Steady-state Boltzmann Equation Research Articles

Study of Electron Energy Distribution Function, Transport Parameters and Instability Analysis of Argon Plasma confined by a Dipole Magnet

This theoretical study is to investigate production and properties of plasma confined by a dipole magnet. A cylindrical permanent magnet (N40 grade) having a surface magnetic field of nearly 1Tesla is simulated to create a dipole magnetic field. Plasma is created by electron cyclotron resonance heating. Heating is accomplished using microwaves operating in continuous mode at 2.45 GHz and electron cyclotron resonance. The investigation is performed in systems of cylindrical geometries at varying pressures (0.5 mTorr – 1.5 mTorr). The electron temperature lies in the range of 1 - 5 eV, is hotter near the magnets and follows a downstream fashion throughout the region.Electron Energy Distribution Function (EEDF) growth investigation throughout the region was done to understand the role of magnetic field confinement. The EEDF was calculated numerically from steady state Boltzmann equation, considering plasma follows druyvesteyn distribution. Followed by analysing transport parameters such as diffusion coefficient, drift velocity, mobility, and recombination rates in plasma. A comparative study for all plots gave been done with respect to pressure. The instability analysis of elastic collisional frequency with inelastic collisions, real frequency and growth factor with wave vector has been also listed in the study.

Testing the predictions of axisymmetric distribution functions of galactic dark matter with hydrodynamical simulations

Signal predictions for galactic dark matter (DM) searches often rely on assumptions regarding the DM phase-space distribution function (DF) in halos. This applies to both particle (e.g. p-wave suppressed or Sommerfeld-enhanced annihilation, scattering off atoms, etc.) and macroscopic DM candidates (e.g. microlensing of primordial black holes). As experiments and observations improve in precision, better assessing theoretical uncertainties becomes pressing in the prospect of deriving reliable constraints on DM candidates or trustworthy hints for detection. Most reliable predictions of DFs in halos are based on solving the steady-state collisionless Boltzmann equation (e.g. Eddington-like inversions, action-angle methods, etc.) consistently with observational constraints. One can do so starting from maximal symmetries and a minimal set of degrees of freedom, and then increasing complexity. Key issues are then whether adding complexity, which is computationally costy, improves predictions, and if so where to stop. Clues can be obtained by making predictions for zoomed-in hydrodynamical cosmological simulations in which one can access the true (coarse-grained) phase-space information. Here, we test an axisymmetric extension of the Eddington inversion to predict the full DM DF from its density profile and the total gravitational potential of the system. This permits to go beyond spherical symmetry, and is a priori well suited for spiral galaxies. We show that axisymmetry does not necessarily improve over spherical symmetry because the (observationally unconstrained) angular momentum of the DM halo is not generically aligned with the baryonic one. Theoretical errors are similar to those of the Eddington inversion though, at the 10–20% level for velocity-dependent predictions related to particle DM searches in spiral galaxies. We extensively describe the approach and comment on the results.

On the Test Particle Monte-Carlo method to solve the steady state Boltzmann equation, the congruity of its results with experiments and its potential for shared memory parallelism

The Test Particle Monte Carlo is a known method to solve the steady state Boltzmann particle transport equation in rarefied gas systems. A description of the Test Particle Monte-Carlo procedure is outlined and the accuracy of this method is investigated by analyzing its consistency to experimental data of steady state effusions in isothermal systems. Computational results present deviations from the experiments that are at most 6.7%. After which, this paper investigates the potential of this method when it is parallelized using multi-core CPUs with shared memory for large Knudsen numbers. Scalability is expected since the method relies on a mean particle field that renders particle trajectories nearly independent from one another, therefore reducing data communication between processing cores. The mean particle distribution field helps linearize the collision term of the Boltzmann equation. Shared Memory Parallelism is an interesting feature when combined with distributed memory parallelism for design optimization of vacuum systems.

Numerical simulations of capsule deformation using a dual time-stepping lattice Boltzmann method.

In this work a quasisteady, dual time-stepping lattice Boltzmann method is proposed for simulation of capsule deformation. At each time step the steady-state lattice Boltzmann equation is solved using the full approximation storage multigrid scheme for nonlinear equations. The capsule membrane is modeled as an infinitely thin shell suspended in an ambient fluid domain with the fluid structure interaction computed using the immersed boundary method. A finite element method is used to compute the elastic forces exerted by the capsule membrane. Results for a wide range of parameters and initial configurations are presented. The proposed method is found to reduce the computational time by a factor of ten.

