Vlasov-Poisson Equations Research Articles

Dispersive estimates for kinetic transport equation in Besov spaces

Although Strichartz estimates have been widely applied for the nonlinear dispersive equations like Schrdinger equation and wave equation in Besov spaces, it is rare in nonlinear kinetic type equations such as Vlasov-Poisson equation. In this paper, we are concerned about the dispersive estimates for the kinetic transport equation in Besov spaces, which could be applied to establish Strichartz estimates in Besov space and further may be exploited to investigate the solutions for Vlasov-Poisson equation and Boltzmann equation in Besov space.

Stochastic variational principles for the collisional Vlasov-Maxwell and Vlasov-Poisson equations.

In this work, we recast the collisional Vlasov–Maxwell and Vlasov–Poisson equations as systems of coupled stochastic and partial differential equations, and we derive stochastic variational principles which underlie such reformulations. We also propose a stochastic particle method for the collisional Vlasov–Maxwell equations and provide a variational characterization of it, which can be used as a basis for a further development of stochastic structure-preserving particle-in-cell integrators.

Trace class properties of the non homogeneous linear Vlasov–Poisson equation in dimension 1+1

We consider the abstract scattering structure of the non homogeneous linearized Vlasov–Poisson equations from the viewpoint of trace class properties which are emblematic of the abstract scattering theory [13, 14, 15, 19]. In dimension 1+1, we derive an original reformulation which is trace class. It yields the existence of the Moller wave operators. The non homogeneous background electric field is periodic with 4 + \varepsilon bounded derivatives.

Landau damping of twisted waves in Cairns distribution with anisotropic temperature

The consideration of orbital angular momentum of an electric field (twisted mode) is applied to the kinetic theory of plasma. The linearized Vlasov–Poisson equation is solved for the anisotropic thermal distributed bi-Maxwellian and Cairns distributions of electrons to obtain the damping rates of twisted waves. The dispersion relation and Landau damping of Langmuir twisted modes are obtained. The presence of twisted modes opens up two more possibilities in Landau damping and dispersion relations. This may generate a mixture with ion sound waves. It seems to play the role of a control parameter of Landau damping.

The generalized Friedmann model as a self-similar solution of Vlasov–Poisson equation system

The generalized Friedmann model as a self-similar solution of Vlasov–Poisson equation system

Second-order modes of transverse oscillations in intense beams: Chernin’s matrix formalism versus Vlasov–Poisson equations

We study an intense space charge dominated beam in the realm of beam optics of particle accelerators. We assume an anisotropic beam with different external focussing constants and with different phase space emittances in the two transverse directions. We consider the root mean square moment equations of the beam with linear transverse coupling of particle motion ( $$x$$ and $$y$$ ). Magnetic focussing fields and the space charge Coulomb’s self-field forces are expressed in a matrix format and describe the governing equations for the beam outer envelope. By linearizing the equations, we investigate the collective response of ensemble of particles. This response is stated as coherent mode frequencies for the second-order even (envelope) and odd (tilting) oscillation modes of the beam cross section. We discuss the main features of the eigenfrequencies in detail. The approach known as Chernin’s matrix formalism is conceptually equivalent to the second-order moments of the full Vlasov–Poisson kinetic equation in four-dimensional phase space. We show that the relations obtained via matrix approach are in full agreement with the eigenfrequencies of transverse oscillations obtained by the linearized kinetic Vlasov–Poisson equations.

A generalized Fourier\u2013Hermite method for the Vlasov\u2013Poisson system

A generalized Fourier–Hermite semi-discretization for the Vlasov–Poisson equation is introduced. The formulation of the method includes as special cases the symmetrically-weighted and asymmetrically-weighted Fourier–Hermite methods from the literature. The numerical scheme is formulated as a weighted Galerkin method with two separate scaling parameters for the Hermite polynomial and the exponential part of the new basis functions. Exact formulas for the error in mass, momentum, and energy conservation of the method depending on the parameters are devised and L^2 stability is discussed. The numerical experiments show that an optimal choice of the additional parameter in the generalized method can yield improved accuracy compared to the existing methods, but also reveal the distinct stability properties of the symmetrically-weighted method.

Debye screening for the stationary Vlasov-Poisson equation in interaction with a point charge

We prove that the Debye screening length emerges in an infinitely extended plasma with uniform background described by the nonlinear Vlasov-Poisson equation, interacting with a point charge. While screened stationary states as well as the time-dependent problem are well-studied for the linearized equation, there are few rigorous results on screening in the nonlinear setting. As such, the results presented here cover the stationary case under the assumptions predicted by the linearized theory.

