Plasma Particle Distribution Functions Research Articles

Quasilinear theory of general electromagnetic fluctuations including discrete particle effects for magnetized plasmas: General analysis

The general quasilinear Fokker–Planck kinetic equation for the gyrophase-averaged plasma particle distribution functions in magnetized plasmas is derived, making no restrictions on the energy of the particles and on the frequency of the electromagnetic fluctuations and avoiding the often made Coulomb approximation of the electromagnetic interactions. The inclusion of discrete particle effects breaks the dichotomy of nonlinear kinetic plasma theory divided into the test particle and the test fluctuation approximation because it provides expression of both the non-collective and collective electromagnetic fluctuation spectra in terms of the plasma particle distribution functions. Within the validity of the quasilinear approach, the resulting full quasilinear transport equation can be regarded as a determining nonlinear equation for the time evolution of the plasma particle distribution functions.

Turbulence and Particle Acceleration in a Relativistic Plasma

Abstract In a collisionless plasma, the energy distribution function of plasma particles can be strongly affected by turbulence. In particular, it can develop a nonthermal power-law tail at high energies. We argue that turbulence with initially relativistically strong magnetic perturbations (magnetization parameter σ ≫ 1) quickly evolves into a state with ultrarelativistic plasma temperature but mildly relativistic turbulent fluctuations. We present a phenomenological and numerical study suggesting that in this case, the exponent α in the power-law particle-energy distribution function, f(γ)d γ ∝ γ −α d γ, depends on magnetic compressibility of turbulence. Our analytic prediction for the scaling exponent α is in good agreement with the numerical results.

Determination of the single particle distribution function in a weakly correlated weakly inhomogeneous plasma

Single particle distribution function of plasma particles has been derived from the first member of the Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy utilising the pair correlation function evaluated in [Bose, Phys. Plasmas 26, 064501 (2019)] from the second member of the BBGKY hierarchy. This distribution function may be employed to probe the thermodynamic properties of the weakly inhomogeneous plasma systems.

Charging time scales and magnitudes of dust and spacecraft potentials in space plasma scenarios

We investigate the interaction of dust with space plasmas and spacecraft with a special focus on the typical values of equilibrium charge and the typical time scales required to reach them. It is well known that objects in space become charged through the combination of a number of different processes: the photoelectric effect, the collection of free electrons and ions from the plasma, and by secondary electron emission due to the impact of highly energetic particles. In the equilibrium state, currents between the plasma and the charged object are balanced. However, perturbations on the orbit of the charged body and perturbations of the distribution of plasma particles may lead to time dependent deviations in charge. In this study, we are interested in order of magnitude estimates of these deviations as well as the time scales on which a charged body in space recovers to the equilibrium charge. Our study includes Maxwellian and Kappa plasma particle distribution functions and the role of motion on charged dust, as well as the effect of dust impacts on time dependent spacecraft potentials. We derive simple relationships on order of magnitude estimates and on time scales of different charging processes and apply our results to charged dust in the heliosphere and spacecraft potential analysis of the Magnetospheric MultiScale Mission around the Earth.

About the cyclotron resonance conditions in magnetized current-carrying plasmas

The resonant cyclotron wave-particle interactions in a cylindrical current-carrying plasma and in two-dimensional axisymmetric toroidal plasma models for tokamaks with D-shaped, elliptic and circular magnetic surfaces are discussed. The corresponding phase resonance conditions are derived by solving the linearized Vlasov equations as a boundary-value problem for perturbed plasma particle distribution functions, accounting for the geometry of a confinement magnetic field in the zero-order over magnetization parameters. It is shown that the Doppler shifts under the cyclotron resonance conditions in the current-carrying plasmas differ from those for a plasma in a uniform magnetic field due to the rotational transformation (including the shear-effects) of the helical magnetic field lines. As for elongated tokamaks, the cyclotron wave-particle resonance conditions in the general case, along with helicity-corrections, involve the bounce-resonances of the untrapped (circulating, passing) and two groups of so-called t-trapped and d-trapped particles.

