assumption of the initial data and the external magnetic field, we prove the global existence of strong solutions and derive a decay rate of such small solutions in L²-norm.     

Nonequilibrium Statistical Operator Method and Generalized Kinetic Equations

We consider some principal problems of nonequilibrium statistical thermodynamics in the framework of the Zubarev nonequilibrium statistical operator approach. We present a brief comparative analysis of some approaches to describing irreversible processes based on the concept of nonequilibrium Gibbs ensembles and their applicability to describing nonequilibrium processes. We discuss the derivation of generalized kinetic equations for a system in a heat bath. We obtain and analyze a damped Schrödinger-type equation for a dynamical system in a heat bath. We study the dynamical behavior of a particle in a medium taking the dissipation effects into account. We consider the scattering problem for neutrons in a nonequilibrium medium and derive a generalized Van Hove formula. We show that the nonequilibrium statistical operator method is an effective, convenient tool for describing irreversible processes in condensed matter.     

Types of convergence for sequences of probability densities and attractors

Some necessary and sufficient conditions on densities convergence for the existence of an ergoidc mixing, or exact dynamical system on a probability space given in [8] are extended to a measure space to obtain an ergodic, mixing or exact dynamical system on this measure space. More general sufficient conditions are given for the existence of those kinds of dynamical systems; as a consequence of these conditions it is obtained an ergodic, exact attractor for the orbits of almost every phase state. This orbits’ behaviour recall the thermodynamical evolution of systems from nonequilibrium to equilibrium states     

Landau-Lifshitz方程派生的球面锥对称族及其演化

直至2008年,尚未见到讨论同时具有外磁场和各向异性场的多维Landau-Lifshitz方程的显式动态解的文献.当缺少外加磁场或各向异性场时,许多作者曾经尝试过Hirota的著名方法,Lie代数结构与Backlund变换,双线性变换等,但精确解仍然没有构造出来.本文首先给出了一种求解具有外磁场和各向异性场的Landa...