Maxwellian Form Research Articles

Lower hybrid drift instability in nonthermal plasmas

Lower hybrid drift instability (LHDI) is driven by the cross-field current and operates in the vicinity of the lower-hybrid frequency, between the ion- and electron-gyro frequencies, and with wavelengths between the electron and ion thermal gyro radii. The free energy source that drives this instability resides in the density gradient associated with an inhomogeneous plasma. The existing literature on LHDI assumes that the charged particle distribution function is given by a Maxwellian form, but the space plasma is pervasively observed to feature nonthermal characteristics. This paper extends the theory of LHDI to nonthermal plasmas. The generalized theory of LHDI is, thus, applicable to various space plasma environments characterized by nonthermal plasma velocity distribution functions.

Maxwellian mirages in general relativity

Maxwellian approximations to linear general relativity are revisited in light of recent results on the degrees of freedom (DoF) in the linear gravitational field. The well-known Maxwellian formalism obtained in harmonic coordinates is compared with a Maxwellian formalism obtained under a coordinate choice where each of the metric components corresponds to each of the coordinate-invariant DoF of the linear gravitational field. The coordinate freedom of general relativity can be exploited to cast the field equations into Maxwellian form, but such forms can be mere mirages of the coordinate choice—mirages such as vector gravitational waves. A coordinate choice that yields perfectly-Maxwellian field equations, will yield a force equation that is not Lorentzian. If field definitions are chosen to obtain Lorentz-like terms in the force equation, then Maxwellian forms are compromised in the field equations. Many treatments of gravito-electromagnetism will make inconsistent ordering choices between the field equations and force equations, or else truncate terms of relevant order from the force equation. Often such mistakes reflect an attempt to force exact Maxwellian analogs simultaneously in both the field equations and the force equation, with the result that terms dropped are as large as those kept.

Time and space-resolved experimental investigation of the electron energy distribution function of a helium capacitive discharge at atmospheric pressure

Using a collisional radiative model coupled with optical emission spectroscopy (OES) of helium n = 3 levels, the electron temperature (Te) of an atmospheric-pressure capacitively coupled radiofrequency (AP-CCRF) discharge in helium is determined with space and time resolution. When the AP-CCRF discharge is sustained in the mode, Te varies from 0.2 to 7.2 eV. In this case, high values of Te (>5 eV) occur only during a brief instant (<10 ns) in the high-voltage sheath. When the AP-CCRF discharge is sustained in the mode, Te varies from 0.3 to 0.4 eV during the complete cycle. The physical meaning of these electron temperatures are then analyzed by considering possible departure from the Maxwellian electron energy distribution function (EEDF). As a first approximation to non-Maxwellian distribution functions, a two-parameter EEDF was used to fit the OES data. This approach yields an overpopulation of high energy electrons with respect to the Maxwellian form in the mode and the opposite trend in the mode.

Ion energy distribution in extremely high electric field discharges

Singly charged ion energy distribution functions (IEDFs) have been investigated by the particle-in-cell/Monte Carlo method in practical discharges at extremely high reduced electric field E/n, i.e. the ratio of electric field and gas density, up to 700 kTd in helium. It is found in high E/n regime that IEDFs deviate from Maxwellian form due to complex processes: (a) ions undergo a transition to runaway regime, (b) the copious energetic fast neutrals generated by charge exchange produce slow ions via ionization collisions, which load in the low-energy part of IEDF, as well as commonly assumed energy-dependent charge exchange cross section. For E/n > 100 kTd, the ratio of the volumetric ionization rate and charge exchange rate exceeds 10% due to heavy particles impact ionization. The ion mean energy close to the cathode is approximately reduced by in comparison with the value obtained by the reduced model with only the charge exchange collision considered. Charge production attributed to fast-neutral impact ionization not only causes the decrease of ion mean energy, but also enhances ion non-equilibrium.

