Properties of the Collision Operators of the Electron Boltzmann Equation for a Weakly Ionized Plasma. I. Representation and General Properties

In the kinetic theory a great variety of physical systems is investigated by means of Boltzmann-like equations. This approach is used for neutral gases, neutron as well as radiation transport, plasmas etc. For many problems the knowledge of the properties of the collision operators is of great importance, especially if eigenvalue problems occur. The paper presents an investigation of the properties of the collision operators of the Boltzmann equation covering elastic, exciting and deexciting processes in a weakly ionized plasma. First, a short survey of the importance of eigenfunctions and eigenvalues in the kinetic theory of various systems is given. Then, properties of the outscattering operator as dependent on the course of the differential cross section are considered. Finally, for the inscattering operator such properties as selfadjointness and rotational invariance are investigated in detail. These considerations provide the basis for the proof of compactness and for first conclusions on the spectral properties of the collision operators in the second part of this paper.     

Remarks on the Calculation of Eigenvalues and Generalized Eigenfunctions Using a Finite Dimensional Approximation for the Inscattering Part of the Boltzmann Collision Operator

From several points of view it is of advantage to know the properties of the collision operators in kinetic equations, e.g. in the well known Boltzmann equation, in particular for the purpose of solving eigenvalue problems. With regard to elastic, exciting and deexciting processes some attemps were recently made to investigate such operators in the Boltzmann equation describing the behaviour of electrons in weakly ionized plasmas. In the following we will prove that in the case of a finite dimensional inscattering operator the eigenvalues and the corresponding eigenfunctions can be represented explicitly. Finite dimensional operators were used successfully in special models to approximate the inscattering operators; they possess the property of compactness and are well suitable for analytical or numerical calculations. The representation has been obtained by solving an adequate linear equation system. The generalized eigenfunctions correspond to the normal solutions used by Case in the neutron transport theory. The regularization of the singular integrals which are necessary to obtain this solution will be given in detail. Further a velocity dependence of the collision frequency which need not be monotonous in the considered case and the dependence on the direction could be included.     

Scattering of long-wavelength acoustic phonons by isotopic impurities

We consider the problem of the relaxation of an arbitrary initial distribution function of a gas of long-wave acoustic phonons scattered by isotopic impurities embedded in a crystalline medium with cubic symmetry. The spectral decomposition of the collision integral of the suitable Boltzmann-Peierls equation is obtained. The spectrum of the collision operator is purely discrete and in addition to the eigenvalue 0 consists of three other eigenvalues. Explicit analytic expressions for these eigenvalues are obtained. Within the Chapman-Enskog approximation we derive the diffusion equation for the density of phonons and obtain the explicit expression for the diffusion coefficient. The dependency of the eigenvalues of the collision operator and the diffusion coefficient on the elastic constants of the medium is studied.     

Elastic scattering of acoustic phonons on point mass defects

The relaxation of homogeneous states of long-wave acoustic phonon gas scattered by point mass defects in transversely—isotropic media is studied. The spectrum of the suitable collision operator of the Boltzmann-Peierls equation is investigated. It consists of a continuous part and several discrete eigenvalues. Both continuous and discrete part of the spectrum depend on the values of components of the elastic constant tensor. For some values of elastic constants the continuous part splits up into two separate intervals and some of the discrete eigenvalues appear in the gap. The number of discrete eigenvalues and their arrangement are also affected by elastic properties of medium.     

Spectral Gap for a Class of Random Billiards

A random billiard is a random dynamical system similar to an ordinary billiard system except that the standard specular reflection law is replaced with a more general stochastic operator specifying the post-collision distribution of velocities for any given pre-collision velocity. We consider such collision operators for certain random billiards that we call billiards with microstructure. Collisions modeled by these operators can still be thought of as elastic and time reversible. The operators are canonically determined by a second (deterministic) billiard system that models “microscopic roughness” on the billiard table boundary. Our main purpose here is to develop some general tools for the analysis of the collision operator of such random billiards. Among the main results, we give geometric conditions for these operators to be Hilbert-Schmidt and relate their spectrum and speed of convergence to stationary Markov chains with geometric features of the microscopic billiard structure. The relationship between spectral gap and the shape of the microstructure is illustrated with several simple examples.     

Linear stability of the spatially homogeneous equilibria of the Vlasov-Poisson system with collisions

In this work we study the stability of the spatially homogeneous solutions of the Vlasov-Poisson system (Vlasov equilibria) when a collision term, in the form of a BGK operator with velocity-dependent collision frequency, is added to the Vlasov equation. Generalizing earlier results, obtained for the same collision model with a constant collision frequency, we find the spectrum and the eigenfunctions of the linear transport operator and derive a new linear dispersion relation for the linearized kinetic equations. Finally, we present some numerical results.     

