Particle Velocity Distribution Function Research Articles

FDTD Solution of the Maxwell–Boltzmann System for Electromagnetic Wave Propagation in a Plasma

In this paper, a self-consistent model is developed of the propagation of an electromagnetic wave into a plasma region, based on the solution of the Maxwell-Boltzmann system of equations. The system is resolved using a finite-difference time-domain approach that gives as results the behavior of both the particle velocity distribution function and the electromagnetic fields. The method is applied to a simple situation in which a wave impinges on a plasma region, but can easily be extended to more complicated configurations.

Fokker-Planck operator for free-molecular, nonspherical, thermally nonequilibrium Brownian particles

The Fokker-Planck collision operator entering into the kinetic equation for the velocity distribution function of convex nonspherical rigid homogeneous Brownian particles in a traveling inhomogeneous monatomic gas is derived. The regime of flow past the particles is free-molecular, i.e., the characteristic particle dimensions are much smaller than the average free path of the gas molecules, the interaction between the particles and their effect on the gas phase can be neglected, and a specular-diffuse law of interaction between the molecules and the particle surface holds. The particle temperatures Tp are the same and differ from the local gas temperature T. Thermal nonequilibrium (Tp ≠ T) leads to violation of the well-known relations between the diffusion coefficients in the spaces of the translational and angular velocities and the coefficients of forces and moments exerted on a particle. The coefficients in the unknown operator are calculated for the particles in the form of bodies of revolution with longitudinal symmetry. Data characterizing the effect of the nonsphericity of the particles, i.e., spheroids and sphere-cylinders, on the degree of violation are given.

Numerical Analysis of Incompressible Flows by the Lattice Boltzmann Method

A lattice Boltzmann method for one-phase incompressible flows is proposed. The conventional lattice Boltzmann equation leads to the compressible Navier-Stokes equations through the Chapman-Enskog expansion. The compressible error is eliminated by applying the Fractional Step method to the conventional LB method. Two particle velocity distribution functions are used in this method : one calculates the momentum, and the other does the pressure in numerical computations. This method is able to compute the fluid flows as the density is constant, that is, the sound velocity is infinity. This method is validated by simulations of the Taylor vortex flow, of the Poiseuille flow, and of the cavity flow. The proposed model eliminates the pressure oscillation observed in the simulations of the Taylor vortex flow. The simulations show that the both errors of the continuity equation and of the fluid velocity reduce to about one-tenth. The numerical results of the proposed model are more accurate than those of the conventional LB method.

Modified gas-kinetic scheme for shock structures in argon

The gas-kinetic scheme (GKS) is a finite-volume method in which the fluxes are constructed from the single particle velocity distribution function f. The distribution function f is obtained from the linearised Boltzmann equation and is retained only to the Navier-Stokes order in terms of the Chapman-Enskog expansion. Higher order non-equilibrium effects are included in a variable local particle collision time depending on local gradients of the hydrodynamic variables up to second order. The fluxes so constructed possess the non-equilibrium information beyond the linear constitutive relations and Fourier law. The modified gas-kinetic scheme with a local particle collision time λ* depending on the gradients of hydrodynamic variables is used to compute shock structures in argon for Mach numbers between 1.2 and 25.0. We validate our results with existing numerical and experimental ones. Our results show that the modified gas-kinetic scheme is effective and efficient for shock structure calculations.

Non-Maxwellian effects in magnetosonic solitons

The role of non-Maxwellian effects on magnetosonic (MS) solitons propagating perpendicular to the external magnetic field in high- β plasmas is analysed. It is shown that they can exist in the form of either humps or holes in the magnetic field in which the field is either increased or decreased relative to the background magnetic field. The shape of the solitary structure depends upon both the form of the ion velocity distribution function and the wave dispersion. A nonlinear equation describing the propagation of MS solitons in high- β plasmas with an arbitrary particle velocity distribution function is derived. It is shown that for Maxwellian and bi-Maxwellian plasmas MS solitons can only exist in the form of the magnetic humps. The same is true for plasmas possessing either a kappa distribution or Kennel–Ashour–Abdalla equilibria. However, plasmas with a ring type ion velocity distribution or a Dory–Guest–Harris distribution with large loss-cone index can support the formation of magnetic holes. The theoretical results obtained are then compared with recent satellite observations.