Electromagnetic fluctuations in magnetized plasmas II: Extension of the theory for parallel wave vectors

Starting from the recently derived general expressions for the electromagnetic fluctuation spectra (electric and magnetic field) from uncorrelated plasma particles in plasmas with an uniform magnetic field, the case of strictly parallel (k⊥=0) oriented wave vectors with the respect to the uniform magnetic field direction is investigated. To derive fluctuation spectra valid in the entire complex frequency plane, the relevant dispersion functions and form factors are analytically continued to negative values of the imaginary part of the frequency for arbitrary gyrotropic plasma particle distribution functions. The generalized fluctuation-dissipation theorems for non-collective fluctuations in isotropic equal-temperature thermal distribution functions for general complex values of the frequency of the fluctuations with parallel wave vectors are derived.

Electromagnetic fluctuations in magnetized plasmas. I. The rigorous relativistic kinetic theory

Using the system of the Klimontovich and Maxwell equations, the general linear fluctuation theory for magnetized plasmas is developed. General expressions for the electromagnetic fluctuation spectra (electric and magnetic fields) from uncorrelated plasma particles in plasmas with a uniform magnetic field are derived, which are covariantly correct within the theory of special relativity. The general fluctuation spectra hold for plasmas of arbitrary composition, arbitrary momentum dependences of the plasma particle distribution functions, and arbitrary orientations of the wave vector with respect to the uniform magnetic field. Moreover, no restrictions on the values of the real and the imaginary parts of the frequency are made. The derived fluctuation spectra apply to both non-collective fluctuations and collective plasma eigenmodes in magnetized plasmas. In the latter case, kinetic equations for the components of fluctuating electric and magnetic fields in magnetized plasmas are derived that include the effect of spontaneous emission and absorption. In the limiting case of an unmagnetized plasmas, the general fluctuation spectra correctly reduce to the unmagnetized fluctuation spectra derived before.

Kinetic (particle-in-cell) simulation of nonlinear laser absorption in a finite-size plasma with a background inhomogeneous magnetic field

In this paper, the effect of an external inhomogeneous magnetic field on the high intensity laser absorption rate in a sub-critical plasma has been investigated by employing a relativistic electromagnetic 1.5 dimensional particle-in-cell code. Relying on the effective nonlinear phenomena such as phase-mixing and scattering, this study shows that in a finite-size plasma the laser absorption increases with inhomogeneity of the magnetic field (i.e., reduction of characteristic length of inhomogeneous magnetic field, λp) before exiting a considerable amount of laser energy from the plasma due to scattering process. On the other hand, the presence of the external inhomogeneous magnetic field causes the maximum absorption of laser to occur at a shorter time. Moreover, study of the kinetic results associated with the distribution function of plasma particles shows that, in a special range of the plasma density and the characteristic length of inhomogeneous magnetic field, a considerable amount of laser energy is transferred to the particles producing a population of electrons with kinetic energy along the laser direction.

A split control variate scheme for PIC simulations with collisions

When the distribution function of plasma particles stays close to some analytically known function, statistical noise inherent to Monte Carlo simulations can be greatly reduced by introducing this function as a control variate in the computation of the velocity moments. Such a method, even though it can be naturally applied to nonlinear simulations, has originally emerged from linearised simulations and is usually called the δf particle-in-cell (PIC) method. In the past, the method has been extended to also handle collisions. This resulted in a two weight scheme which is known to produce a pronounced weight growth problem which rapidly makes it inefficient as a control variate method for variance reduction. In this work we analyse the weight growth problem within a simple example, which allows us to overcome its pathological behaviour. We also introduce a new split algorithm based on switching the control variate for PIC simulations with collisions. A key element of our algorithm is a new weight smoothing operator which enables us to obtain a significant noise reduction both in the presence of collisions and in the deep nonlinear phase of PIC simulations.

Effect of the radio frequency discharge on the dust charging process in a weakly collisional and fully ionized plasma

A close analysis of dust charging process in the presence of radio frequency (RF) discharge on low pressure and fully ionized plasma for both weak and strong discharge's electric field is considered. When the electromagnetic waves pass throughout fully ionized plasma, the collision frequency of the plasma is derived. Moreover, the disturbed distribution function of plasma particles in the presence of the RF discharge is obtained. In this article, by using the Krook model, we separate the distribution function in two parts, the Maxwellian part and the perturbed part. The perturbed part of distribution can make an extra current, so-called the accretion rate of electron (or ion) current, towards a dust particle as a function of the average electron-ion collision frequency. It is proven that when the potential of dust grains increases, the accretion rate of electron current experiences an exponential reduction. Furthermore, the accretion rate of electron current for a strong electric field is relatively smaller than that for a weak electric field. The reasons are elaborated.

