Why and how to avoid standard kinetic equations II. Time-convolution or convolutionless equations and hopping

Review of ideas questioning applicability of standard approximate kinetic equations derived from Generalized Master Equations (GME) with time-independent projector, in particular in the long-time (low-frequency) domain except for special situations, is given. Attention is concentrated on the time-convolutionless GME for systems interacting with a reservoir with applications to the phonon-assisted hopping problem. As compared to the previous work of the series, the treatment is extended to any power of the external field. Inapplicability of mathematical methods justifying standard markovian treatments in the Van Hove limit is found to be owing to finite coupling constants encountered in nature. It is argued that performing (for finite coupling constants usually necessary) approximations in a consistent way (preserving some exact identities derived here) means, in fact, avoiding the approximate kinetic equation method. The existence of an alternative stemming from GME is argued.Part of the work was done during the author's stay at the University of Stuttgart. Hospitality of the University and, in particular, of Prof. M. Wagner as well as a financial grant of the Heraeus Stiftung is gratefully acknowledged.     

Closure Approximations for Passive Scalar Turbulence: A Comparative Study on an Exactly Solvable Model with Complex Features

Some standard closure approximations used in turbulence theory are analyzed by examining systematically the predictions these approximations produce for a passive scalar advection model consisting of a shear flow with a fluctuating cross sweep. This model has a general geometric structure of a jet flow with transverse disturbances, which occur in a number of contexts, and it encompasses a wide variety of possible spatio-temporal statistical structures for the velocity field, including strong long-range correlations. Even though the Eulerian and Lagrangian velocity statistics are not equal and the passive scalar statistics exhibit broader-than-Gaussian intermittency, this model is nevertheless simple enough so that many passive scalar statistics can be computed exactly and compared systematically with the predictions of the closure approximations. Our comparative study illustrates the strength and weaknesses of the closure approximations and points out the physical phenomena that these approximations are able or not able to describe properly. In particular it is shown that the direct interaction approximation (DIA), one of the most sophisticated closure approximations available, fails to reproduce adequately the statistical features of the scalar and may even lead to absdurd predictions, even though the equations it produces are rather complicated and difficult to analyze. Two alternative closure approximations, the Modified DIA (MDIA) and the Renormalized Lagrangian Approximation (RLA), with different levels of sophistication, both are simpler to use than the DIA and perform better. In particular, it is shown that both closure approximations always reproduce exactly the second order statistics for the scalar and that the MDIA is even able to capture intermittency effects.     

Transport coefficients of QCD matter

Transport coefficients of QCD matter are discussed in the framework of relativistic kinetic theory. Expressions for shear and bulk viscosities and heat conductivity are derived in lowest order in deviations from local thermal equilibrium. Pure gluon matter is considered both at low temperatures in the glueball phase and at high temperatures in the gluon plasma phase, and quark-gluon matter is considered in the high-temperature plasma phase. In the critical region nonperturbative calculations based on Kubo formulas should be attempted. Applications to ultra-relativistic nucleus-nucleus collisions are discussed.     

The thermoelectric properties of the small polaron

The high temperature behaviour of the thermoelectric transport coefficients for the Holstein small polaron model is investigated. An expression for the energy flux operator is derived, which includes the flow of the electron-lattice interaction energy. This inclusion is necessary for the small polaron model, because the electron-phonon coupling is assumed to be strong, while the kinetic energy of the electrons is understood as a small perturbation. Kubo formulae are used and approximated in lowest order perturbation theory. It follows that there are no dynamical corrections to the thermoelectric power. But the strong electron-lattice interaction leads to a contribution to the thermal conductivity.     

On the approach to kinetic theory due to Gross

A classical kinetic theory introduced by Gross is explored in further detail. The theory consists of a sequence of approximations to the Liouville distribution function, with each approximation leading to a truncation of the BBGKY hierarchy at successively higher order. We formulate the truncation scheme at general order in terms of a set of time-dependent equilibrium correlation functions. It has the correct symmetries and, as is implied by the work of Gross with the first two approximations, is such that the interparticle potential appears only implicitly via static equilibrium correlation functions. We arrange the theory as a sequence of linear kinetic equations for the phase-space density correlation function, and solve for the collision kernels which result in each order. The collision kernel of the second approximation, which involves only binary dynamics, is shown to be a mean-field generalization of the known low-density kernel. The third approximation gives a similar generalization of the triple-collision kernel. The nth approximation also reproduces the frequency moments of S(kω) through order ω²ⁿ. More generally, the approximations are shown to give a continued-fraction expansion of the collision kernel, with the levels governed by the dynamics of successively larger numbers of particles. This is a renormalized kinetic theory in the sense that the potential is eliminated and clusters of particles are never isolated.     

