Equations Of Kinetic Theory Research Articles

On the spatially homogeneous landau equation for hard potentials part i : existence, uniqueness and smoothness

We study the Cauchy problem for the homogeneous Landau equation of kinetic theory, in the case of hard potentials. We prove that for a large class of initial data, there exists a unique weak solution to this problem, which becomes immediately smooth and rapidly decaying at infinity.

A Covariant Generalization of the Real-Time Green's Functions Method in the Theory of Kinetic Equations

A generalized quantum relativistic kinetic equation (RKE) of the Kadanoff–Baym type is obtained on the basis of the Heisenberg equations of motion where the time evolution and space translation are separated from each other by means of the covariant method. The same approach is used also for a covariant modification of the real-time Green's functions method based on the Wigner representation. The suggested approach does not contain arbitrariness' elements and uncertainties which often arise from derivation of RKE on the basis of the motion equations of the Kadanoff–Baym type for the correlation functions in the case of systems with inner degrees of freedom. Possibilities of the proposed method are demonstrated by examples of derivation of RKE of the Vlasov type and collision integrals of the Boltzmann– Uehling–Uhlenbeck (BUU) type in the frame of the σω-version of quantum hadrodynamics, for the simplest case of spin saturated nuclear matter without antinuclear component. Here, the quasiparticle approximation in a covariant performance is used. A generalization of the method for the description of strong non-equilibrium states based on the non-equilibrium statistical operator is then proposed as well.

Brownian dynamics of bead–rod–nugget–spring polymer chains with hydrodynamic interactions

With Brownian dynamics simulations of solutions containing complex biological macromolecules in mind we have formulated a polymer chain model consisting of a selectable sequence of spherical subunits (beads) and nonspherical subunits of arbitrary shapes (nuggets) connected by rigid rods, rigid ball–socket joints, or springs. Many physical properties of both subunits and connections are selectable. First a very general diffusion equation of polymer kinetic theory is employed. Next the equivalent Itô stochastic differential equation of motion is established and finally an effective Brownian dynamics simulation algorithm is presented. In all these steps it is assumed that the numerical values of the subunit steady state mobility tensor are obtainable. The important and previously unanswered question of how to include translational–rotational hydrodynamic interactions in Brownian dynamics simulations of polymer chains containing nonspherical subunits linked by rigid constraints is resolved rigorously here. This means that during the formal derivation of the presented Brownian dynamics simulation algorithm all subunit hydrodynamic interactions of the proposed bead–rod–nugget–spring polymer chain model have in principle been taken fully into account. In addition the model includes rigid constraints to fixed points in the laboratory coordinate system, solvent flow, external forces, excluded volume effects, and bending and torsional stiffness between polymer chain subunits. Bead–spring, bead–rod–spring, needle–spring, needle polymer chains and liquid crystals are all special cases of the bead–rod–nugget–spring polymer chain model. The validity of the algorithm presented is limited to the diffusion time domain.

Transition from collision-dominated to collisionless regions in a plasma: a global distribution function for outflowing light ions in polar regions of the ionosphere – an analytical approach

Analytical expressions for the lowest-order distribution functions of a plasma that are valid throughout the plasma are generally not easily obtainable when conditions shift from collision dominance in some regions of the plasma to absence of collisions in other regions. Indeed, the reason for the breakdown of classical approximation methods for distribution functions and the corresponding macroscopic equations in plasma kinetic theory is often the lack of such global lowest-order distribution functions. The main purpose of the present paper is to derive an approximate global distribution function for an outflowing dilute light ion component in a polar-region model of the ionosphere where conditions for the light ions shift from collision dominance in low regions to collisionless conditions high up. For this, a simplified collision integral of a form given by Chandrasekhar will be used. The proposed global solution is given after simpler solutions, as a collisionless solution, an exact collision-dominated solution and a solution corresponding to classical transport theory are derived, for reason of comparison, in the first parts of the paper.

Hamiltonian Structure of Benney Type Hydrodynamic and Boltzmann-Vlasov Kinetic Equations on an Axis and Some Applications to Manufacturing Science

Some years ago Zakharov and Gibbon observed a very nice relation between the Benney type equation in hydrodynamics and the Vlasov equation of kinetic theory. These equations are generalized and put into the framework of infinite-dimensional Lie algebras associated to Lie algebra structures on rings of functions on finite-dimensional manifolds. This gives rise to a complete description of the Hamiltonian structure of both types of equations under consideration. In particular, their Lax type representations together with an infinite involutive hierarchy of conservation laws are obtained in an exact form. Some applications to chaotic manyparticle dynamical systems, turbulent fluid flows and swept volume analysis are considered.

