Collisionless Kinetic Theory Research Articles

Collisionless kinetic theory for parametrically pumped magnons

We discuss collisionless kinetic equations describing the non-equilibrium dynamics of magnons in a ferromagnet exposed to an oscillating microwave field. Previously, this problem has been treated within the so-called "S-theory" where the collision integral in the kinetic equation for the magnon distribution is either neglected or taken into account phenomenologically via an effective relaxation time. However, the possibility of magnon condensation has not been included in S-theory. Moreover, the momentum integrations appearing in the magnon self-energies are usually decoupled by retaining only the term where the loop momentum is equal to the external momentum. In this work we critically examine the accuracy of these approximations and develop the proper extensions of S-theory. We show that these extensions can significantly modify the time evolution of the magnon distribution.

Kinetic flux ropes: Bernstein–Greene–Kruskal modes for the Vlasov–Poisson–Ampère system

Electrostatic structures have been observed in many regions of space plasmas, including the solar wind, the magnetosphere, and the auroral acceleration region. One possible theoretical description of some of these structures is the concept of Bernstein–Greene–Kruskal (BGK) modes, which are exact nonlinear steady-state solutions of the Vlasov–Poisson system of equations in collisionless kinetic theory. We generalize the exact solutions of two-dimensional BGK modes in a magnetized plasma with a finite magnetic field strength [Ng et al., Phys. Plasmas 13, 055903 (2006)] to cases with azimuthal magnetic fields so that these structures carry the electric current as well as the steady electric and magnetic fields. Such nonlinear solutions now exactly satisfy the Vlasov–Poisson–Ampère system of equations. Explicit examples with either positive or negative electric potential structure are provided.

An introductory guide to fluid models with anisotropic temperatures. Part 2. Kinetic theory, Padé approximants and Landau fluid closures

In Part 2 of our guide to collisionless fluid models, we concentrate on Landau fluid closures. These closures were pioneered by Hammett and Perkins and allow for the rigorous incorporation of collisionless Landau damping into a fluid framework. It is Landau damping that sharply separates traditional fluid models and collisionless kinetic theory, and is the main reason why the usual fluid models do not converge to the kinetic description, even in the long-wavelength low-frequency limit. We start with a brief introduction to kinetic theory, where we discuss in detail the plasma dispersion function $Z(\unicode[STIX]{x1D701})$ , and the associated plasma response function $R(\unicode[STIX]{x1D701})=1+\unicode[STIX]{x1D701}Z(\unicode[STIX]{x1D701})=-Z^{\prime }(\unicode[STIX]{x1D701})/2$ . We then consider a one-dimensional (1-D) (electrostatic) geometry and make a significant effort to map all possible Landau fluid closures that can be constructed at the fourth-order moment level. These closures for parallel moments have general validity from the largest astrophysical scales down to the Debye length, and we verify their validity by considering examples of the (proton and electron) Landau damping of the ion-acoustic mode, and the electron Landau damping of the Langmuir mode. We proceed by considering 1-D closures at higher-order moments than the fourth order, and as was concluded in Part 1, this is not possible without Landau fluid closures. We show that it is possible to reproduce linear Landau damping in the fluid framework to any desired precision, thus showing the convergence of the fluid and collisionless kinetic descriptions. We then consider a 3-D (electromagnetic) geometry in the gyrotropic (long-wavelength low-frequency) limit and map all closures that are available at the fourth-order moment level. In appendix A, we provide comprehensive tables with Padé approximants of $R(\unicode[STIX]{x1D701})$ up to the eighth-pole order, with many given in an analytic form.

New results from exact solution of the dispersion relation for nonextensive electron-ion plasma in the framework of collisionless kinetic theory

The electrostatic (ES) oscillations, spatio-temporal Landau damping and growth in an electron-ion (EI) plasma is investigated by taking into account the effect of ion (electron) dynamics on the electron (ion) oscillations in the context of nonextensive particle distributions. The dispersion relation is exactly solved without employing approximations on the phase velocity, in contrast to the procedure developed by Landau where some approximations are applied on the phase velocity. This enables us to obtain some results which are not shown previously duo to approximate numerical and analytical studies. It is remarked that four kinds of ES modes; a high frequency mode (well-known Langmuir wave), two low and high intermediate frequency (LIF and HIF) modes and a very low frequency mode (well-known IA wave) may propagate in desired plasma system. In particular, these ES modes represent very different behavior when moving to left and right directions, from the point of view of spatio-temporal damping and growth. In addition, it is shown that exact solution of the dispersion relation gives rise to some interesting phenomena; such as the existence of non-acoustic electron and ion modes, the frequency and wavelength cutoffs and the existence of backward waves.