Kinetic Fokker-Planck equation for free-molecular, thermally nonequilibrium Brownian particles in an inhomogeneous gas

A kinetic equation for the translational and angular velocity distribution function of spherical rigid Brownian particles in an inhomogeneous monatomic gas is derived. The particle diameters are much smaller than the average free path of the gas molecules and the interaction between the particles and their effect on the carrier (gas) phase are neglected. The particle temperatures T p are the same and differ from the local gas temperature T. The molecular velocity distribution function is specified by the first approximation of the Chapman-Enskog method. The difference between the characteristic phase velocities is small as compared with the mean thermal molecular velocity. The dependences of the diffusion coefficients in velocity space on the ratio T p/T, which characterize the effect of thermal nonequilibrium, i.e., violation of the thermodynamic equilibrium between the phases of the disperse system, are found using a specular-diffuse law of reflection of the molecules from the particle surface.

Diffusion plateaus in the velocity distributions of fast solar wind protons

In a collisionless plasma, such as the fast solar wind, wave‐particle interactions play the decisive role in determining the shape of particle velocity distribution functions (VDFs). In this paper we provide observational evidence for cyclotron‐resonant interaction of ion cyclotron Alfvén waves, which propagate outward from the Sun along the interplanetary magnetic field, with fast solar wind protons. According to quasi‐linear theory, the protons thereby diffuse in velocity space, a process leading to the formation of plateaus in the VDF. This diffusion plateau formation naturally explains the observed thermal anisotropies in the core of the proton VDFs. In this respect, we investigated a large number of data from several distinct fast solar wind streams between 0.3 and 1 AU. All measurements were made on Helios 2 during the solar minimum in 1976 and 1977. The proton VDFs as provided by the plasma instrument are modeled by a superposition of multiple Gaussians, such that the plasma dispersion relation for parallel propagating cyclotron waves can readily be solved numerically. Thus the details of the proton VDFs are well represented and are reflected in the dispersion relation, which makes our analysis as self‐consistent as possible. Proton thermal effects on the wave dispersion relation are naturally taken into account.

Whistler wave excitation by relativistic electrons in coronal loops during solar flares

Whistler wave packets propagating along magnetic loops are the standard model for the generation of the fiber bursts that are observed in dynamical radio spectra during solar flares. The whistlers are believed to be generated by electron-loss cone distributions that form in the loop, if energetic electrons that have been injected by the flare are mirrored at the loop footpoints. A relativistic method for calculating plasma wave growth/damping rates for given particle velocity distribution functions (VDFs) is used to study the wave generation in dependence on the wave frequency and on the wave propagation angle with respect to the background magnetic field. Due to a Taylor expansion of the plasma dispersion function, it yields better accuracy for small growth rates than does a direct solution of the dispersion relation for a complex frequency. The results show that the whistler wave generation is limited to small propagation angles with respect to the background magnetic field. The wave growth rates are found to be high enough to account for the properties of fiber bursts observed in solar radio data.

One-Step Aeroacoustics Simulation Using Lattice Boltzmann Method

The lattice Boltzmann method (LBM) is a numerical simplification of the Boltzmann equation of the kinetic theory of gases that describes fluid motions by tracking the evolution of the particle velocity distribution function based on linear streaming with nonlinear collision. If the Bhatnagar‐Gross‐Krook (BGK) collision model is invoked, the velocity distribution function in this mesoscopic description of nonlinear fluid motions is essentially linear. This intrinsic feature of LBM can be exploited for convenient parallel programming, which makes itself particularly attractive for one-step aeroacoustics simulations. It is shown that the compressible Navier‐Stokes equations and the ideal gas equation of state can be correctly recovered by considering the translational and rotational degrees of freedom of diatomic gases in the internal energy and using a multiscale Chapman‐Enskog expansion. Assuming two relaxation times in the BGK model allows the temperature dependence of the first coefficient of viscosity of diatomic gases to be replicated. The modified LBM model is solved using a two-dimensional 9-discretized and a twodimensional 13-discretized velocity lattices. Three cases are selected to validate the one-step LBM aeroacoustics simulation. They are the one-dimensional acoustic pulse propagation, the circular acoustic pulse propagation, and the propagation of acoustic, vorticity, and entropy pulses in a uniform stream. The accuracy of the LBM is established by comparing with direct numerical simulation (DNS) results obtained by solving the governing equations using a finite difference scheme. The tests show that the proposed LBM and the DNS give identical results, thus suggesting that the LBM can be used to simulate aeroacoustics problems correctly.