Dynamic interfacial tensiometry of biologic liquids — does it have an impact on medicine

Dynamic surface tension measurements of biological liquids with an impact on medicine are scarce. One reason for this deficit is that technologies, e.g. dynamic interfacial tensiometry, for the measurement of surface tension characteristics of a given fluid under dynamic conditions are not easily available. Another reason is that the theoretical background is not introduced to medicine. Finally, normal values of surface tension characteristics of biological liquids using dynamic interfacial tensiometry are not established. We will therefore summarize experimental techniques for measuring dynamic surface tension characteristics regarding its medical application and present some of the theoretical background for these measurements. We will also summarize surface tension characteristics of reference fluids and of serum and urine that were obtained from healthy and sick humans. This data were collected using a modified maximum bubble pressure tensiometer (MPT2, LAUDA, Germany) and applying asymptotic equations of adsorption kinetic theory. We found that surface tension characteristics of serum and urine depends in part on gender and age. In addition, surface tension characteristics changes during pregnancy. Finally, some changes of the dynamic tensiograms as found in serum and urine are related to specific kidney diseases. Our study shows promising results that might become relevant in medicine but points to the need that more research is necessary to establish dynamic surface tension measurements as a diagnostic tool.

Projected moments in relativistic kinetic theory

In this paper a new set of moment equations in relativistic kinetic theory is presented. The moments under consideration are the projections of particle 4-flux and energy momentum tensor with respect to the Eckart velocity or the Landau–Lifshitz velocity, alternatively. The moment equations follow from integrations of the relativistic Boltzmann equation in which the interactions of the particles are described by the relativistic BGK model for reasons of simplicity. The projected moment formalism is extended to an arbitrary number of moments and moment equations and it is shown that the non-relativistic limit of moments and moments equations leads to the so-called central moments of non-relativistic theory. The moment equations may be closed by means of the entropy maximum principle. After this method has been outlined, the closure is performed for the case of 14 moments, i.e. the projections of particle 4-flux and energy momentum tensor. Moreover local thermal equilibrium is considered where the projected moment formalism is used for the derivation of the relativistic Navier–Stokes and Fourier laws. Different choices of moment equations for this task are compared and it is shown that the proper choice of moment equations depends on the interaction term in the relativistic Boltzmann equation.

New developments in kinetic schemes

Kinetic schemes for the numerical solution of PDE of fluid dynamics are a new class of upwind schemes. The moment method strategy used in constructing these schemes is based on the connection between the Boltzmann equation of kinetic theory of gases and governing equations of fluid dynamics. Any upwind scheme for the solution of the Boltzmann equation becomes an upwind scheme for the solution of PDE of fluid dynamics by taking suitable moments. This idea has been exploited in the present paper in developing Least Squares Kinetic Upwind Method (LSKUM) and K inetic F lux vector splitting method on M oving G rid (KFMG). The robustness and versatility of LSKUM has been demonstrated by applying it to 2-D flow problems of inviscid gas dynamics. The LSKUM has been found to operate on any type of grid and can be called a grid fault tolerant scheme in view of the use of least squares approximation to the space derivatives. The KFMG is a promising new method showing how easily the kinetic schemes lend themselves easily to problems involving moving grids which are generally employed while solving problems in unsteady aerodynamics.

THEORY OF CORRELATION EFFECTS FOR DIFFUSION IN A TWO SUBLATTICE STRUCTURE

The tracer diffusion correlation factor has been studied for a simple cubic lattice gas with two energetically inequivalent sublattices and sites restricted to accommodate a maximum of one atom. The model serves as a paradigm for solids with site inequivalence but no algebraic formulae have previously been derived and compared with available Monte Carlo simulation data. An expression based on a kinetic equation theory which neglects the highest order in concentration fluctuations is shown to lead to a good description of the simulation results. The time dependence of the associated time correlation function appears to be unusually complex for diffusion in a two sublattice system. © 1997 Elsevier Science Ltd. All rights reserved

On a functional equation in mathematical kinetic theory

A functional equation arising firom kinetic theory of dense gases is considered. The solutions to this equation represent the quillbrium distniutions for the revised Enskog Equation. Indeed, they make the Rdsiiois-Polewczak Liapounov fUncti0∼1 constant m time. The class of all the above mentioned distriiutions is completely characterized. This class is wider than that characterized by Risibois only consisting of the distributions which are periodic with respect to the space variable.

