Collisionless Kinetic Equations Research Articles

Initial kinetics of electrons, ions and electric field in planar vacuum diode with plasma cathode

The paper presents the results of a 1D1V theoretical study of the initial stage of plasma expansion into a planar vacuum gap 1 cm long with an applied voltage of 500–5000 V. Based on the collisionless kinetic and Maxwell equations, the motion of a two-component electron–ion plasma in a self-consistent electric field is described. The fundamental mechanism of ‘superthermal’ velocities of the emission edge of the plasma and anode-directed velocities of positive ions is demonstrated in detail. The general regularities of this process are established for variations in the number density of the initial plasma and the applied voltage.

Exponential methods for solving hyperbolic problems with application to collisionless kinetic equations

The efficient numerical solution of many kinetic models in plasma physics is impeded by the stiffness of these systems. Exponential integrators are attractive in this context as they remove the CFL condition induced by the linear part of the system, which in practice is often the most stringent stability constraint. In the literature, these schemes have been found to perform well, e.g., for drift-kinetic problems. Despite their overall efficiency and their many favorable properties, most of the commonly used exponential integrators behave rather erratically in terms of the allowed time step size in some situations. This severely limits their utility and robustness.Our goal in this paper is to explain the observed behavior and suggest exponential methods that do not suffer from the stated deficiencies. To accomplish this we study the stability of exponential integrators for a linearized problem. This analysis shows that classic exponential integrators exhibit severe deficiencies in that regard. Based on the analysis conducted we propose to use Lawson methods, which can be shown not to suffer from the same stability issues. We confirm these results and demonstrate the efficiency of Lawson methods by performing numerical simulations for both the Vlasov–Poisson system and a drift-kinetic model of a ion temperature gradient instability.

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.

Invariant density and time asymptotics for collisionless kinetic equations with partly diffuse boundary operators

This paper deals with collisionless transport equations in bounded open domains Ω⊂Rd(d⩾2) with C1 boundary ∂Ω, orthogonally invariant velocity measure m(dv) with support V⊂Rd and stochastic partly diffuse boundary operators H relating the outgoing and incoming fluxes. Under very general conditions, such equations are governed by stochastic C0-semigroups (UH(t))t⩾0 on L1(Ω×V,dx⊗m(dv)). We give a general criterion of irreducibility of (UH(t))t⩾0 and we show that, under very natural assumptions, if an invariant density exists then (UH(t))t⩾0 converges strongly (not simply in Cesarò means) to its ergodic projection. We show also that if no invariant density exists then (UH(t))t⩾0 is sweeping in the sense that, for any density φ, the total mass of UH(t)φ concentrates near suitable sets of zero measure as t→+∞. We show also a general weak compactness theorem of interest for the existence of invariant densities. This theorem is based on several results on smoothness and transversality of the dynamical flow associated to (UH(t))t⩾0.

Onsager-type conjecture and renormalized solutions for the relativistic Vlasov–Maxwell system

In this paper we give a proof of an Onsager type conjecture on conservation of energy and entropies of weak solutions to the relativistic Vlasov--Maxwell equations. As concerns the regularity of weak solutions, say in Sobolev spaces $W^{\alpha,p}$, we determine Onsager type exponents $\alpha$ that guarantee the conservation of all entropies. In particular, the Onsager exponent $\alpha$ is smaller than $\alpha = 1/3$ established for fluid models. Entropies conservation is equivalent to the renormalization property, which have been introduced by DiPerna--Lions for studying well-posedness of passive transport equations and collisionless kinetic equations. For smooth solutions renormalization property or entropies conservation are simply the consequence of the chain rule. For weak solutions the use of the chain rule is not always justified. Then arises the question about the minimal regularity needed for weak solutions to guarantee such properties. In the DiPerna--Lions and Bouchut--Ambrosio theories, renormalization property holds under sufficient conditions in terms of the regularity of the advection field, which are roughly speaking an entire derivative in some Lebesgue spaces (DiPerna--Lions) or an entire derivative in the space of measures with finite total variation (Bouchut--Ambrosio). In return there is no smoothness requirement for the advected density, except some natural a priori bounds. Here we show that the renormalization property holds for an electromagnetic field with only a fractional space derivative in some Lebesgue spaces. To compensate this loss of derivative for the electromagnetic field, the distribution function requires an additional smoothness, typically fractional Sobolev differentiability in phase-space. As concerns the conservation of total energy, if the macroscopic kinetic energy is in $L^2$, then total energy is preserved.