A Panoramic Plasma Spectrometer: An All-Sky Camera for Charged Particles

The necessity of the high temporal resolution measurements of the complete velocity distribution function of charged particles in space leads to the invention of analyzers of new types including top-hat analyzers [2] for operation onboard rotating spacecraft and instruments with electrostatic scanning [3] for operation onboard slowly rotating spacecraft or onboard spacecraft with tri-axial stabilization. The instrument SKA-1 with electrostatic scanning [3] mounted on the Interball Tail Probe satellite had a temporal resolution of 10 s. There were two electrostatic mirrors and 36 Thomson parabola analyzers in the mass-spectrometer FONEMA for the Mars-96 mission [4], which guaranteed full coverage of 4 π angle and a temporal resolution of ~0.2 s. However, the failure of launching the Mars96 project did not allow one to use this unique instrument. The highest temporal resolution in measurements of the three-dimensional velocity distribution function of charged particles ( ~3 s) is achieved by employing a tophat analyzer on a rotating spacecraft. In this case, the temporal resolution is determined by the velocity of rotation of a spacecraft around its axis. Further improvement of the temporal resolution requires either arrangement of many detectors (which in turn causes many technical problems) or employment of new approaches. Earlier [1] we proposed a miniature electro-optic system that guarantees complete 2 π field of view ( 4 π with two analyzers) with a single coordinate-sensitive detector: an all-sky camera for charged particles. The feed optics can be modified for specific conditions of experiments and for different analyzers of the second stage. The suggested concept can guarantee very fast, flexible, and complete three-dimensional measurements of charged particles (ions or electrons).

Kinetic Alfvén wave driven by the density inhomogeneity in the presence of loss-cone distribution function—particle aspect analysis

Kinetic Alfven waves (KAWs) driven by the diamagnetic drift instability that is excited by the density inhomogeneity in low- β plasmas, such as plasmas in the auroral region, are investigated by adopting the particle aspect analysis and loss-cone distribution function. The results obtained in this paper indicate that the propagation and evolution of kinetic Alfven waves decrease and the kinetic Alfven wave excitation becomes not easier with increasing loss-cone index J . But the spatial scales of the perpendicular perturbation driving kinetic Alfven waves have a decreasing tendency with the larger values of J , which perhaps is in relation with the decreasing width of loss-cone. A single hump appears in the plots of the growth rate of the instability when J = 2 . But the hump cannot emerge when J = 0 or J = 1 . The density inhomogeneity of ions plays an important role in driving KAWs and it cannot be ignored. KAWs can be easier driven and KAWs can propagate and evolve faster with the increasing level of density inhomogeneity. However, the range of the perpendicular wave number of the wave instability decreases, namely, the longer the scale of perpendicular disturbance the easier the excitation of KAW. As the density inhomogeneity increases, the tendency of numerical solutions of the dispersion relation is similar to that obtained by the kinetic theory and Maxwellian distribution function ( Duan and Li, 2004). But the profiles of the plots of numerical solutions are different. This means that the velocity distribution function of particles is important for KAW driven in magnetoplasmas, especially in the active regions of the magnetosphere, such as auroral region, and plasma sheet boundary.

Prediction of particle motion in a two-dimensional bubbling fluidized bed using discrete hard-sphere model

The solids motion in a gas–solid fluidized bed was investigated using a discrete hard-sphere model. Detailed collision between particles and a nearest list method are presented. The turbulent viscosity of gas phase was predicted by subgrid scale (SGS) model. The interaction between gas and particles phases was governed by Newton's third law. The distributions of concentration, velocity and granular temperature of particles are obtained. The radial distribution function is calculated from the simulated spatio-temporal particle distribution. The normal and shear stresses of particles are predicted from the simulated instantaneous particle velocity. The pressure and viscosity of particles are obtained from both the kinetic theory of granular flow and the calculated stresses of particles. For elastic particles the individual lateral and vertical particle velocity distribution functions are isotropic and Maxwellian. The observed anisotropy becomes more pronounced with increasing degree of inelasticity of the particles.

Velocity distribution of Brownian particles at the input of a closed detector

The velocity distribution function of passive-tracer particles in a gas flow on a closed boundary of a given spatial region is found for various relations between the regular drift and diffusion. An example of calculation of the velocity probability density on the boundary of a region comprising a source of particles is given.

Correlation effects and nonlocal velocity distribution functions

In the present paper we consider the functional equation describing the collisionless particle velocity distribution function f ( V ) in the framework of probabilistic approach. The key element of the collisionless particles description is using the waiting time distribution ψ ( t ) . The solution of the considered functional is obtained for several model functions ψ ( t ) and it leads to the power form tails of the velocity distribution f ( V ) . It is possible to adopt considered functional to the Laplace transformation form that allows us to accord “collision” and “collisionless” description. This Laplace form of the functional yields the Levy–Smirnov velocity distribution function with the characteristic exponent a L = 1 2 . The possibility to accord the model functional equation and the non-local Einstein–Smoluchowski equation is discussed.