Solution to the linear BGK equation via the principles of invariance

The solution to the linearized BGK equation of kinetic theory is obtained via the principles of invariance as formulated by Chandrasekhar. The solution represents an alternative to the singular eigenfunction method featuring regular integral equations which are more easily evaluated. Laplace transforms are obtained for the net flow velocity and velocity distribution for both the albedo and Milne problems with wall reflection. The transforms are then numerically inverted to a high degree of accuracy. Results are compared to those of Kainz and Titulaer.

Spectral velocity discretizations for the Vlasov-Maxwell equations

Abstract Two Hermite based spectral methods are examined. A method based on an asymmetric expansion in Hermite polynomials offers theoretical advantages over other spectral basis sets for the treatment of the velocity variables in the Vlasov-Maxwell equations of plasma kinetic theory. It provides for exact conservation of energy, momentum, and particles (in a fully discrete system, not just as the discretization is refined), and this conservation is not effected by the use of velocity space filters during time-stepping. The asymmetric Hermite method exactly solves the spatially uniform problem (plasma oscillations), and both Hermite based methods exactly recover the temporal behavior of retained moments for free streaming in a periodic system. Yet the cost of these Hermite methods scales only linearly with the number of degrees for freedom, just like a finite difference method, and in fact the number of runtime operations required per degree of freedom in the asymmetric method can be essentially identical...

Statistical interpretation of topographies and dynamics of multidimensional potentials

A statistically based method of characterizing the topography of a multidimensional potential surface classifies not only local minima and saddles but entire basins containing many minima, and divides separating basins and monotonic sequences of local minima within each basin. The data, so classified, fold readily into the formalisms of chemical kinetic isomerization theory and master equations to provide a connection between that topography and the dynamics on the surface. This analysis, in particular, permits an interpretation of the glass-forming or ‘‘focusing’’ character of the surface. The method is illustrated with a model system derived, with simplifications, from the 19-atom Lennard-Jones cluster. The method also leads naturally to control problems including the determination of optimum conditions for forming glasses or selected structures, such as particular crystal structures or folded protein structures.

Topography and Dynamics of Multidimensional Interatomic Potential Surfaces.

A statistically based characterization of the topography of a multidimensional potential surface classifies not only local minima and saddles but entire basins containing many minima, and divides separating basins and monotonic sequences of local minima within each basin. The data fold readily into the formalisms of chemical kinetic isomerization theory and master equations to provide a connection between that topography and the dynamics on the surface. This analysis permits an interpretation of the glass-forming or ``focusing'' character of the surface. The method is illustrated with a model system derived from the 19-atom Lennard-Jones cluster.

Kinetic theory of low-frequency cross-field instability in a weakly ionized plasma. II

The consistent kinetic approach developed in Paper I [Ya. S. Dimant and R. N. Sudan, Phys. Plasmas 2, 1157 (1995)] is applied to obtain the general dispersion relation of the two-stream E×B instability in collisionally dominated weakly ionized plasmas for wave frequencies small compared to the ion–neutral collision frequency. This dispersion relation covers the whole low-frequency band from the asymptotic short-wave limit studied in Paper I to the long-wave limit. Previous theories employing simplified kinetic theory or fluid equations for electron behavior are only correct in the long-wave limit. The principal new results are that the threshold conditions for this instability and the growth rates are altered from those predicted by earlier simplified theories.

Brownian dynamics of rigid polymer chains with hydrodynamic interactions.

With the idea of a broad investigation of the flow behavior of dilute polymer solutions in mind, the dynamics of polymer chains with rigid constraints and hydrodynamic interaction is formulated in various equivalent ways. Starting from a very general diffusion equation of polymer kinetic theory, equivalent stochastic differential equations of motion both in terms of generalized coordinates and in terms of constraint conditions are derived. Then an efficient Brownian dynamics simulation algorithm is constructed rigorously, and a convenient expression for evaluating stresses in simulations is suggested. Furthermore, a modified simulation algorithm, which is appropriate for infinitely stiff rather than rigid systems, is discussed.