Lattice Boltzmann models based on half-range Gauss–Hermite quadratures

We discuss general features of thermal lattice Boltzmann models based on half-range Gauss quadratures, specialising to the half-range Gauss-Hermite and Gauss-Laguerre cases. The main focus of the paper is on the construction of high order half-range Hermite lattice Boltzmann (HHLB) models. The performance of the HHLB models is compared with that of Laguerre lattice Boltzmann (LLB) and full-range Hermite lattice Boltzmann (HLB) models by conducting convergence tests with respect to the quadrature order on stationary profiles of the particle number density, macroscopic velocity, temperature and heat fluxes in the two-dimensional Couette flow. The Bhatnagar-Gross-Krook (BGK) collision term is used throughout the paper. To reduce the computational costs of the numerical simulations, we use mixed lattice Boltzmann models, constructed using different quadrature methods on each Cartesian axis. For Kn ? 0.01 , the HLB models require the least number of velocities to satisfy our convergence test. When Kn ? 0.05 , the HLB models are outperformed in terms of number of velocities employed by both the LLB and the HHLB models. Moreover, we find that the HHLB models require less quadrature points than the LLB models at all tested values of Kn, which we attribute to the Maxwellian form of the weight function for the half-range Hermite polynomials.

Calculation of equation of state of a material mixture

The problem of calculating the equation of state (EOS) of a material mixture often comes from fluid-dynamical system containing multiple materials. Generally, the EOS of a material mixture is a system of nonlinear equations which are usually solved by the tabular method and the Newton iterative method. However, the former has poor accuracy, and the later has a finite radius of convergence and hence will converge only if the initial guess is sufficiently close to the final solution. So, a procedure different from the above two method is presented for calculating the EOS of a material mixture whose constituents are in pressure equilibrium and temperature equilibrium. An imbedding method is used to determine the constituent partial thermodynamic variables subjected to the constraints that the total volume and energy of mixture and the constituent mass fractions are specified. The imbedding method has a large radius of convergence, introducing a parameter defined in the interval [0, 1] and a system of imbedding equations which is linearly composed of the to-be-solved EOS of a material mixture and the easy-to-solve EOS of a material mixture. While the parameter changes continuously from 0 to 1, the imbedding method continuously changes the solution of the to-be-solved EOS which the easy-to-solve EOS of a material mixture is continuously converted into. The system of imbedding equations can be changed into a system of ordinary differential form by taking the parameter as independent variable, easily solved by a matured computational method such as trapezoidal rule. By using thermodynamic formulae, two equations in the generalized Maxwellian form are obtained, relating respectively the pressure rate and temperature rate to the strain rate and the constituent mass fraction rate. Finally, the computational method is verified by calculating the EOS of various mass fractions of lead and tin mixture.

On the non-Gaussian corrections in the self dynamics of semi-quantum fluids

Abstract This paper is devoted to the study of the limits of the well-known Gaussian approximation in the self dynamics of quantum systems. After introducing the basic formalism and shortly reviewing the methods used in classical systems to apply corrections to the Gaussian approximation, an extension to quantum fluids is devised, with a particular interest in the so-called semi-quantum fluids, i.e. those in which the single particle momentum distribution approximately retains its Maxwellian form (but not its classical width). In this case a detailed correction scheme for both the short- and the long-time behaviors of the intermediate scattering function is proposed. Subsequently, a practical test of this approach is performed on a high resolution neutron scattering spectrum derived from liquid parahydrogen at T = 14.1 K. Extracting the spectral deviations from the Gaussian approximation with the help of an accurate centroid molecular dynamics simulation, we are able to describe them precisely and to derive the first two correction coefficients in this system by means of a simple fitting procedure. These experimental findings confirm the validity of our approach and show that a description of the self dynamics beyond the Gaussian approximation is necessary even in simple liquids affected by mild quantum effects.