Transport equation for a gas of bosons

A kinetic equation for the Wigner density function of a Bose gas of quasiparticles (magnons) is derived in the region of linear response. The method uses a simple factorization procedure to transform the Hierarchy equations into a system which can be solved iteratively. The first nontrivial approximation is examined in the limit of slow time and space variation. It gives the linearized Boltzmann equation in the form assumed in the Landau theory. The correction of lowest order in (coupling constant × density) is shown to be a correction to the Born approximation of the scattering cross section.Work supported by the Fonds zur Förderung der wissenschaftlichen Forschung.     

Connection between the Dielectric and the Ballistic Treatment of Collisional Absorption

I this work two important models of treating collisional absorption in a laser driven plasma are compared, the dielectric and the ballistic model. We will see that there exists a remarkable connection between these basic approaches which could give a hint how to overcome the inherent limitations. The approximations made in the models are not identical and lead to different advantages and disadvantages. We notice that the dieletric model is able to handle screening in a selfconsistent manner, but is limited to first order in the electron‐ion interaction. The ballistic model calculates the electron‐ion collision exactly in each order of the interaction, but has to introduce a cut‐off to incorporate screening effects. This means in the context of kinetic theory that the electron‐ion correlation has to be calculated either in random phase or in ladder approximation, or, in other words, the linearized Lenard‐Balescu or Boltzmann collision term has to be used.     

Quantum statistical theory of chemical reactions

Exact rate equations for chemical reactions taking place in a closed homogeneous system arbitrarily far from equilibrium are derived by use of a recently developed method. The possibly different kinetic temperatures of the species are taken into account. The diagram expansion of the integral kernels of the rate equations is given and the renormalization of the interaction carried out. In lowest order non-linear non-markovian equations are obtained which reduce after suitable approximations to the usual collision theory results.     

Non-Abelian Kubo Formula and the Multiple Time-Scale Method

The non-Abelian Kubo formula is derived from the kinetic theory. That expression is compared with the one obtained using the eikonal for a Chern–Simons theory. The multiple time-scale method is used to study the non-Abelian Kubo formula, and the damping rate for longitudinal color waves is computed.     

Hamiltonian and Brownian systems with long-range interactions: IV. General kinetic equations from the quasilinear theory

We develop the kinetic theory of Hamiltonian systems with weak long-range interactions. Starting from the Klimontovich equation and using a quasilinear theory, we obtain a general kinetic equation that can be applied to spatially inhomogeneous systems and that takes into account memory effects. This equation is valid at order 1/N in a proper thermodynamic limit and it coincides with the kinetic equation obtained from the BBGKY hierarchy. For N→+∞, it reduces to the Vlasov equation governing collisionless systems. We describe the process of phase mixing and violent relaxation leading to the formation of a quasistationary state (QSS) on the coarse-grained scale. We interpret the physical nature of the QSS in relation to Lynden-Bell’s statistical theory and discuss the problem of incomplete relaxation. In the second part of the paper, we consider the relaxation of a test particle in a thermal bath. We derive a Fokker-Planck equation by directly calculating the diffusion tensor and the friction force from the Klimontovich equation. We give general expressions of these quantities that are valid for possibly spatially inhomogeneous systems with long correlation time. We show that the diffusion and friction terms have a very similar structure given by a sort of generalized Kubo formula. We also obtain non-Markovian kinetic equations that can be relevant when the auto-correlation function of the force decreases slowly with time. An interesting factor in our approach is the development of a formalism that remains in physical space (instead of Fourier space) and that can deal with spatially inhomogeneous systems.     

Theory of Stark broadening—I soluble scalar model as a test

The purpose of the present paper is to test, on an exactly soluble model, the validity of standard approximations used in the theory of Stark broadening of atomic lines. The mathematical treatment involves no approximation; in particular, overlapping and incomplete collisions are correctly taken into account. In setting up the model, standard physical approximations are made: the dipole approximation, and uncorrelated perturbers with classical paths. In addition, in order to permit exact solution, the problem is taken to be scalar.     

Generalized Master Equations and Boltzmann-like transport in solids

Solution of the linearized Generalized Master Equations and that of the linearized Boltzmann equation are compared in the vicinity of the dc limit. In addition to straightforward higher-order corrections, the former solution is shown to differ from the latter by an additional term which is made explicit for the dc conductivity to the lowest order in coupling to static random short-range impurities. Numerical estimate shows that this term (proportional to the first power of impurity concentration) may become non-negligible in realistic situations.     

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.     