Rates of convergence to equilibrium for collisionless kinetic equations in slab geometry

This work deals with free transport equations with partly diffuse stochastic boundary operators in slab geometry. Such equations are governed by stochastic semigroups in L1 spaces. We prove convergence to equilibrium at the rate O(t−k2(k+1)+1)(t→+∞) for L1 initial data g in a suitable subspace of the domain of the generator T where k∈N depends on the properties of the boundary operators near the tangential velocities to the slab. This result is derived from a quantified version of Ingham's tauberian theorem by showing that Fg(s):=limε→0+⁡(is+ε−T)−1g exists as a Ck function on R\\{0} such that ‖djdsjFg(s)‖≤C|s|2(j+1) near s=0 and bounded as |s|→∞(0≤j≤k). Various preliminary results of independent interest are given and some related open problems are pointed out.

On Asymptotic Stability and Sweeping of Collisionless Kinetic Equations

We study a collisionless kinetic equation describing density distribution function of the position and velocity of particles moving in a slab with finite thickness and with a partly diffusive boundary reflection. In particular, we deal with existence of an invariant density and with the convergence to the equilibrium. We also study the long time behavior of densities when the equilibrium does not exists.

An improved control-variate scheme for particle-in-cell simulations with collisions

Particle-in-cell methods combined with a δf approach constitute an established and powerful method for simulating collisionless kinetic equations in e.g. plasma physics. Including collisions in such simulations requires a modified approach leading to a two-weight scheme, which has the drawback of giving a statistical error that increases with time. As in the collisionless case, this scheme can be interpreted as an application of an ordinary control variate. Using an enhanced control variate approach, an improved scheme is constructed. This approach has been applied to a model problem with the result of a much better behaviour of the error, which, instead of growing indefinitely, becomes bounded by the error of a full- f scheme. In addition, the application of the enhanced control variate is illustrated for a collisionless simulation of ITG turbulence. Here it can be used both as a diagnostic tool and as a means to eliminate the spurious violation of particle number conservation inherent to δf simulations due to statistical noise.

A local existence result for a plasma physics model containing a fully coupled magnetic field

A local existence theorem is proved for classical solutions of theVlasov-Poisswell system, a set of collisionless equations used in plasmaphysics.Although the method employed is standard,there are several technical difficulties in the treatment of this systemthatarise mainly from the, compared to related systems, special form of theelectric-field term.Furthermore, uniquenessof classical solutions is proved and a continuation criterion forsolutions well knownfor other collisionless kinetic equations is established. Finally, aglobal existence result for a regularized versionof the system is derived and comments are given on the problem ofobtaining global weak solutions.

The multi-water-bag equations for collisionless kinetic modeling

In this paper we consider the multi-water-bag model for collisionless kinetic equations. The multi-water-bag representation of the statistical distribution function of particles can be viewed as a special class of exact weak solution of the Vlasov equation, allowing to reduce this latter into a set of hydrodynamic equations while keeping its kinetic character. After recalling the link of the multi-water-bag model with kinetic formulation of conservation laws, we derive different multi-water-bag (MWB) models, namely the Poisson-MWB, the quasineutral-MWB and the electromagnetic-MWB models. These models are very promising because they reveal to be very useful for the theory and numerical simulations of laser-plasma and gyrokinetic physics. In this paper we prove some existence and uniqueness results for classical solutions of these different models. We next propose numerical schemes based on Discontinuous Garlerkin methods to solve these equations. We then present some numerical simulations of non linear problems arising in plasma physics for which we know some analytical results.