Maxwellian theory of gravitational waves and their mechanical properties

We present a theory in Maxwellian form for gravitational waves in a flat background. This requires us to identify the gravitational analogues of the electric and magnetic fields for light. An important novelty, however, is that our analogues are not vector fields but rather rank-two tensor fields; in place of a three-component vector at each point in space, as in electromagnetism, our fields are three by three symmetric matrices at each point. The resulting Maxwell-like equations lead directly to a Poynting theorem for the local energy density associated with a gravitational wave and to associated local properties including densities of momentum and angular momentum.

A continuum constitutive model and computational method of explosive detonation

Some basic thermodynamic relationships are used to develop a theroretical framework for modeling the detonation in explosives, on the assumption that explosive and detonation product are in a local thermodynamic equilibrium state, i.e., their pressures and temperatures are identical. Using this framework, a continuum constitutive model for explosive detonation is composed of a group of ordinary differential equations including the state equations of explosive and its product, simple mixing law, reaction rate equation and energy conservation equation, which are easily solved by a mature computational method such as trapezoidal rule. A group of nonlinear constitutive equations in a generalized Maxwellian form describe the relationship among the time evolution rates of pressure and temperature, the strain rate, and the chemical reaction rate. Coefficients appearing in the constitutive equations are determined only by parameters of the explosive and the product through using simple mixing rule. The continuum constitutive model and the corresponding computational method are verified by simulating the detonation behaviour of PBX9404 impacted by high velocity Cu flyer, and in the simulation the JWL equation of state for unreacted explosive and detonation product and the two-term Lee-Tarver reaction rate equation are adopted.

Tachyonic Cherenkov emission from Jupiterʼs radio electrons

Abstract Tachyonic Cherenkov radiation from inertial relativistic electrons in the Jovian radiation belts is studied. The tachyonic modes are coupled to a frequency-dependent permeability tensor and admit a negative mass-square, rendering them superluminal and dispersive. The superluminal radiation field can be cast into Maxwellian form, using 3D field strengths and inductions, and the spectral densities of tachyonic Cherenkov radiation are derived. The negative mass-square gives rise to a longitudinal flux component. A spectral fit to Jupiterʼs radio spectrum, inferred from ground-based observations and the Cassini 2001 fly-by, is performed with tachyonic Cherenkov flux densities averaged over a thermal electron population.

Deriving the velocity distribution of Galactic dark matter particles from the rotation curve data

The velocity distribution function (VDF) of the hypothetical weakly interacting massive particles (WIMPs), currently the most favored candidate for the dark matter in the Galaxy, is determined directly from the circular speed (``rotation'') curve data of the Galaxy assuming isotropic VDF. This is done by ``inverting''---using Eddington's method---the Navarro-Frenk-White universal density profile of the dark matter halo of the Galaxy, the parameters of which are determined by using the Markov chain Monte Carlo technique from a recently compiled set of observational data on the Galaxy's rotation curve extended to distances well beyond the visible edge of the disk of the Galaxy. The derived most-likely local isotropic VDF strongly differs from the Maxwellian form assumed in the ``standard halo model'' customarily used in the analysis of the results of WIMP direct-detection experiments. A parametrized (non-Maxwellian) form of the derived most-likely local VDF is given. The astrophysical ``g factor'' that determines the effect of the WIMP VDF on the expected event rate in a direct-detection experiment can be lower for the derived most-likely VDF than that for the best Maxwellian fit to it by as much as 2 orders of magnitude at the lowest WIMP mass threshold of a typical experiment.

Solar Wind Electron Acceleration via Langmuir Turbulence

The solar wind electrons observed at 1 AU are characterized by velocity distribution functions (VDF) that deviate from the Maxwellian form in a high energy regime. Such a feature is often modeled by a kappa distribution. In the present paper a self-consistent theory of quiet-time solar wind electrons that contain a power-law tail component, , fv a is discussed. These electrons are assumed to be in dynamic equilibrium with enhanced electrostatic fluctuations with peak frequency near the plasma frequency (i.e., the Langmuir turbulence). In order to verify the theoretical prediction, the solar wind electrons in the high-energy range known as the super-halo distribution detected by WIND and STEREO spacecraft are compared against the theoretical model where it was found that the theoretical power-law index is intermittent with regard to the observed range of indices, thus indicating that the turbulent equilibrium model of suprathermal solar wind electrons may be valid.