The analysis of dynamic surface tension of sodium alkyl sulphate solutions, based on asymptotic equations of adsorption kinetic theory

We analysed existing and newly derived asymptotic solutions of the adsorption kinetic equations for the liquid phase interface in the regions of infinitely small and infinitely great surface lifetimes ( t) for the cases of one and of a few surfactants, on non-deforming and deforming surfaces, under non-stationary, stationary and quasi-stationary conditions, assuming either the diffusion adsorption mechanism or mixed adsorption mechanism. It was proved that in the region t → ∞, the adsorption barrier does not influence the dynamic surface tension σ, but the role of surface-active contaminants is significant. In contrast, in the region t → 0, the role of contaminants is small, but the adsorption barrier influences the dynamic surface tension substantially. The dynamic surface tension of sodium alkyl sulphate solutions was measured by the maximum bubble pressure method, in the t range 0.001–10 s. In the region t → ∞ we obtained good agreement of experimental results with asymptotic formulae. The diffusion adsorption mechanism of the surfactant solutions studied was confirmed and we also estimated the concentration values of the surfactant admixtures. Small additions of the more active surfactant sodium tetradecyl sulphate to sodium dodecyl sulphate substantially influences the shape of the σ— t curve in the region t → ∞, increasing (in full accordance with theoretical considerations) the tangent value of the curve inclination of the dependence of σ on t −1/2. In the regions t → 0, long-chained high molecular weight sodium alkyl sulphates adsorb according to the diffusion mechanism, whereas for sodium decyl and dodecyl sulphates the existence of the adsorption barrier was confirmed. We corroborated experimentally the absence of any influence of surfactant admixtures on the values of dynamic surface tension at t → 0.

On the evolution operators for some equations of mathematical physics with variable coefficients

It is shown that, with the help of a relatively simple operator technique, it is possible to solve, from a common point of view, the Cauchy problem for many important equations of mathematical physics with variable coefficients. This result is applied to the equations of kinetic theory, and diffusion and heat conduction equations. We discuss the problem of equivalence of different schemes of expansion according to the Hausdorff formula.

SEGREGATED SOLUTION TECHNIQUE FOR SIMULATING THE TRANSIENT OPERATION OF HEAT PIPES

Simulating the transient operation of fully thawed heat pipes involves solving a highly nonlinear homogeneous two-phase flow problem, which necessitates the development of a stable and efficient numerical technique. In this work, various segregated numerical techniques are implemented, and their accuracy and computation time requirement are examined using experimental data of a water heat pipe. Best results are obtained using an iterative SIMPLEC-type segregated solution technique (HPTAM-Revised), which includes two internal iterative steps to resolve the pressure-velocity coupling and reduce the linearization errors of the kinetic theory relationship and equations of state. While all solution techniques examined performed the same in terms of accuracy, the HPTAM-Revised is 90 times faster than the basic noniterative SIMPLE-type approach in terms of CPU time. Also, using the iterative SIS solver, instead of the banded Gauss-elimination solver, for the discretized energy and momentum equations resulted in more than 50% savings in the CPU time for computational grids larger than 20 × 30.

Systems of coupled Riccati equations in discrete kinetic theory: New self-similar solutions, shock profiles, stability

Abstract In the kinetic theory of the discrete Boltzmann models (DBMs) the standard similarity shock waves solutions are, like the Broadwell shock waves, solutions of scalar Riccati equations. The corresponding microscopic densities, as well as the macroscopic conservative quantities: mass, momentum and energy, are monotonic. In the first part of the paper we construct new exact similarity shock waves solutions of systems of coupled Riccati equations, which arise for the DBMs with n conservation laws and n + k independent densities (here with n = 3, k = 2). The former standard shock waves are rational functions of two linear polynomials with an exponential variable while for the new waves the polynomials are quadratic and consequently nonmonotonic behaviours are possible. We stress that from the mathematical point of view we enter into a new domain of exact self-similar solutions associated to the systems of coupled Riccati equations. Due to the fact that, for the considered models, there exists a linear differential relation between two microscopic densities we obtain two different classes of solutions. For one class these two densities are necessarily of the former standard type while for the other all densities are of the new type of self-similar solutions. Furthermore we prove an important result: Let us consider microscopic densities associated to opposite velocities along the shock-axis. To any solution, there exists another one, called partner solution, such that those densities are exchanged with opposite similarity variable. This means that from any solution with mass, energy and pressure given, there exists a partner solution where these macroscopic quantities have opposite similarity variable. The new exact solutions are not the most general self-similar solutions of the considered models. In the second part we reduce the problem to the dynamical system of the coupled Riccati equations, with the limit conditions corresponding to the equilibrium states on both sides of the shocks. We analyse the structure of singularities corresponding to the equilibria, and obtain new nonmonotonic profiles of the microscopic densities by solving numerically the relevant two-point BVPs. For the new class of models studied in this paper we observe overshoots on one density for the exact solutions. In addition in the second part we find undershoots on one density and/or overshoot on one density accompanied by undershoots on another one. Contrary to the previously studied models, no nonmonotonic effects have been observed for the macroscopic conserved quantities: mass and energy.