Non-local velocity distribution function and one-flight approximation

The functional equation describing the collisionless particle velocity distribution function f ( V ) is considered in the framework of probabilistic approach. The key element of the collisionless particles description is using the waiting time distribution ψ ( t ) . The solution of the considered functional is obtained for several model functions ψ ( t ) and it leads to the power form tails of the velocity distribution f ( V ) . It is possible to adopt considered functional to the Laplace transformation form that allows us to accord “collision” and “collisionless” description. This Laplace form of the functional yields the Levy–Smirnov velocity distribution function with the characteristic exponent a L = 1 / 2 .

Characteristics of Hyperconcentrated Sediment-Laden Flows

Microscopic and macroscopic characteristics of hyperconcentrated sediment-laden flows are investigated using kinetic theory for differing concentrations of non-cohesive sediment particles. Variations in the microscopic particle velocity distribution function and in derived macroscopic variables such as the particle mean velocity, particle fluctuating velocity, and particle number per unit volume are discussed. Analysis of the microscopic characteristics shows that the peak value of the particle velocity probability density distribution function decreases with increasing vertically averaged volumetric (or mean) particle concentration Cv¯, particularly when Cv¯ exceeds a certain threshold. A case study indicates that the characteristic velocity uc is hardly influenced by the particle concentration, but the standard deviation σ is significantly affected. For given particle concentrations, both uc and σ tend to become larger with increasing relative elevation from the bottom. From the derived macroscopic char...

A particle‐based model of size or anion exclusion with application to microbial transport in porous media

This paper presents a novel approach to the representation of pore‐scale exclusion processes, based on the truncation of the distribution of local dispersive displacements in a random‐walk particle model. This approach increases the mean velocity of colloidal‐sized particles relative to inert solute tracers, and decreases the apparent dispersion. An equivalent continuum model, with modified velocity and dispersion parameters, is also derived. Both the particle and the equivalent continuum models are applied to the results of laboratory experiments on bacterial transport in intact cores from a research site near Oyster, Virginia. The particle‐based model requires only modest truncation (8% maximum) of the particle velocity distribution function to closely reproduce the significant observed decrease in bacterial arrival times relative to a bromide tracer. The approach provides a conceptually appealing and consistent means of incorporating the exclusion process into groundwater transport models.

Modeling a Pairwise Peculiar Velocity Distribution Function of Dark Matter from Halo Density Profiles

Abstract We derive the pairwise peculiar velocity distribution function of dark matter particles by applying a dark-matter halo approach. Unlike previous work, we do not assume a Gaussian velocity distribution function of dark matter in a single halo, but compute it self-consistently with the assumed density profile for the dark-matter halo. The resulting distribution function is well approximated by an exponential distribution which is consistent with the previous observational, numerical, and theoretical results. We also compute the pairwise peculiar velocity dispersion for different density profiles, and provide a practical fitting formula. We apply an empirical biasing scheme into our model and present a prediction for a pairwise peculiar velocity dispersion of galaxies, and reproduce the previous results of simulations using our semi-analytical method.

Hydrodynamic modelling of dense gas-fluidised beds: Comparison of the kinetic theory of granular flow with 3D hard-sphere discrete particle simulations

A novel technique to sample particle velocity distributions and collision characteristics from dynamic discrete particle simulations of intrinsically unsteady, non-homogeneous systems, such as those encountered in dense gas-fluidised beds, is presented. The results are compared to the isotropic Maxwellian particle velocity distribution and the impact velocity distribution that constitute the zeroth-order Enskog approximation for the kinetic theory of granular flow. Excellent agreement with the kinetic theory is obtained for elastic particles. The individual particle velocity distribution function is isotropic and Maxwellian. A good fit of the collision velocity distribution and frequency is obtained, using the radial distribution function proposed by Carnahan and Starling (J. Chem. Phys. 51 (1969) 635). However, for inelastic and rough particles an anisotropic Maxwellian velocity distribution is obtained. It is concluded that the formation of dense particle clusters disturbs spatial homogeneity and results in collisional anisotropy. Analysis of the impact velocity shows that, in dense gas-fluidised beds, not all impact angles are of equal likelihood. The observed anisotropy becomes more pronounced with increasing degree of inelasticity of the particles.

Measurement of velocity distribution functions of neutral particles in ion source plasma

The velocity distribution function of the gas effusing from a 5.5-cm-diam, 5-cm-long small ion source was measured by the mass separated time-of-flight method. The measured distribution function could be fitted to a Maxwellian distribution before the commencement of a discharge, when the deuterium pressure in the ion source was below 0.6 Pa. The effect of discharge upon the distribution function of neutral deuterium below this operating pressure was investigated by changing the arc power. The deuterium gas temperature determined by fitting the measured distribution function to a Maxwellian showed a maximum gas temperature of 800 K at discharge power density of 6 W/cm3. Effects of discharge upon argon and helium gas temperature were also measured, which showed a monotonic increase against the discharge power.