Acceleration for microflow simulations of high-order moment models by using lower-order model correction

We study the acceleration of steady-state computation for microflow, which is modeled by the high-order moment models derived recently from the steady-state Boltzmann equation with BGK-type collision term. By using the lower-order model correction, a novel nonlinear multi-level moment solver is developed. Numerical examples verify that the resulting solver improves the convergence significantly thus is able to accelerate the steady-state computation greatly. The behavior of the solver is also numerically investigated. It is shown that the convergence rate increases, indicating the solver would be more efficient, as the total levels increases. Three order reduction strategies of the solver are considered. Numerical results show that the most efficient order reduction strategy would be ml−1=⌈ml/2⌉.

Kinetic study on non-thermal volumetric plasma decay in the early afterglow of air discharge generated by a short pulse microwave or laser

This paper reports a kinetic study on non-thermal plasma decay in the early afterglow of air discharge generated by short pulse microwave or laser. A global self-consistent model is based on the particle balance of complex plasma chemistry, electron energy equation, and gas thermal balance equation. Electron-ion Coulomb collision is included in the steady state Boltzmann equation solver to accurately describe the electron mobility and other transport coefficients. The model is used to simulate the afterglow of microsecond to nanosecond pulse microwave discharge in N2, O2, and air, as well as femtosecond laser filament discharge in dry and humid air. The simulated results for electron density decay are in quantitative agreement with the available measured ones. The evolution of plasma decay under an external electric field is also investigated, and the effect of gas heating is considered. The underlying mechanism of plasma density decay is unveiled through the above kinetic modeling.

A nonlinear multigrid steady-state solver for 1D microflow

We develop a nonlinear multigrid method to solve the steady state of 1D microflow, which is modeled by the high order moment system derived recently for the steady-state Boltzmann equation with ES-BGK collision term. The solver adopts a symmetric Gauss–Seidel iterative scheme nested by a local Newton iteration on grid cell level as its smoother. Numerical examples show that the solver is insensitive to the parameters in the implementation thus is quite robust. It is demonstrated that expected efficiency improvement is achieved by the proposed method in comparison with the direct time-stepping scheme.

Equilibrium statistical mechanics for self-gravitating systems: local ergodicity and extended Boltzmann-Gibbs/White-Narayan statistics

The long-standing puzzle surrounding the statistical mechanics of self-gravitating systems has not yet been solved successfully. We formulate a systematic theoretical framework of entropy-based statistical mechanics for spherically symmetric collisionless self-gravitating systems. We use an approach that is very different from that of the conventional statistical mechanics of short-range interaction systems. We demonstrate that the equilibrium states of self-gravitating systems consist of both mechanical and statistical equilibria, with the former characterized by a series of velocity-moment equations and the latter by statistical equilibrium equations, which should be derived from the entropy principle. The velocity-moment equations of all orders are derived from the steady-state collisionless Boltzmann equation. We point out that the ergodicity is invalid for the whole self-gravitating systems, but it can be re-established locally. Based on the local ergodicity, using Fermi-Dirac-like statistics, with the nondegenerate condition and the spatial independence of the local microstates, we rederive the Boltzmann-Gibbs entropy. This is consistent with the validity of the collisionless Boltzmann equation, and should be the correct entropy form for collisionless self-gravitating systems. Apart from the usual constraints of mass and energy conservation, we demonstrate that the series of moment or virialization equations must be included as additional constraints on the entropy functional when performing the variational calculus; this is an extension to the original prescription by White & Narayan. Any possible velocity distribution can be produced by the statistical-mechanical approach that we have developed with the extended Boltzmann-Gibbs/White-Narayan statistics. Finally, we discuss the questions of negative specific heat and ensemble inequivalence for self-gravitating systems.

A steady-state lattice Boltzmann model for incompressible flows

In this paper, we propose a steady-state lattice Boltzmann equation which is independent of time for steady incompressible flows. On the basis of the steady-state lattice Boltzmann equation, we find a way of modeling some steady-state problems using the lattice Boltzmann model. In further study, we investigate steady-state incompressible flows with the method of the higher-order moment of equilibrium distribution functions and a series of partial differential equations for different space scales. By assuming ρ = constant , we obtain the momentum equation for incompressible steady-state flow. The numerical results show that the new method is effective.