On Spectral Analysis of an Integral-Difference Operator

A complete spectral analysis of an integral-difference operator arising as a collision operator in some nonequilibrium statistical physics models is presented. Eigenfunctions of both discrete and continuous spectrum are constructed.     

Poincaré's theorem and unitary transformations for classical and quantum systems

Poincaré's celebrated theorem on the nonexistence of analytical invariants of motion is extended to the case of a continuous spectrum to deal with large classical and quantum systems. It is shown that Poincaré's theorem applies to situations where there exist continuous sets of resonances. This condition is equivalent to the nonvanishing of the asymptotic collision operator as defined in modern kinetic theory. Typical examples are systems presenting relaxation processes or exhibiting unstable quantum levels. As the result of Poincaré's theorem, the unitary transformation, leading to a cyclic Hamiltonian in classical mechanics or to the diagonalization of the Hamiltonian operator in quantum mechanics, diverges. We obtain therefore a dynamical classification of large classical or quantum systems. This is of special interest for quantum systems as, historically, quantum mechanics has been formulated following closely the patterns of classical integrable systems. The well known results of Friedrichs concerning the coupling of discrete states with a continuum are recovered. However, the role of the collision operator suggests new ways of eliminating the divergence in the unitary transformation theory.     

A stochastic derivation of the multivariate master equation describing reaction diffusion systems

We derive the multivariate master equation describing reaction diffusion systems from a discrete form master equation in phase space, assuming that the elastic collisions of the chemically active substances with the inert carrier gas have relaxed. In this state of collisional equilibrium the stochastic operator modelling the displacement of the particles between spatial cells reduces to the random wall operator and the reactive collision term yields the usual birth and death operator. Correlation functions are derived and their validity is discussed.     

Bulk viscosity of low-temperature strongly interacting matter

We study the bulk viscosity of a pion gas in unitarized Chiral Perturbation Theory at low and moderate temperatures, below any phase transition to a quark-gluon plasma phase.We argue that inelastic processes are irrelevant and exponentially suppressed at low temperatures. Since the system falls out of chemical equilibrium upon expansion, a pion chemical potential must be introduced, so we extend the existing theories that include it. We control the zero modes of the collision operator and Landau?s conditions of fit when solving the Boltzmann equation with the elastic collision kernel.The dependence of the bulk viscosity with temperature is reminiscent of the findings of Fernández-Fraile and Gómez Nicola (2009) [1], while the numerical value is closer to that of Davesne (1996) [2]. In the zero-temperature limit we correctly recover the vanishing viscosity associated to a non-relativistic monoatomic gas.     

The collision operator for a lattice model in a uniform field

We discuss the properties of the collision operator, as defined by the Brussels group, for a simple lattice model. The existence of a non-trivial asymptotic collision operator is related to the change of the spectrum of the free Hamiltonian due to the external field.     

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     

The role of the gluonic $gg\leftrightarrow{ggg}$ interactions in early thermalization in ultrarelativistic heavy-ion collisions

We “quantify” the role of elastic as well as inelastic pQCD processes in kinetic equilibration within a pQCD inspired parton cascade. The contributions of different processes to kinetic equilibration are manifested by the transport collision rates. We find that in a central Au+Au collision at RHIC energy pQCD bremsstrahlung processes are much more efficient for momentum isotropization compared to elastic scatterings. For the parameters chosen the ratio of their transport collision rates amounts to 5:1. PACS  05.60.-k, 25.75.-q, 24.10.Lx     

Understanding the accuracy of Nanbu’s numerical Coulomb collision operator

We investigate the accuracy of and assumptions underlying the numerical binary Monte Carlo collision operator due to Nanbu [K. Nanbu, Phys. Rev. E 55 (1997) 4642]. The numerical experiments that resulted in the parameterization of the collision kernel used in Nanbu’s operator are argued to be an approximate realization of the Coulomb–Lorentz pitch-angle scattering process, for which an analytical solution for the collision kernel is available. It is demonstrated empirically that Nanbu’s collision operator quite accurately recovers the effects of Coulomb–Lorentz pitch-angle collisions, or processes that approximate these (such interspecies Coulomb collisions with very small mass ratio) even for very large values of the collisional time step. An investigation of the analytical solution shows that Nanbu’s parameterized kernel is highly accurate for small values of the normalized collision time step, but loses some of its accuracy for larger values of the time step. Careful numerical and analytical investigations are presented, which show that the time dependence of the relaxation of a temperature anisotropy by Coulomb–Lorentz collisions has a richer structure than previously thought, and is not accurately represented by an exponential decay with a single decay rate. Finally, a practical collision algorithm is proposed that for small-mass-ratio interspecies Coulomb collisions improves on the accuracy of Nanbu’s algorithm.     