Coupled Hartree-Fock-Bogoliubov kinetic equations for a trapped Bose gas

Using the Kadanoff-Baym nonequilibrium Green's-function formalism, we derive the self-consistent Hartree-Fock-Bogoliubov (HFB) collisionless kinetic equations and the associated equation of motion for the condensate wave function for a trapped Bose-condensed gas. Our work generalizes earlier work by Kane and Kadanoff for a uniform Bose gas. We include the off-diagonal (anomalous) pair correlations, and thus we have to introduce an off-diagonal distribution function in addition to the normal (diagonal) distribution function. This results in two coupled kinetic equations. If the off-diagonal distribution function can be neglected as a higher-order contribution, we obtain the semiclassical kinetic equation recently used by Zaremba, Griffin, and Nikuni (based on the simpler Popov approximation). We discuss the static local equilibrium solution of our coupled HFB kinetic equations within the semiclassical approximation. We also verify that a solution is the rigid in-phase oscillation of the equilibrium condensate and noncondensate density profiles, oscillating with the trap frequency.

Closure conditions for classical and quantum moment hierarchies in the small-temperature limit

Abstract We analyze closure conditions in the small initial temperature limit for classical and quantum mechanical moment hierarchies of corresponding collisionless kinetic equations. Euler equations with a nondiagonal pressure tensor are obtained in the classical case. In the quantum case we consider the cases of fixed and small (scaled) Planck constant and derive Quantum Hydrodynamic equations.

Collisionless kinetic ballooning equations in the comparable frequency regime

The collisionless kinetic ballooning mode equations are obtained from the gyrokinetic and Maxwell equations for the comparable frequency regime (i.e., for frequency of the modes comparable with the ion acoustic frequency along the field lines). A two length-scale analysis is made to reduce the kinetic ballooning mode equations to a fluid-like one. The finite Larmor radius (FLR) effect and particle–wave resonance (Landau damping) are taken into account. The trapped-particle effect is proven to be negligible in the ordering scheme adopted in this paper. It is emphasized that the non-FLR kinetic effects, especially the parallel electric field effect, are significant in the comparable frequency regime rather than the intermediate one. The non-FLR kinetic effects are found to be coupled to the vorticity equation through the apparent mass term, which suggests that the toroidal effect is essential to the kinetic mechanism in the singular layer (i.e., the outer region of ballooning mode representation space). The applications of the equations are discussed. The comparison with ideal magnetohydrodynamic (MHD) theory is made.

Collisionless kinetic equations for massive particles with spins 1/2 and 0 in gravitational and electromagnetic fields

We define a covariant and gauge-invariant generalization of the Wigner functions of particles with spins 1/2 and 0. The collisionless kinetic equations are obtained for these particles in external gravitational and electromagnetic fields in the quasiclassical approximation; also obtained are the momentum representations of the energy-momentum tensor, current, and spin tensor, taking into account the effects of the spin's interaction with the gravitational field an electromagnetic field. The following notation is used: e and m are the charge and mass of the particles; is Planck's constant; γμ(x) are the covariant-fixed Dirac matrices; σα,β=(1/4)[γα, γβ]: a(μbν)=(1/2) (a μ b ν +aνb μ); [A, B]=A·B − B·A; A,B=A·B+B·A; g(x)=det(g αβ(x));R α αμν=∂μΓ βν βν −...; the speed of light c=1.

Statistical description of an ensemble of ultrarelativistic particles in a spatially flat universe

The kinetic equation for massless particles in an unperturbed Friedman universe with gravitational interactions taken into account is obtained by averaging the collisionless kinetic equations over local background gravitational field fluctuations. Uniformly distributed spherically symmetric fluctuations of an ultrarelativistic fluid serve as local inhomogeneities in a spatially flat universe. It is shown that the collision integral of our kinetic equation does not vanish when the distribution function is homogeneous and isotropic.

Collisionless internal kinks

The paper presents an investigation of internal kink modes in high temperature plasmas. The relevant dynamics within the mode resonant layer is described by the collisionless kinetic equations. The approach covers a wide range of plasma parameters and, in particular, applies to situations in which the ion gyroradius is comparable to the resonant layer width and the temperature gradient scales are comparable to the density gradient scale. Several dispersion relations applicable to different plasma parameter regimes are derived.

Nonlinear gravitational waves in plasma

An exact solution is obtained for the relativistic collisionless kinetic equations describing a test plasma in the field of a strong plane gravitational wave. The gravitational wave induces in the plasma a longitudinal electric current whose amplitude is maximum at temperatures Te ip mec2. The interaction of gravitational waves with a system consisting of Boltzmann ions and degenerate electrons is also considered.