On the Ion Distribution Function in Degenerate Electron Plasmas

In highly-compressed plasmas as realized in inertial confinement fusion, the wave nature of electrons becomes noticeable and Pauli’s exclusion principle restricts the energy transition of electrons remarkably. Such a state is called “electron degeneracy”. In addition, the electron degeneracy may affect the energy distribution and temperature of coexisting ions through Coulombic ion-electron interaction. In order to evaluate these effects, we developed and solved the model equation for the distribution function of ions in degenerate electron plasmas. As a result, it is shown that the ion distribution function maintains a Maxwellian form at a temperature equal to that of degenerate electrons in thermal equilibrium because two effects of electron degeneracy—spectral hardening and Pauli blocking—counteract each other. Furthermore the electron degeneracy slows temperature relaxation between ions and electrons in non-thermal equilibrium. c

Optical diagnostics for characterization of electron energy distributions: argon inductively coupled plasmas

The assumption of thermodynamic equilibrium in low-temperature plasmas can lead to errors in determination of electron energy distribution functions (EEDFs) using optical diagnostics. Significant improvements in the accuracy of EEDFs determined using optical diagnostics on argon-containing inductively coupled plasmas (ICPs) are possible by fitting an analytic representation of the EEDF with two adjustable parameters (x and Tx) to recorded optical emission spectra. This so-called x-form of the EEDF captures the well-known suppression of the high-energy portion of the EEDF (compared with the Maxwellian form) observed in ICPs. A review of electron kinetics summarizes the physical mechanisms that lead to this form, and explains the weak dependence of x on operating pressure. For the 1–25 mTorr pressure range of argon-containing mixtures in the system examined, the fixed value x = 1.2 is found to represent the EEDF very well, followed by a near linear increase in x with pressure from x = 1.2 to 1.6 in the 25–50 mTorr range. Significantly improved agreement between predictions of effective electron temperature with a global model and both probe and optical measurements when a Maxwellian EEDF is replaced by the x-form further highlights the benefit of the x-form in improved accuracy in determining the EEDF with optical diagnostics.

Electron-ion Coulomb scattering and the electron Landau damping of Alfvén waves in the solar wind

[1] All Alfven waves in the solar wind have parallel electric fields, which enable Landau damping. The Alfven waves' Landau resonate with very low energy electrons; low-energy electrons are easily trapped in the Alfven waves, and at low energies electron-ion Coulomb scattering is very rapid. Analytic fluid theory and numerical solutions to the linear Vlasov equation are used to determine the properties of Alfven waves (and kinetic Alfven waves) in the solar wind. Electrostatic potentials associated with the waves' parallel electric fields are found to be relatively large. Owing to the large potentials, electrons over a broad range of velocities interact with the wave to produce Landau damping. Because of this broad range, linear Vlasov theory is invalid and the Landau-damping rates computed via linear Vlasov theory may not be accurate. Electron velocity diffusion owed to electron-ion Coulomb scattering is analyzed. Electron diffusion times in the Alfven wave Landau resonance are calculated and are found to be faster than wave periods and much faster than Landau-damping time scales. Coulomb collisions should prevent the electron distribution function from evolving away from a Maxwellian form, and thereby Coulomb collisions act to maintain Landau damping (although at a rate different than that given by linear Vlasov theory). Some complications of this Landau-damping picture arise from the interplanetary electric field competing with the Alfven wave parallel electric fields and from ion beams near the Landau resonance.