On a density matrix equation for atomic systems subject to an external radiation field interaction and collision processes

Approximations are examined which are necessary for a density matrix equation of Rautian's form describing the interaction of a gas medium with an external radiation field by considering collision processes between active atoms as well as active and perturber atoms. The collision integrals of the kinetic density matrix equation calculated by the Bogolyubov method involve exitation processes, inelastic and elastic collisions. The features of phase- and velocity changing collisions are discussed by specializing the collision terms. Some phenomenological collision models for velocity changing collisions are examined and conditions for their validity given.     

Kinetic Description of Elastic Processes in Hydrogen-Helium Plasmas

Recent experiments with high density divertor plasma operation in tokamaks and future tokamak design projects have revived interest in the much earlier work on elastic collision processes between neutral and charged components in hot plasmas. In revisiting these papers, we employ classical methods to calculate deflection functions, cross sections and collision rates for the most important of such elastic collision processes. A phyiscally motivated cut-off procedure for the model interaction potential function is introduced. An algorithm for implementation of such processes into kinetic Monte Carlo neutral gas transport models is described, which accurately accounts also for possible singularities in the deflection function. Data fits for the relevant quantities are provided for hydrogenic and helium species. The relevance of elastic neutral-ion collisions is illustrated by a numerical simulation of an ITER relevant divertor model.     

Collision operator for relativistic electrons in a cold gas of atomic particles

The collision operator of relativistic electrons with a cold gas of atomic particles is derived consistently taking into account elastic interactions, excitation of electron shells, and ionization. The creation of secondary electrons is described accurately. In the range of energies exceeding the binding energy of atomic electrons, the operator implicates only the angular scattering by nuclei and the ionization integral that automatically allows for scattering by atomic electrons. The collision operator used earlier for studying the kinetics of avalanches of relativistic runaway electrons is analyzed. A more exact operator derived in the present study is simpler in form and saves time in computer calculations.     

A Proof of Factorization Theorem of Drell–Yan Process at Operator Level

An alternative proof of factorization theorem for Drell–Yan process that works at operator level is presented in this paper. Contributions of interactions after the hard collision for such inclusive processes are proved to be canceled at operator level according to the unitarity of time evolution operator. After this cancellation, there are no longer leading pinch singular surface in Glauber region in the time evolution of electromagnetic currents. Effects of soft gluons are absorbed into Wilson lines of scalar-polarized gluons. Cancelation of soft gluons is attribute to unitarity of time evolution operator and such Wilson lines.     

Resonance blocking of traveling waves by a system of cracks in an elastic layer

Wave processes that occur in an elastic layer when waves traveling in it are diffracted by a system of horizontal cracks are investigated. Integral representations of wave fields are constructed in terms of the convolution of Green’s matrices and unknown jumps of displacements at the cracks. The displacement jumps are determined from the boundary integral equations, which are obtained from the initial boundary-value problem with the boundary conditions at crack faces being satisfied. The spectrum of the integral operator is studied for different variants of mutual crack arrangement and is compared with the spectrum of the corresponding operators for individual cracks; the relationship between the spectrum and the blocking effects is analyzed. The possibility of obtaining an extended frequency band of waveguide blocking in the case of groups of cracks is demonstrated.     

Global Existence of Classical Solutions to the Fokker-Planck-BGK Equation

A kinetic model of the Fokker-Planck-Boltzmann equation is introduced by replacing the original Boltzmann collision operator with the Bhatnagar-Gross-Krook collision model (BGK collision model). This model equation, which we call the Fokker-Planck-BGK equation, has many physical features that the Fokker-Planck-Boltzmann equation possesses. We first establish an L ^∞ existence result for this equation, by which we construct the approximate solutions. Then, by means of the regularizing effects of the linear Fokker-Planck operator and L ^p estimates of local Maxwellians, we obtain some uniform estimates of the approximate solutions. Finally, combining those estimates and regularizing effects, we prove by a compactness argument that the equation has a global classical solution under rather general initial conditions. Supported by the Scientific Research Foundation of Huazhong University of Science and Technology (HUST-SRF).