Zum Einfluß der Energieabhängigkeit der Elektronenstoßfrequenz auf die elektrische Leitfähigkeit schwach ionisierter Plasmen

With regard to experimental applications in plasma diagnostics numerical approximations are given for the Gurevic-type correction functions, which in the kinetic theory of weakly ionized plasmas describe the deviations from the Lorentzian electrical conductivity. The approximations are based either upon the dependance of the collision frequency on the power of the electron velocity, or on a first order Taylor expansion around the most probable electron velocity of a Maxwellian distribution. For all assumptions regarding the velocity dependance of the collision frequency the influence of temperature (and pressure) on the effective collision frequency is indicated.     

Magnon–magnon interactions in O(3) ferromagnets and equations of motion for spin operators

The method of equations of motion for spin operators in the case of O(3) Heisenberg ferromagnet is systematically analyzed starting from the effective Lagrangian. It is shown that the random phase approximation and the Callen approximation can be understood in terms of perturbation theory for type B magnons. Also, the second order approximation of Kondo and Yamaji for one dimensional ferromagnet is reduced to the perturbation theory for type A magnons. An emphasis is put on the physical picture, i.e. on magnon–magnon interactions and symmetries of the Heisenberg model. Calculations demonstrate that all three approximations differ in manner in which the magnon–magnon interactions arising from the Wess–Zumino term are treated, from where specific features and limitations of each of them can be deduced.     

Molecular dynamics studies of thermal conductivity time correlation functions

The Green–Kubo time correlation function for the thermal conductivity in liquid argon is studied for a thermodynamic state close to the triple point by standard molecular dynamics simulations. The collective heat flux vector has been separated into contributions originated at the kinetic energy, the intermolecular potential and the pair virial function. Furthermore, the Green–Kubo time correlation functions have been broken down into partial n-body terms (n=1,2,3,4). The most important contribution to the thermal conductivity is represented by the auto correlation of the virial term. In contrast to other collective phenomena described by time correlation functions involving n-body terms, the partial Green–Kubo time correlation functions for the thermal conductivity are not affected by exponential long-time tails.     

Accurate models of collisions in glow discharge simulations

Very detailed, self-consistent kinetic glow discharge simulations are used to examine the effect of various models of collisional processes. The effects of allowing anisotropy in elastic electron collisions with neutral atoms instead of using the momentum transfer cross-section, the effects of using an isotropic distribution in inelastic electron-atom collisions, and the effects of including a Coulomb electron-electron collision operator are all described. It is shown that changes in any of the collisional models, especially the second and third described above, can make a profound difference in the simulation results. This confirms that many discharge simulations have great sensitivity to the physical and numerical approximations used. Our results reinforce the importance of using a kinetic theory approach with highly realistic models of various collisional processes     

Power-law corrections to the Kubo formula vanish in quantum Hall systems

In first order perturbation theory conductivity is given by the Kubo formula, which in a Quantum Hall System equals the first Chern class of a vector bundle. We apply the adiabatic theorem to show that these topological constraints quantize the averaged conductivity to all orders of perturbation theory. Hence the Kubo formula is valid to all orders.     

Validity of the Markov approximation in ocean acoustics

Moment equations and path integrals for wave propagation in random media have been applied to many ocean acoustics problems. Both these techniques make use of the Markov approximation. The expansion parameter, which must be less than one for the Markov approximation to be valid, is the subject of this paper. There is a standard parameter (the Kubo number) which various authors have shown to be sufficient. Fourth moment equations have been successfully used to predict the experimentally measured frequency spectrum of intensity in the mid-ocean acoustic transmission experiment (MATE). Yet, in spite of this success, the Kubo number is greater than 1 for the measured index of refraction variability for MATE, arriving at a contradiction. Here, that contradiction is resolved by showing that the Kubo parameter is far too pessimistic for the ocean case. Using the methodology of van Kampen, another parameter is found which appears to be both necessary and sufficient, and is much smaller than the Kubo number when phase fluctuations are dominated by large scales in the medium. This parameter is shown to be small for the experimental regime of MATE, justifying the applications of the moment equations to that experiment.     

Phonon-assisted hopping-conductivity of disordered systems II: Discussion of approximations and results

A theory of the frequency dependent hopping-conductivity of disordered systems which has been developed in a previous paper is shortly reviewed in connection with other approximations. In lowest order, the probability of finding a carrier on the same site some time later (Verweil-function) is calculated and discussed. After deriving an explicit expression for the lowest order conductivity its dependence on the three variables temperature, density of sites, and frequency is studied. In particular, the characteristic exponents for vanishing temperature and density differ from those predicted by Mott. The results are also compared with a nearest-neighbour-treatment and show good agreement in certain limiting cases.