Optical emission measurements of electron energy distributions in low-pressure argon inductively coupled plasmas

Optical modeling of emissions from low-temperature plasmas provides a non-invasive technique to measure the electron energy distribution function (EEDF) of the plasma. While many models assume the EEDF has a Maxwell–Boltzmann distribution, the EEDFs of numerous plasma systems deviate significantly from the Maxwellian form. In this paper, we present an optical emission model for the Ar(3p54p → 3p54s) emission array which is capable of capturing details of non-Maxwellian distributions. Our model combines previously measured electron-impact excitation cross sections with Ar(3p54s) number density measurements and emission spectra. The model also includes corrections for radiation trapping of the Ar(3p54p → 3p54s) emission lines. Results obtained with this optical technique are compared with corresponding Langmuir probe measurements of the EEDF for Ar and Ar/N2 inductively coupled plasma systems operating under a wide variety of source conditions (1–25 mTorr, 20–1000 W, %N2 admixture). Both the optical emission method and probe measurements indicate the EEDF shapes are Maxwellian for low electron energies, but with depleted high energy tails.

Sensitivity of an exothermic chemical wave front to a departure from local equilibrium

We study the propagation of an exothermic chemical wave front in a reactive dilute gas and show that the particle velocity distribution departs from the Maxwellian form in the front zone. The analytical corrections to the balance equations for concentrations, temperature, and stream velocity induced by the departure from local equilibrium are derived from a perturbative solution of the Boltzmann equation. Our analytical predictions of the front properties, including its propagation speed, compare well with microscopic simulations of the particle dynamics.

Results from an Early Polarization Model Based on Maxwell's Invariant Multipole Form.

This paper reviews the cooperative water model of Campbell and Mezei based on the Maxwellian form of multipole interaction. The Maxwellian form is described, and the algorithms and software for their implementation in both disordered and ordered phases are presented, followed by the specifics of the model. The model has been used in a number of calculations on various water clusters, liquid, and crystal models. The results of these calculations are briefly summarized, and their implications, relevant to polarization model in general, are discussed.

Brownian motion of black holes in stellar systems with non-Maxwellian distribution of the field stars

AbstractA massive black hole at the center of a dense stellar system, such as a globular cluster or a galactic nucleus, is subject to a random walk due gravitational encounters with nearby stars. It behaves as a Brownian particle, since it is much more massive than the surrounding stars and moves much more slowly than they do. If the distribution function for the stellar velocities is Maxwellian, there is a exact equipartition of kinetic energy between the black hole and the stars in the stationary state. However, if the distribution function deviates from a Maxwellian form, the strict equipartition cannot be achieved.The deviation from equipartition is quantified in this work by applying the Tsallis q-distribution for the stellar velocities in a q-isothermal stellar system and in a generalized King model.

Nuclear astrophysical plasmas: ion distribution functions and fusion rates

This article illustrates how very small deviations from the Maxwellian exponential tail, while leaving unchanged bulk quantities, can yield dramatic effects on fusion reaction rates and discuss several mechanisms that can cause such deviations. Fusion reactions are the fundamental energy source of stars and play important roles in most astrophysical contexts. Since the beginning of quantum mechanics, basic questions were addressed such as how nuclear reactions occur in stellar plasmas at temperatures of few keV (1 keV ≈ 11.6× 10 ◦K) against Coulomb barriers of several MeV and what reactions or reaction networks dominate the energy production. It was soon realized that detailed answers to such questions involved not only good measurements or quantum mechanical understanding of the relevant fusion cross sections, but also the use of statistical physics for describing energy and momentum distributions of the ions and their screening [1]. Gamow understood that reacting nuclei penetrate Coulomb barriers by means of the quantum tunnel effect and Bethe successfully proposed the CNO and then the pp cycle as candidates for the stellar energy production: this description has been directly confirmed by several terrestrial experiments that have detected neutrinos produced by pp and CNO reactions in the solar core [2]. In the past only few authors (e.g., d’E. Atkinson, Kacharov, Clayton, Haubold) examined critically the energy distribution and proposed that such distribution could deviate from the Maxwellian form. In fact, it is commonly accepted that main-sequence stars like the Sun have a core, i.e. an electron nuclear plasma, where the ion velocity distribution is Maxwellian. In the following, we first discuss why even tiny deviations from the Maxwellian distribution can have important consequences and then what can originate such deviations.