Nonlocal Boltzmann Equation Research Articles

Simulating glow discharge positive column parameters in neon

The nonlocal self-consistent model for the low temperature plasma parameters in a glow discharge positive column is presented. The model implies relaxation method for the solving of the nonlocal Boltzmann equation for electron energy distribution function, continuity equation for ions distribution and Poisson equation for radial electric field. Calculations are provided for a cylindrical discharge tube used in the Plasma Kristall-4 experiments for neon as a plasma forming gas for the gas pressures of 30-70 Pa and discharge currents of 0.5-2.5 mA. Axial electric field strength, electron temperature and electron density are calculated as functions of the discharge current and the gas pressure. The results are in good agreement with the data provided by other authors.

Simulations of the glow discharge positive column parameters in helium

The self-consistent model for the positive column of a glow discharge is presented. The model is based on the simultaneous solution of the nonlocal Boltzmann equation for electron energy distribution function, continuity equation for ions and Poisson equation for electric field. Calculations were performed for a discharge tube similar to the Plasma Kristall-4 experiments for plasma forming gas helium for the range of pressures of 20-70 Pa and the range of discharge currents of 0.5-2.5 mA. Axial electric field strength, electron temperature and electron density dependences on the discharge current and the gas pressure are obtained. The results show a good correlation with the data obtained by the other authors.

Local conservation laws in ultracold Fermi systems with time-dependent interaction potential.

In the context of an ultracold Fermi gas, we derive conservation laws for mass, energy and momentum based on a generalized nonlocal Boltzmann equation with gradient corrections in the collision integral. The corrections are expressed in terms of effective collision duration, particle displacement and changes of total momentum and energy. Their origin is in the in-medium T matrix. Using variations of the optical theorem, we show that in the collision integral the particle-hole symmetry can be recast into a form of collision symmetry amenable to semiclassical simulation. Pauli-blocked collisions are distinguished from Bose-stimulated nondissipative ones; the latter are not present in the absence of gradient corrections. Consolidating with the microscopic theory, we extract local conservation laws for a general time-dependent interaction potential, and demonstrate how both types of collisions affect densities and flows of conserving quantities. Comparison is made with the approach of Nozières and Schmitt-Rink in the limit of thermal equilibrium. Under approximations used for normal-state ultracold Fermi gases interacting via Feshbach resonances we demonstrate the effect of the collision delay on the shear viscosity.

Experimental and numerical study of a low-frequency ferromagnetic enhanced inductively coupled plasma

Low-frequency ferromagnetic-enhanced inductively coupled plasma (FMICP) has been investigated under conditions typical for large-scale plasma processing. Radial profiles of ion density and plasma floating potential were determined with a Langmuir probe. A self-consistent radial kinetic model of argon FMICP was developed, based on the simultaneous solution of a non-local Boltzmann equation for the electron energy distribution function, balance equations for the ion and metastable argon atom densities, the thermal balance equation and the Poisson equation for a self-consistent radial electric field. A satisfactory agreement between the numerical and experimental results was found that confirms the validity of the presented approach to the description of the FMICP.

Experimental and numerical study of a dust cloud formation in the stratified positive column of a dc glow discharge in helium

The experimental and theoretical investigations of the formation of dust particle clouds in the stratified positive column of a dc glow discharge in helium were performed. The size and shape of a dust cloud that levitated in the strong electric field of the striations in a vertically oriented discharge tube were measured under different helium pressures. Axial electric field strength was also experimentally estimated. A model for radial distributions of all dusty plasma parameters of the positive column of the dc glow discharge was developed to describe the obtained experimental results. The model is based on the solution of a non-local Boltzmann equation for an electron energy distribution function, drift-diffusion equations for ions and dust particles, and a Poisson equation for a self-consistent radial electric field. The experimental and calculated results show that the size of the dust cloud decreases with the increase in the gas pressure, and the axial electric field strength pressure dependencies have minima. At low gas pressures, the dust particle charge number density increases and exceeds the electron density that strongly influences the plasma parameters.