The Fluid-like and Kinetic Behavior of Kinetic Alfvén Turbulence in Space Plasma

Abstract Kinetic Alfvén waves (KAWs) are the short-wavelength extension of the magnetohydrodynamics Alfvén-wave branch in the case of highly oblique propagation with respect to the background magnetic field. Observations of space plasma show that small-scale turbulence is mainly KAW-like. We apply two theoretical approaches, a collisional two-fluid theory and a collisionless linear kinetic theory, to obtain predictions for the KAW polarizations depending on β p (the ratio of the proton thermal pressure to the magnetic pressure) at the ion gyroscale in terms of fluctuations in density, bulk velocity, and pressure. We perform a wavelet analysis of Magnetospheric Multiscale magnetosheath measurements and compare the observations with both theories. We find that the two-fluid theory predicts the observations better than the kinetic theory, suggesting that the small-scale KAW-like fluctuations exhibit a fluid-like behavior in the magnetosheath although the plasma is weakly collisional. We also present predictions for the KAW polarizations in the inner heliosphere that are testable with Parker Solar Probe and Solar Orbiter.

Collisionless kinetic theory of oblique tearing instabilities

The linear dispersion relation for collisionless kinetic tearing instabilities is calculated for the Harris equilibrium. In contrast to the conventional 2D geometry, which considers only modes at the center of the current sheet, modes can span the current sheet in 3D. Modes at each resonant surface have a unique angle with respect to the guide field direction. Both kinetic simulations and numerical eigenmode solutions of the linearized Vlasov-Maxwell equations have recently revealed that standard analytic theories vastly overestimate the growth rate of oblique modes. We find that this stabilization is associated with the density-gradient-driven diamagnetic drift. The analytic theories miss this drift stabilization because the inner tearing layer broadens at oblique angles sufficiently far that the assumption of scale separation between the inner and outer regions of boundary-layer theory breaks down. The dispersion relation obtained by numerically solving a single second order differential equation is found to approximately capture the drift stabilization predicted by solutions of the full integro-differential eigenvalue problem. A simple analytic estimate for the stability criterion is provided.

Finite-dimensional collisionless kinetic theory

A collisionless kinetic plasma model may often be cast as an infinite-dimensional noncanonical Hamiltonian system. I show that, when this is the case, the model can be discretized in space and particles while preserving its Hamiltonian structure, thereby producing a finite-dimensional Hamiltonian system that approximates the original kinetic model. I apply the general theory to two example systems: the relativistic Vlasov-Maxwell system with spin and a gyrokinetic Vlasov-Maxwell system.

Linear instabilities driven by differential rotation in very weakly magnetized plasmas

We study the linear stability of weakly magnetized differentially rotating plasmas in both collisionless kinetic theory and Braginskii's theory of collisional, magnetized plasmas. We focus on the very weakly magnetized limit that is important for understanding how astrophysical magnetic fields originate and are amplified at high redshift. We show that the single instability of fluid theory - the magnetorotational instability mediated by magnetic tension - is replaced by two distinct instabilities, one associated with ions and one with electrons. Each of these has a different way of tapping into the free energy of differential rotation. The ion instability is driven by viscous transport of momentum across magnetic field lines due to a finite ion cyclotron frequency (gyroviscosity); the fastest growing modes have wavelengths significantly longer than MHD and Hall MHD predictions. The electron instability is a whistler mode driven unstable by the temperature anisotropy generated by differential rotation; the growth time can be orders of magnitude shorter than the rotation period. The electron instability is an example of a broader class of instabilities that tap into the free energy of differential rotation or shear via the temperature anisotropy they generate. We briefly discuss the application of our results to the stability of planar shear flows and show that such flows are linearly overstable in the presence of fluid gyroviscosity. We also briefly describe the implications of our results for magnetic field amplification in the virialized halos of high redshift galaxies.

Collisionless kinetic theory of rolling molecules

We derive a collisionless kinetic theory for an ensemble of molecules undergoing nonholonomic rolling dynamics. We demonstrate that the existence of nonholonomic constraints leads to problems in generalizing the standard methods of statistical physics. In particular, we show that even though the energy of the system is conserved, and the system is closed in the thermodynamic sense, some fundamental features of statistical physics such as invariant measure do not hold for such nonholonomic systems.Nevertheless, we are able to construct a consistent kinetic theory using Hamilton's variational principle in Lagrangian variables, by regarding the kinetic solution as being concentrated on the constraint distribution. A cold fluid closure for the kinetic system is also presented, along with a particular class of exact solutions of the kinetic equations.