Quantum corrections to the momentum distribution in a laser plasma

The evolution of the momentum distribution of particles in a nonrelativistic laser plasma is studied in the context of the kinetic theory of plasma based on the construction of propagators for the plasma particle distribution functions. Damping of plasma waves and induced braking absorption accelerate particles in a plasma. Quantum indeterminacy in energy, which is associated with the interaction of particles with plasma waves and their collisions with one another, also affects their momentum distribution. In the semiclassical approximation, general analytic expressions are obtained for the momentum distribution functions.

Marginal instability threshold condition of the aperiodic ordinary mode in equal-mass plasmas

The purely growing ordinary (O) mode instability for counter-streaming bi-Maxwellian plasma particle distribution functions has recently received renewed attention due to its importance for the solar wind plasma. Here, the analytical marginal instability condition is derived for magnetized plasmas consisting of equal-mass charged particles, distributed in counter-streams with equal temperatures. The equal-mass composition assumption enormously facilitates the theoretical analysis due to the equality of the values of the electron and positron (positive and negative ion) plasma and gyrofrequencies. The existence of a new instability domain of the O-mode at small plasma beta values is confirmed, when the parallel counter-stream free energy exceeds the perpendicular bi-Maxwellian free energy.

On the marginal instability threshold condition of the aperiodic ordinary mode

The purely growing ordinary (O) mode instability has recently received renewed attention owing to its potential applicability to the solar wind plasma. Here, an analytical marginal instability condition is derived for counter-streaming bi-Maxwellian plasma particle distribution functions. The derived marginal instability condition as a function of the temperature anisotropy and plasma beta agrees remarkably well with the numerically determined instability condition. The existence of a new instability domain of the O-mode at small plasma beta values is confirmed with the leading A∝β∥−1-dependence, if the counter-stream parameter Pe exceeds a critical value. At small plasma beta values at large enough counter-stream parameter, the O-mode also operates for temperature anisotropies A = T⊥/T∥ > 1 even larger than unity, as the parallel counter-stream free energy exceeds the perpendicular bi-Maxwellian free energy.

Spontaneous electromagnetic fluctuations in unmagnetized plasmas. IV. Relativistic form factors of aperiodic Lorentzian modes

Closed analytical expressions for the electromagnetic fluctuation spectra in unmagnetized plasmas are derived using fully relativistic dispersion functions and form factors for the important class of isotropic form-invariant Lorentzian plasma particle distribution functions. Such distribution functions occur frequently in cosmic plasmas due to the presence of suprathermal charged particles and energetic cosmic ray particles. The results are illustrated for the important special case of aperiodic fluctuations. The collective, transverse, damped aperiodic mode, discovered before in nonrelativistic Maxwellian particle distributions, also exists in Lorentzian electron-proton particle distributions, now with the damping rate γ∝−k3 for all wavenumber values, resulting from the presence of relativistic particles in the tail of the Lorentzian distribution. For longitudinal electric field, fluctuations no damped or growing aperiodic collective mode exists in Lorentzian plasmas. The existence of a damped, collective, transverse, aperiodic mode is not in conflict with earlier general instability studies excluding the existence of growing aperiodic collective modes in isotropic plasmas.

Spontaneous electromagnetic fluctuations in unmagnetized plasmas I: General theory and nonrelativistic limit

Using the system of the Klimontovich and Maxwell equations, general expressions for the electromagnetic fluctuation spectra (electric and magnetic field, charge and current densities) from uncorrelated plasma particles are derived, which are covariantly correct within the theory of special relativity. The general expressions hold for arbitrary momentum dependences of the plasma particle distribution functions and for collective and non-collective fluctuations. In this first paper of a series, the results are illustrated for the important special case of nonrelativistic isotropic Maxwellian particle distribution functions providing in particular the thermal fluctuations of weakly amplified modes and aperiodic modes.