The main properties of microwave argon plasma at atmospheric pressure

Plasma torch sustained by surface wave at atmospheric pressure is theoretically studied by means of 1D model. A steady-state Boltzmann equation in an effective field approximation coupled with a collisional-radiative model for high-pressure argon discharge is numerically solved together with Maxwell's equations for an azimuthally symmetric TM surface wave. The axial dependences of the electrons, excited atoms, atomic and molecular ions densities as well as the electron temperature, the mean power per electron and the effective electron-neutral collision frequency are determined. A strong dependence of the plasma properties on the discharge conditions and the gas temperature is obtained.

Inhomogeneous microwave discharge in oxygen

Results are given of one-dimensional quasi-static simulation of steady microwave discharge in a system of electrodes with spherical symmetry in oxygen at pressures of 1 to 8 torr. The model includes the equation for electric field intensity in quasi-static approximation, Poisson equation, balance equations defining the kinetics of charged and neutral particles of plasma, and steady-state homogeneous Boltzmann equation for electrons. The well-known analytical expression for vibrational distribution of molecules, derived in diffusion approximation, is used for including the processes which involve vibrationally excited particles. It is demonstrated that the presence of negative ions does not result in differences of the electrodynamics of discharge from those of discharge in nitrogen. The composition of heavy component of plasma is investigated and, in particular, it is demonstrated that the concentration of molecules in the O2(a1Δ) state may be as high as 20%.

Addendum to “Entropy in Nonequilibrium Statistical Mechanics”

In a previous paper I have proposed a principle of maximum entropy for nonequilibrium steady states. However, the argument based on the H theorem in Introduction is not appropriate, because there is net outflow of entropy in nonequilibrium steady states. Thus, the principle is to be regarded as a pure conjecture. It should also be noted that the principle is identical in the dilute classical limit to the information theory developed by Jou et al. Kim and Hayakawa tested the information theory for dilute classical hard-core and Maxwell molecules with heat conduction by comparing its predictions with those from the steady-state Boltzmann equation. They found that the two approaches can yield identical results only up to the first-order deviations from the local equilibrium generally, and discrepancies emerge at the second order for all the physical quantities except entropy. This implies that the above principle cannot be valid beyond the first order. It is possible, however, that it generally holds within this lowest order, as exemplified by Kim and Hayakawa and also by ref. 4 on Rayleigh–Benard convection. Another support to this conjecture is that the nonequilibrium statistical operator of Zubarev corresponds to an extremum of the information entropy. Note finally that, unlike the linear response theory which is an expansion from the global equilibrium, the first-order expansion from the local equilibrium can describe a wide range of nonequilibrium nonlinear phenomena, as may be realized by the fact that the Navier–Stokes equations belong to this order of approximation. We certainly need further investigations to clarify the applicable range of the conjecture.

Modelling of microwave sustained capillary plasma columns at atmospheric pressure

In this work we present a model of argon microwave sustained discharge at high pressure (1 atm), which includes two self-consistently linked parts - electrodynamic and kinetic ones. The model is based on a steady-state Boltzmann equation in an effective field approximation coupled with a collisional-radiative model for high-pressure argon discharge numerically solved together with Maxwell's equation for an azimuthally symmetric TM surface wave and wave energy balance equation. It is applied for the purpose of theoretical description of the discharge in a stationary state. The phase diagram, the electron energy distribution function as well as the dependences of the electron and heavy particles densities and the mean input power per electron on the electron number density and wave number are presented.

Multigrid solution of the steady-state lattice Boltzmann equation

Efficient solution strategies for the steady-state lattice Boltzmann equation are investigated. Stable iterative methods for the linearized lattice Boltzmann equation are formulated based on the linearization of the lattice Boltzmann time-stepping procedure. These are applied as relaxation methods within a linear multigrid scheme, which itself is used to drive a Newton solver for the full non-linear problem. Although the linear multigrid strategy provides rapid convergence, the cost of a linear residual evaluation is found to be substantially higher than the cost of evaluating the non-linear residual directly. Therefore, a non-linear multigrid approach is adopted, which makes use of the non-linear LBE time-stepping scheme on each grid level. Rapid convergence to steady-state is achieved by the non-linear algorithm, resulting in one or more orders of magnitude increase in solution efficiency over the LBE time-integration approach. Grid-independent convergence rates are demonstrated, although degradation with increasing Reynolds number is observed, as in the case of the original LBE time-stepping scheme. The multigrid solver is implemented in a modular fashion by calling an existing LBE time-stepping routine, and delivers the identical steady-state solution as the original LBE time-stepping approach.