Kinetic instability of twisted ion-acoustic mode with superthermal species of plasma

The electrostatic twisted ion-acoustic waves with finite orbital angular momentum states and associated kinetic instability are investigated in an electron-ion plasma. The plasma consisting of superthermal electrons and ions is modeled by using a non-gyrotropic Kappa distribution function in which the free energy source for wave excitation is provided by the relative directed motion of streaming electrons with respect to the ions. In the frame work of kinetic theory, the Vlasov-Poisson equations are employed to derive the expressions for dispersion relation and Landau damping rate under paraxial approximation. The results are analyzed for threshold condition of wave dispersion and instability growth rate in the presence of helical electric field structure. The relevance of study to the observed situations is also described.

Conservative semi-Lagrangian schemes for kinetic equations Part I: Reconstruction

In this paper, we propose and analyze a reconstruction technique which enables one to design high-order conservative semi-Lagrangian schemes for kinetic equations. The proposed reconstruction can be obtained by taking the sliding average of a given polynomial reconstruction of the numerical solution. A compact representation of the high order conservative reconstruction in one and two space dimension is provided, and its mathematical properties are analyzed. To demonstrate the performance of proposed technique, we consider implicit semi-Lagrangian schemes for kinetic-like equations such as the Xin-Jin model and the Broadwell model, and then solve related shock problems which arise in the relaxation limit. Applications to BGK and Vlasov-Poisson equations will be presented in the second part of the paper.

On instability of three-dimensional dynamic equilibrium states of self-gravitating Vlasov-Poisson gas

The Vlasov-Poisson kinetic equations describe the dynamics of point masses in a self-consistent gravitational field. For example, these equations describe clusters of stars, galaxies or interstellar gas, large-scale processes in the Universe. For stationary solutions to the Vlasov-Poisson kinetic equations, a sufficient stability condition has already been obtained previously. However, it has not been conversed until now (neither for small perturbations, nor, especially, for finite ones). The Vlasov-Poisson kinetic equations are related to equations of hydrodynamic type, for which, in turn, there are methods for conversing sufficient stability conditions (at least, in the linear approximation). In this paper, absolute instability for spatial dynamic equilibrium states of Vlasov-Poisson gas with respect to small three-dimensional (3D) perturbations will be proved by the direct Lyapunov method.

Fizička priroda "anomalnih" elektrona u visoko-vakuumskim diodama

Introduction/purpose: A fundamental theoretical explanation is given for the fact that in subnanosecond vacuum diodes there exists a group of electrons with kinetic energies much higher than the applied voltage (multiplied by the value of the elementary charge) qUmax. Methods: A mathematical method is used based on the numerical solution of the Vlasov-Poisson differential equations system for one-dimensional vacuum diodes of various designs. Results: It is shown in detail that the so-called "anomalous" electrons appear in the transient time domain characterizing the processes of establishing current flow in vacuum diodes. Conclusion: It has been convincingly shown that the presence of "anomalous" electrons is not associated with either the diode design or the presence of additional current carriers. In vacuum diodes with a subnanosecond leading edge of the voltage pulse, the excess of energy over qUmax can be over 20%.

Nonlinear echoes and Landau damping with insufficient regularity

We prove that the theorem of Mouhot and Villani on Landau damping near equilibrium for the Vlasov-Poisson equations on $\mathbb{T}_x \times \mathbb{R}_v$ cannot, in general, be extended to high Sobolev spaces in the case of gravitational interactions. This is done by showing in every Sobolev space, there exists background distributions such that one can construct arbitrarily small perturbations that exhibit arbitrarily many isolated nonlinear oscillations in the density. These oscillations are known as plasma echoes in the physics community. For the case of electrostatic interactions, we demonstrate a sequence of small background distributions and asymptotically smaller perturbations in $H^s$ which display similar nonlinear echoes. This shows that in the electrostatic case, any extension of Mouhot and Villani's theorem to Sobolev spaces would have to depend crucially on some additional non-resonance effect coming from the background -- unlike the case of Gevrey-$\nu$ with $\nu < 3$ regularity, for which results are uniform in the size of small backgrounds. In particular, the uniform dependence on small background distributions obtained in Mouhot and Villani's theorem in Gevrey class is false in Sobolev spaces.