General properties of small-amplitude fluctuations in magnetized and unmagnetized collision poor plasmas. I. The dielectric tensor

The dielectric tensor of a collision poor plasma determines all the physical properties of small-amplitude fluctuations for given initial plasma particle distribution functions, as it enters the Maxwell operator as the only plasma-specific quantity. For the important class of gyrotropic initial particle distribution functions fa(0)(p∥,p⊥) in a uniform magnetic field, we rigorously investigate the general properties of the plasma fluctuations without specifying the explicit momentum dependence of the gyrotropic distribution function. Two alternative forms of the relativistically correct dielectric tensor are derived which differ from nonrelativistic expressions in the literature. The first standard form is expressed in terms of infinite series of Bessel functions, whereas in the second form these infinite series are calculated with the Lerche–Newberger sum rules, yielding products of Bessel functions with complex indices for the individual elements of the Maxwell operator. The second form of the dielectric tensor is well suited to simplify the tensor in the special cases of parallel wave vectors and unmagnetized plasmas. For unmagnetized plasmas it is shown that aperiodic electrostatic and transverse fluctuations can only exist in symmetric distribution functions with f(−p∥,p⊥)=f(p∥,p⊥). Because this includes isotropic distribution functions, the more thorough investigation of this special case reveals that no electrostatic and fluctuations with positive growth rates γ=kcS>0 exist in an isotropic unmagnetized plasma, excluding both aperiodic (with R=0) and wave-like (with R≠0) instabilities, where R=ωR/(kc) denotes the real part of the phase speed. The second form of the dielectric tensor is also most appropriate to investigate fluctuations in magnetized equal mass plasmas, such as electron-positron-pair and/or proton-antiproton plasmas. Here for arbitrary wave vector orientation the dispersion relation factorizes into three separate modes. For fluctuations with parallel wave vectors in isotropic plasmas of arbitrary composition, the electromagnetic stability of such isotropic plasma populations is proven, relativistically generalizing the known corresponding nonrelativistic theorem.

Nonlinear propagation of intense electromagnetic waves in a hot electron–positron plasma

AbstractThe creation and annihilation of relativistically hot electron–positron (EP) pair plasmas in the presence of intense electromagnetic (EM) waves, which are not in thermal equilibrium, are studied by formulating a new plasma particle distribution functions, which are valid for both relativistic temperatures and relativistic amplitudes of the EM waves. It is found that intense EM waves in a collisionless EP plasma damp via nonlinear Landau damping. Accounting for the latter, we have obtained relativistic kinetic nonlinear Schrödinger equation (NLSE) with local and non-local nonlinearities. The NLSE depicts nonlinear Landau damping rates for intense EM waves. The damping rates are examined for dense and tenuous pair plasmas. Furthermore, we have studied the modulational instabilities of intense EM waves in the presence of nonlinear Landau damping. Our results reveal a new class of the modulational instability that is triggered by the inverse Landau damping in a relativistically hot EP plasma. Finally, we discuss localization of intense EM waves due to relativistic electron and positron mass increase in a hot pair plasma.

Generation of energetic electrons and ions by interaction of relativistic laser pulses with a supercritical plasma

Generation of hot electrons and fast ions under the action of a relativistic femtosecond laser pulse propagating through a supercritical-density plasma is studied in terms of the kinetic theory by constructing propagators for plasma particle distribution functions. Calculations are carried out for different laser pulse intensities, different sorts of multicharged ions, and different degrees of plasma inhomogeneity. Mechanisms of electron and ion acceleration in the plasma are explored.

Interaction of picosecond laser pulses with a thin target

Interaction between a picosecond laser pulse of nonrelativistic intensity with a thin target is studied in terms of the kinetic theory of laser plasma, which is based on constructing propagators for the plasma particle distribution functions. Allowance is made for both the self-consistent plasma field and plasma particle correlations (collisions). The interaction causes charge separation in the target and strong heating of electrons.

Magnetized Fokker–Planck equation with quasilinear diffusion for nonaxially symmetric velocity distribution functions

The kinetic equation governing the evolution of a nonaxially symmetric velocity distribution function of plasma particles in a uniform magnetic field, including the effects of collisions as well as quasilinear diffusion, is presented and analyzed. It is shown that, if the test particles’ orbits are straight lines, the kinetic equation reduces to a standard Fokker–Planck form with friction, collisional diffusion, and quasilinear diffusion coefficients all expressible in terms of scalar potentials. These potentials are simple generalizations of Rosenbluth potentials and can easily be used in numerical solutions of the Fokker–Planck equation.