Time-nonlocal kinetic equations, jerk and hyperjerk in plasmas and solar physics

The simulation and analysis of nonlocal effects in fluids and plasmas is an inherently complicated problem due to the massive breadth of physics required to describe the nonlocal dynamics. This is a multi-physics problem that draws upon various miscellaneous fields, such as electromagnetism and statistical mechanics. In this paper we strive to focus on one narrow but motivating mathematical way: the derivation of nonlocal plasma-fluid equations from a generalized nonlocal Liouville derivative operator motivated from Suykens's nonlocal arguments. The paper aims to provide a guideline toward modeling nonlocal effects occurring in plasma-fluid systems by means of a generalized nonlocal Boltzmann equation. The generalized nonlocal equations of fluid dynamics are derived and their implications in plasma-fluid systems are addressed, discussed and analyzed. Three main topics were discussed: Landau damping in plasma electrodynamics, ideal MHD and solar wind. A number of features were revealed, analyzed and confronted with recent research results and observations.

Low-pressure low-frequency inductive discharge with ferrite cores for large-scale plasma processing

Electrophysical characteristics of a low–frequency (100 kHz) low–pressure (10–100 Pa) argon inductive discharge with ferrite cores have been investigated for discharge currents of 1–50 A, discharge chamber diameter of 230 mm. The dependencies of electric field strength on the argon pressure and discharge current were measured. Radial profiles of electron density and temperature were determined with double electric probes. Experimental results were compared with numerical results obtained by a self-consistent kinetic model of the low-frequency ICP, based on the simultaneous solution of a non-local Boltzmann equation for electron energy distribution function and balance equations for the metastable argon atoms density and gas temperature. Comparison of numerical and experimental results is presented.

Influence of ion drag force on radial distribution of dust particles and void formation in a DC glow discharge

A model of the positive column of a DC glow discharge in argon with dust particles is presented. The model is based on the solution of the non-local Boltzmann equation for the electron energy distribution function (EEDF), the drift-diffusion equations for ions, and the Poisson equation for a self-consistent electric field. The radial distribution of dust particles density in a dust cloud is calculated according to an equilibrium Boltzmann distribution in the electrostatic field and under the action of the ion drag force. It is shown that if in the center of the discharge tube the ion drag force is larger than the electrostatic force, a dust-free region (void) in the central part of the dust cloud is formed. Otherwise (for a smaller ion drag force) the dust cloud is formed around the axis of the discharge tube without a void. In both cases, in the region of the dust cloud the ionization and recombination rates become almost equal to each other, and the radial component of the electric field is expelled from the dust cloud.

Dust particle radial confinement in a dc glow discharge

A self-consistent nonlocal model of the positive column of a dc glow discharge with dust particles is presented. Radial distributions of plasma parameters and the dust component in an axially homogeneous glow discharge are considered. The model is based on the solution of a nonlocal Boltzmann equation for the electron energy distribution function, drift-diffusion equations for ions, and the Poisson equation for a self-consistent electric field. The radial distribution of dust particle density in a dust cloud was fixed as a given steplike function or was chosen according to an equilibrium Boltzmann distribution. The balance of electron and ion production in argon ionization by an electron impact and their losses on the dust particle surface and on the discharge tube walls is taken into account. The interrelation of discharge plasma and the dust cloud is studied in a self-consistent way, and the radial distributions of the discharge plasma and dust particle parameters are obtained. It is shown that the influence of the dust cloud on the discharge plasma has a nonlocal behavior, e.g., density and charge distributions in the dust cloud substantially depend on the plasma parameters outside the dust cloud. As a result of a self-consistent evolution of plasma parameters to equilibrium steady-state conditions, ionization and recombination rates become equal to each other, electron and ion radial fluxes become equal to zero, and the radial component of electric field is expelled from the dust cloud.

Non-local effects in a stratified glow discharge with dust particles

The work is aimed at describing non-local effects in the positive column of a low-pressure stratified dc glow discharge in argon with dust particles in a vertical cylindrical discharge tube. Numerical calculations of plasma parameters in the axis of the discharge tube were performed with the help of a hybrid model based on the solution of a non-local Boltzmann equation for electron energy distribution function (EEDF). Axial distributions of optical emission from striations with dust particles were measured experimentally. Negatively charged dust particles in a low-pressure stratified gas discharge should levitate at the anode-side branch of an electric field distribution above its maximum. At the same time the experiments showed that the dust particles levitate at the cathode side of a stratum. This paradox is explained by the fact that in a low-pressure striated discharge the optical emission distribution is displaced relative to the electric field distribution that was shown both by numerical simulations and experimental measurements.