Drift wave dispersion relation for arbitrarily collisional plasma

The standard local linear analysis of drift waves in a plasma slab is generalized to be valid for arbitrarily collisional electrons by considering the electrons to be governed by the drift-kinetic equation with a BGK-like (Bhatnagar-Gross-Krook) collision operator. The obtained dispersion relation reduces to that found from collisionless kinetic theory when the collision frequency is zero. Electron temperature fluctuations must be retained in the standard fluid analysis in order to obtain good quantitative agreement with our general solution in the highly collisional limit. Any discrepancies between the fluid solution and our general solution in this limit are attributed to the limitations of the BGK collision operator. The maximum growth rates in both the collisional and collisionless limits are comparable and are both on the order of the fundamental drift wave frequency. The main role of the destabilizing mechanism is found to be in determining the parallel wave number at which the maximum growth rate will occur. The parallel wave number corresponding to the maximum growth rate is set by the wave-particle resonance condition in the collisionless limit and transitions to being set by the real frequency being on the order of the rate for electrons to diffuse a parallel wavelength in the collisional limit.

Absolute Stability of Axisymmetric Perturbations in Strongly Magnetized Collisionless Axisymmetric Accretion Disk Plasmas

This Letter presents a kinetic description of low-frequency and long-wavelength axisymmetric electromagnetic perturbations in nonrelativistic, stronglymagnetized, and gravitationallybound axisymmetric accretion-disk plasmas in the collisionless regime. The analysis, carried out within the framework of the Vlasov-Maxwell description, relies on stationary kinetic solutions which allow for the simultaneous treatment of nonuniform fluid fields, stationary accretion flows, and temperature anisotropies. It is demonstrated that these stationary configurations are actually stable against axisymmetric kinetic instabilities of this type. As a fundamental consequence, this rules out the possibility of having the axisymmetric magnetorotational or thermal instabilities to arise in these systems.

Studies of particle wake potentials in plasmas

A detailed understanding of electron stopping and scattering in plasmas with variable values for the number of particles within a Debye sphere is still not at hand. Presently, there is some disagreement in the literature concerning the proper description of these processes. Theoretical models assume electrostatic (Coulomb force) interactions between particles and neglect magnetic effects. Developing and validating proper descriptions requires studying the processes using first-principle plasma simulations. We are using the particle-particle particle-mesh (PPPM) code ddcMD and the particle-in-cell (PIC) code BEPS to perform these simulations. As a starting point in our study, we examine the wake of a particle passing through a plasma in 3D electrostatic simulations performed with ddcMD and BEPS. In this paper, we compare the wakes observed in these simulations with each other and predictions from collisionless kinetic theory. The relevance of the work to Fast Ignition is discussed.

Klein–Gordon equation in hydrodynamical form

We follow and modify the Feshbach–Villars formalism by separating the Klein–Gordon equation into two coupled time-dependent Schrödinger equations for particle and antiparticle wave function components with positive probability densities. We find that the equation of motion for the probability densities is in the form of relativistic hydrodynamics where various forces have their classical counterparts, with the additional element of the quantum stress tensor that depends on the derivatives of the amplitude of the wave function. We derive the equation of motion for the Wigner function and we find that its approximate classical weak-field limit coincides with the equation of motion for the distribution function in the collisionless kinetic theory.

Effect of ion–neutral collision on the deduction of Mach number in collisional plasmas

A new unmagnetized collisional Mach probe theory is developed in order to resolve the collisional effect on Mach probe analysis by including ionization, charge and momentum transfer of ions in the perturbation region of a Mach probe. A fluid model is established by assuming Boltzmann electrons and taking the moments of the one-dimensional Boltzmann transport equation, which contains the two-dimensional transport information as a source, after adding a collisional term. A new relation between the flow velocity and the ratio of the ion sheath current densities is obtained, and is compared with those by a collisionless kinetic theory and a particle-in-cell simulation in the applicable range of these theories. A new relation between flow velocity and the ratio of the ion sheath current densities shows that ion–neutral collisions have a very strong effect to produce much smaller Mach numbers from the same ratio than those by collisionless models.

Spectral properties of general advection operators and weighted translation semigroups

We investigate the spectral properties of a class of weighted shift semigroups $(\mathcal{U}(t))_{t \geq 0}$ associated to abstract transport equations with a Lipschitz continuous vector field $\mathcal{F}$ and no--reentry boundary conditions. Generalizing the results of [25], we prove that the semigroup $(\mathcal{U}(t))_{t \geq 0}$ admits a canonical decomposition into three $C_0$-semigroups $(\mathcal{U}_1(t))_{t \geq 0}$, $(\mathcal{U}_2(t))_{t \geq 0}$ and $(\mathcal{U}_3(t))_{t \geq 0}$ with independent dynamics. A complete description of the spectra of the semigroups $(\mathcal{U}_i(t))_{t \geq 0}$ and their generators $\mathcal{T}_i$, $i=1,2$ is given. In particular, we prove that the spectrum of $\mathcal{T}_i$ is a left-half plane and that the Spectral Mapping Theorem holds: $\mathfrak{S}(\mathcal{U}_i(t))=\exp$ {$t \mathfrak{S}(\mathcal{T}_i)$}, $i=1,2$. Moreover, the semigroup $(\mathcal{U}_3(t))_{t \geq 0}$ extends to a $C_0$-group and its spectral properties are investigated by means of abstract results from positive semigroups theory. The properties of the flow associated to $\mathcal{F}$ are particularly relevant here and we investigate separately the cases of periodic and aperiodic flows. In particular, we show that, for periodic flow, the Spectral Mapping Theorem fails in general but $(\mathcal{U}_3(t))_{t \geq 0}$ and its generator $\mathcal{T}_3$ satisfy the so-called Annular Hull Theorem. We illustrate our results with various examples taken from collisionless kinetic theory.

