Why and how to avoid standard kinetic equations I. Principles and band conduction problem

A review of ideas leading to full rejection of any finite or partially-infinite order kinetic equation linearized in external field is given on grounds of the time-convolution Generalized Master Equations (GME). By two examples (two-level and band conduction problem), it is shown how standard kinetic equations result from GME in the lowest order approximations which obscure, however, a direct correspondence with the Kubo linear response theory. Without approximations, on the other hand, the rigorous approach is shown to be fully equivalent with the Kubo results. It is argued and illustrated that usual technical simplicity and seeming physical lucidity of standard theories (connected with the presence of field-independent transfer or scattering rates in the fielddependent linearized theory) are just owing to structural features which are solely due to the lowest order approximations involved. These features (i.e. also the possibility of standard physical interpretation of kinetic phenomena) are proved to disappear completely as far as the theory goes properly to higher orders.     

On the derivation of quantum kinetic equations. II. Nonlocal Uehling-Uhlenbeck equation

The binary and triple collision terms of the quantum kinetic equation derived previously are analyzed in the weak coupling approximation. In this approximation the equation appears to be a nonlocal Markovian extension of the kinetic equation due to Uehling and Uhlenbeck. After linearization, its relationship with non-Markovian formulations found in the literature is studied.This investigation is part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (FOM), which is financially supported by the Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek (ZWO).     

Method of green's functions in the theory of quantum kinetic equations. II

The kinetic and antikinetic equations are obtained for the single-particle Wigner function in the context of the method of Green's time-temperature functions for an inhomogeneous system of weakly interacting particles situated in a time-dependent electric field. The kinetic equation is derived here from the equation of motion for Green's function, satisfying the causality condition.     

A unified approach for deriving kinetic equations in nonequilibrium statistical mechanics. II. Approximate results

The exact form for the kinetic equation derived by Mori, Fujisaka, and Shigematsu (MFS) is used to obtain several approximations better suited to be compared with macroscopic transport equations. Three approximations are discussed, namely, those known as the diagonal, the slow process, and the Markovian. The corresponding results are emphasized and their relationship is established. In particular, the Kramers-Moyal expansion for the Markovian kinetic equation is obtained from a microscopic basis.     

A novel approach to the kinetic theory and hydrodynamic turbulence. II

A novel approach to the kinetic theory of gases is suggested through certain correlation functions known as product densities. This approach is shown to be very useful in explaining transitions from gaseous to liquid state as well as laminar to turbulent flows in gas dynamics. For large deviations from the equilibrium configuration, it is shown that the system obeys a set of generalised Navier-Stokes equation where some stochastic features are present.     

Concerning Einstein's linear equations. II

Two sets of solutions of the system of equations of a weak perturbation of a non-Galilean metric are constructed, and some of their properties are noted.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 3–6, April, 1982.     

On the equivalence of convergent kinetic equations for hot dilute plasmas: II. Generating functions for collision brackets

The generating functions for the collision brackets associated with two alternative convergent kinetic equations are derived for small values of the plasma parameter. It is shown that the first few terms in the asymptotic expansions of these generating functions are identical. Consequently, both kinetic equations give rise to the same transport coefficients in arbitrarily high order of the Chapman-Cowling truncation scheme.     

Microscopic modes in a Fermi superfluid. I. Linearized kinetic equations

This is the first of two papers in which microscopic expressions for the amplitudes and dispersion relations for hydrodynamic modes in an isotropic Fermi superfluid are derived. In this first paper we derive closed, decoupled, linearized kinetic equations for the bogolon spin density and total density in a Fermi superfluid with fluctuating superfluid velocity, and we discuss the form of the hydrodynamic equations that result from these equations.     

Boolean delay equations. II. Periodic and aperiodic solutions

Boolean delay equations (BDEs) areevolution equations for a vector of discrete variables x(t). The value of each componentX _i (t), 0 or 1. depends on previous values of all componentsx _j (t– t _ij ), x _i (t)=f _i (x₁(t–t _i1),...,x _n (t –t _in )). BDEs model the evolution of biological and physical systems with threshold behavior and nonlinear feedbacks. The delays model distinct interaction times between pairs of variables. In this paper, BDEs are studied by algebraic, analytic, and numerical methods. It is shown that solutions depend continuously on the initial data and on the delays. BDEs are classified intoconservative anddissipative. All BDEs with rational delays only haveperiodic solutions only. But conservative BDEs with rationally unrelated delays haveaperiodic solutions of increasing complexity. These solutions can be approximated arbitrarily well by periodic solutions of increasing period.Self-similarity andintermittency of aperiodic solutions is studied as a function of delay values, and certain number-theoretic questions related toresonances and diophantine approximation are raised. Period length is shown to be a lower semicontinuous function of the delays for a given BDE, and can be evaluated explicitly for linear equations. We prove that a BDE isstructurable stable if and only if it has eventually periodic solutions of bounded period, and if the length of initial transients is bounded. It is shown that, for dissipative BDEs, asymptotic solution behavior is typically governed by areduced BDE. Applications toclimate dynamics and other problems are outlined.     

Asymptotic behavior of solutions to the coagulation-fragmentation equations. II. Weak fragmentation

The discrete coagulation-fragmentation equations are a model for the kinetics of cluster growth in which clusters can coagulate via binary interactions to form larger clusters or fragment to form smaller ones. The assumptions made on the fragmentation coefficients have the physical interpretation that surface effects are important. Our results on the asymptotic behavior of solutions generalize the corresponding results of Ball, Carr, and Penrose for the Becker-Doring equation.Dedicated to Oliver Penrose on the occasion of his 65th birthday     

Mode coupling from linear and nonlinear kinetic equations

The calculation of mode coupling contributions to equilibrium time correlation functions from the nonlinear Boltzmann equation is reconsidered. It is suggested that the use of a nonlinear kinetic equation is not appropriate in this context, but instead such calculations should be reinterpreted in terms of the Klimontovich equation for the microscopic phase space density. For hard spheres the Klimontovich equation is formally similar to the nonlinear Boltzmann equation, and this similarity is exploited to explain the successful calculation of mode coupling effects from the latter. The relationship of the Klimontovich formulation to the linear ring approximation is also established.     

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.     

Boltzmann and Vlasov kinetic equations in anisotropic cosmological models

The structure of the Boltzmann and Vlasov Kinetic equations is considered for spatially uniform cosmological models taking into account the Einstein equations. A successive approximation technique is proposed for solving Boltzmann's equation at the intensely anisotropic vacuum stage of expansion for the type I Bianci cosmological model. It is shown that during the hadron stage the collisional gas can be found in a highly nonequilibrium state.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 28–33, 1986.     

Generalized equations of the gravitoinertial field. II

A generalized Lagrangian of the gravitational and inertial fields is introduced. Modified field equations are obtained.     

Anisotropic solutions of the Einstein-Boltzmann equations. II. Some exact properties of the equations

A covariant harmonic analysis is used to determine exact properties of the Einstein-Boltzmann equations. In particular, it is shown that if there are a finite number of harmonic components, or if the first and second harmonic components are zero, then the solutions are kinematically very restricted in many circumstances. Implications for the understanding of the microscopic foundations of perfect fluids and of transport coefficients are discussed.