Squeezing mode in nuclear collisions

The time dependent Schroedinger equation is solved analytically for a simplified model of moving infinite walls. A new knock-out mode is described which might occur during heavy ion collisions. The outer shell-nucleons are ionised due to the increase of level energy when two nuclei are approaching fast enough. This is analogous to the Mott effect but in contrast occurs only if the reaction time is short enough that no common ionisation threshold in the compound system is established. To demonstrate this pure nonequilibrium effect a simulation of realistic heavy ion collision by a nonlocal Boltzmann equation is performed.

NONLOCAL KINETIC THEORY

The short time behavior of a disturbed system is influenced by off-shell motion and characterized by the reduced density matrix possessing high energetic tails. After this short time regime the time evolution is controlled by small gradients. This leads to a nonlocal Boltzmann equation for the quasiparticle distribution and a functional relating the latter one to the reduced density matrix. The nonlocalities are presented as time and space shifts arising from gradient expansion and are leading to virial corrections in the thermodynamical limit.

Nonlocal kinetic theory. II. Transport and virial corrections

After the short time regime the time evolution is controlled by small gradients. This leads to a nonlocal Boltzmann equation for the quasiparticle distribution and a functional relating the latter one to the reduced density matrix. The nonlocalities are presented as time and space shifts arising from gradient expansion and are leading to virial corrections in equilibrium.

Analytic model of the hollow cathode effect

An analytic model of the hollow cathode effect (HCE) is suggested. The model is based on an approximate kinetic treatment of fast electrons which are responsible for ionization processes in the cathode cavity. The difference between exponential electron multiplication in the cathode sheath and linear multiplication in the plasma is shown to be responsible for the HCE. The HCE takes place when the fast electrons from one side of the cathode sheath are capable of reaching the opposite side of the sheath and produce a substantial fraction of the total ionization in the sheath. Simple analytic solutions of the non-local Boltzmann equation in a spatially inhomogeneous electric field are obtained for two different cases which correspond to the onset of the HCE and to the well developed oscillatory motion of electrons. The main approximations which lead to analytical solutions are a continuous slowing down approximation and assumption of predominantly forward electron scattering. The calculated electron distribution functions are in good agreement with Monte Carlo (MC) numerical simulations. The electron multiplication coefficient is obtained and used for a quantitative description of the HCE.

Conversion between kinetic energy and potential energy in the classical nonlocal Boltzmann equation

An important property of the classical Boltzmann equation is that kinetic energy is conserved. This is closely connected to the fact that the Boltzmann equation describes the nonequilibrium properties of an “ideal” gas. Generalizations of the Boltzmann equation to higher density involve, among other things, allowing the colliding particles to be at different positions. This spatial nonlocality is known to contribute to the density corrections of gas transport properties. For soft potentials such a spatial separation of the particles also leads to a conversion between kinetic and potential energy. In evaluating these effects the classical dynamics of the whole collision trajectory must be taken into account, involving also the time for the collision process. The resulting time nonlocality has usually been reinterpreted in terms of a spatial nonlocality. However, for a homogeneous system this is not possible and only the time nonlocality remains, this then being responsible for the conversion between kinetic and potential energy. This paper aims to clarify these properties of the nonlocal corrections to the classical mechanical Boltzmann collision term. Comments on the corresponding problem for the quantum Boltzmann equation are also made.

On a unified treatment of kinetics and diffusion, and a connection to a nonlocal boltzmann equation

AbstractFick's law of diffusion has been generalized to include kinetic processes, the transport term of the Boltzmann equation, and nonlocal interaction processes. It is shown that the collision interaction term can be obtained by the introduction of a quantum stochastic potential equation. Some approximations of a nonlocal Boltzmann equation can be solved exactly. The solutions can be applied to problems of molecular pattern in biology.