Nonlinear mirror waves in non‐Maxwellian space plasmas

A theory of finite‐amplitude mirror type waves in non‐Maxwellian space plasmas is developed. The collisionless kinetic theory in a guiding center approximation, modified for accounting of the finite ion Larmor radius effects, is used as the starting point. The model equation governing the nonlinear dynamics of mirror waves near instability threshold is derived. In the linear approximation it describes the classical mirror instability that is valid for a wide class of the velocity distribution functions. In the nonlinear regime the mirror waves form solitary structures that have the shape of magnetic holes. The formation of such structures and their nonlinear dynamics has been analyzed both analytically and numerically. It is suggested that the main nonlinear mechanism responsible for mirror instability saturation is associated with modification (flattening) of the shape of the background ion distribution function in the region of small parallel particle velocities. The width of this region is of the order of the particle trapping zone in the mirror hole. Near the mirror instability threshold the saturation arises before its width reaches the ion thermal velocity. The nonlinear mode coupling effects in this approximation are smaller and unable to take control over evolution of the space profile of saturated mirror waves or lead to their magnetic collapse. This results in the appearance of quasi‐stable solitary mirror structures having the form of deep magnetic depressions. A phenomenological description of this process is formulated. The relevance of the theoretical results to recent satellite observations is stressed.

Neutrino Kinetics in Dense Astrophysical Plasmas

Starting from the effective potentials describing the interaction of a single neutrino with a background species and the concept of ponderomotive force of neutrinos, we develop a collisionless relativistic kinetic theory for neutrinos coupled with a background medium. This formalism provides the basic tool for the physics of collective neutrino-plasma interactions. Comparison with equivalent kinetic equations derived using the methods of finite-temperature quantum field theory is presented. Astrophysical scenarios in which collective neutrino-plasma interactions can play an important role are discussed.

On the auroral current‐voltage relationship

A simple model of collisionless plasma kinetic theory [Knight, 1973] has been used to predict the relationship between the upward parallel current and the parallel potential drop. The DE 1/DE 2 pair offers a unique opportunity to test this relationship because the DE 1 spacecraft can measure the high altitude plasma parameters without contamination from auroral heating. The field‐aligned current density from Knight's formula is almost linear in the ratio of j∥/eΦ∥ for values of eΦ∥/kT between 1 and 10. We find, using measured values of J∥ (mapped to the surface) and Φ∥ (inferred from measurements either at high altitudes or low altitudes), the ratio of J∥/eΦ∥ varies considerably but with a mean value about 0.5∼2.2×10−9 mho/m². Using either the full Knight's formula or the linear simplification, we can predict the parallel potential drop (given the high‐altitude density, temperature, current, and magnetic mirror factor) to test against the measured potential drop, or we can predict the current given the potential drop and the high‐altitude plasma parameters. The agreement is generally quite good and the linear simplification is reasonable; however, the Knight current, since it does not include upward ionospheric electrons, is not valid in the downward return current region. We also show that suprathermal electron bursts are observed in the diffuse aurora at the same invariant latitudes, both at high and at low altitudes. Thus we suggest that these “bursts” are more properly described as a spatial rather than temporal phenomenon.

A kinetic theory of the magnetic islands merging instability

Collisionless kinetic theory of the merging instability of magnetic islands is presented. Merging instability results in a very fast coalescence of attracting magnetic islands and may play an important role in magnetic energy release in solar flares and laboratory plasma devices. The general picture of merging instability regimes dependent on the value of magnetic shear and magnetic island width w is sketched. The new kinetic regime of collisionless coalescence with gamma approximately w1/2 which operates in a very wide parameter range is discussed in detail.

Kinetic Guiding-Center Equations for the Theory of Drift Instabilities and Anomalous Transport

Littlejohn's guiding-center mechanics, with the polarization drift included, is rederived from a Lagrangian. The respective collisionless kinetic guiding-center theory is then established. While single-guiding-center energy is exactly conserved here in time-independent fields, the self-consistent kinetic guiding-center theory presented here conserves the total energy of the guiding-center plasma and its self-consistent electromagnetic fields only to leading order in ϵ ≡ rg/L. The present formalism seems to be relevant to the theory of drift waves and anomalous transport.