Two-dimensional nonlinear nonequilibrium kinetic theory under steady heat conduction

The two-dimensional steady-state Boltzmann equation for hard-disk molecules in the presence of a temperature gradient has been solved explicitly to second order in density and the temperature gradient. The two-dimensional equation of state and some physical quantities are calculated from it and compared with those for the two-dimensional steady-state Bhatnagar-Gross-Krook equation and information theory. We have found that the same kind of qualitative differences as the three-dimensional case among these theories still appear in the two-dimensional case.

Surface effects on nanowire transport: a numerical investigation using the Boltzmannequation

A direct numerical solution of the steady-state Boltzmann equation in a cylindricalgeometry is reported. Finite-size effects are investigated in large semiconducting nanowiresusing the relaxation-time approximation. A nanowire is modelled as a combination of aninterior with local transport parameters identical to those in the bulk, and a finite surfaceregion across whose width the carrier density decays radially to zero. The roughness of thesurface is incorporated by using lower relaxation times there than in the interior.An argument supported by our numerical results challenges a commonly used zero-widthparametrization of the surface layer (Chambers 1950 Proc. R. Soc. A 202 378). In thenon-degenerate limit, appropriate for moderately doped semiconductors, a finite surfacewidth model does produce a positive longitudinal magneto-conductance, in agreementwith existing theory of Chambers. However, the effect is seen to be quite small(a few per cent) for realistic values of the wire parameters even at the highestpractical magnetic fields. Physical insights emerging from the results are discussed.

Test of Information Theory on the Boltzmann Equation

We examine information theory using the steady-state Boltzmann equation. In a nonequilibrium steady-state system under steady heat conduction, the thermodynamic quantities from information theory are calculated and compared with those from the steady-state Boltzmann equation. We have found that information theory is inconsistent with the steady-state Boltzmann equation.

Experimental and theoretical studies of the electron temperature in nitrogen afterglow

In this paper, results of joint experimental and theoretical studies of the electron temperature in nitrogen afterglow at pulse-periodical excitation are presented. Electron energy distribution function (EEDF) in an afterglow of a pulsed direct current discharge has been measured by means of a time-resolved Langmuir probe technique. Electron concentration, vibrational temperature, and population of lower metastable electronic state of N/sub 2/ molecules have also been experimentally estimated at different delays after the discharge pulse. The results show that electron temperature in afterglow decreases with time, while the vibrational temperature remains almost constant. The EEDF has been calculated numerically from a steady-state Boltzmann equation, taking into account electron-electron collisions as well as superelastic collisions with vibrationally and electronically excited molecules. The vibrational distribution function was found numerically by solving a system of kinetic equations. Calculations show that the vibrational distribution function weakly varies within a cycle and is controlled by an average discharge power. Electron temperature in nitrogen afterglow for given populations of vibrational levels and of lower electronic level essentially depends on the electron concentration. Finally, a comparison of the theoretical and experimental results is performed.

Kinetic Theory of a Dilute Gas System under Steady Heat Conduction

The velocity distribution function of the steady-state Boltzmann equation for hard-core molecules in the presence of a temperature gradient has been obtained explicitly to second order in density and the temperature gradient. Some thermodynamical quantities are calculated from the velocity distribution function for hard-core molecules and compared with those for Maxwell molecules and the steady-state Bhatnagar-Gross-Krook(BGK) equation. We have found qualitative differences between hard-core molecules and Maxwell molecules in the thermodynamical quantities, and also confirmed that the steady-state BGK equation belongs to the same universality class as Maxwell molecules.

Contributions of steady heat conduction to the rate of chemical reaction

We have derived the effect of steady heat flux on the rate of chemical reaction based on the line-of-centers model using the explicit velocity distribution function of the steady-state Boltzmann equation for hard-sphere molecules to second order. It is found that the second-order velocity distribution function plays an essential role for the calculation of it. We have also compared our result with those from the steady-state Bhatnagar–Gross–Krook (BGK) equation and information theory, and found no qualitative differences among them.