On Solutions to the Vlasov Equations

In this study we consider the mixed boundary value problem for the Vlasov-Poisson equations in an infinite cylinder, a problem describing the evolution of the density distribution of ions and electrons in a high temperature plasma under an external magnetic field. We examine the way to find a solution presented by Skubachevskii in his articles. We then present the alternative way consisting in the following. The system is reduced to an inhomogeneous form by replacing the unknown distribution function. After this, the fixed-point iteration method is used twice. First, we find function $f^{\beta }( x,p,t )$ as the limit for the sequence $f_{n}^{\beta }(x,p,t )$, then we use it to construct the solution $\varphi ( x,t )$ as the limit for $\varphi_{n}(x,t)$. The found classical solution for which the supports of the charged-particle density distributions are at a distance from the cylindrical boundary is shown to exist and to be unique in some neighbourhood of the stationary solution.

A general way to confined stationary Vlasov-Poisson plasma configurations

We address the existence of stationary solutions of the Vlasov-Poisson system on a domain $\Omega\subset\mathbb{R}^3$ describing a high-temperature plasma which due to the influence of an external magnetic field is spatially confined to a subregion of $\Omega$. In a first part we provide such an existence result for a generalized system of Vlasov-Poisson type and investigate the relation between the strength of the external magnetic field, the sharpness of the confinement and the amount of plasma that is confined measured in terms of the total charges. The key tools here are the method of sub-/supersolutions and the use of first integrals in combination with cutoff functions. In a second part we apply these general results to the usual Vlasov-Poisson equation in three different settings: the infinite and finite cylinder, as well as domains with toroidal symmetry. This way we prove the existence of stationary solutions corresponding to a two-component plasma confined in a Mirror trap, as well as a Tokamak.

Effect of quantization and positron concentration on shielding potentials in electron‐positron‐ion plasma

AbstractThe kinetic theory of plasma has been employed to compute the test‐charge potential distributions accounting for quantization effects in magnetized electron‐positron‐ion (EPI) plasmas. In this regard, the degenerate positrons and electrons are assumed to follow the Fermi‐Dirac distribution, while inertial ions are modelled by Maxwellian velocity distribution. By solving the Fourier‐transformed Vlasov–Poisson equations, a modified dielectric function and electrostatic potential is obtained. By imposing various constraints on the test‐charge speed, the potential profile has been analysed in terms of Debye–Hückel (DH), far‐field (FF), and wake‐field (WF) potentials. It has been found that the amplitude of DH and FF potentials increases by the inclusion of quantization effects, and it becomes the opposite for the WF potential profile. Furthermore, the variation of positron concentration significantly affects the DH, FF, and WF potentials. The present findings are important to understand the shielding phenomenon in degenerate multi‐species plasmas.

Cosmological Vlasov–Poisson Simulations of Structure Formation with Relic Neutrinos: Nonlinear Clustering and the Neutrino Mass

Abstract We present the results of cosmological simulations of large-scale structure formation with massive neutrinos. The phase-space distribution of the cosmic relic neutrinos is followed, for the first time, by directly integrating the six-dimensional Vlasov–Poisson equations. Our novel approach allows us to represent free streaming and clustering of neutrinos, and their gravitational interaction with cold dark matter accurately. We thus obtain solutions for the collisionless dynamics independent of conventional N-body methods. We perform a suite of hybrid N-body/Vlasov simulations with varying the neutrino mass, and systematically examine the dynamical effects of massive neutrinos on large-scale structure formation. Our simulations show characteristic large-scale clustering of the neutrinos and their coherent streaming motions relative to dark matter. The effective local neutrino “temperature” around massive galaxy clusters varies by several percent with respect to the cosmic mean; the neutrinos in clusters can be hotter or colder depending on the neutrino mass. We study a number of statistics of the large-scale structure and of dark matter halos in comparison with those obtained by N-body simulations and/or by perturbation theory. Our simulations mark an important milestone in numerical cosmology, and pave a new way to study cosmic structure formation with massive neutrinos.

Semi-classical Limit of Quantum Free Energy Minimizers for the Gravitational Hartree Equation

For the gravitational Vlasov-Poisson equation, Guo and Rein constructed a class of classical isotropic states as minimizers of free energies (or energy-Casimir functionals) under mass constraints. For the quantum counterpart, that is, the gravitational Hartree equation, isotropic states are constructed as free energy minimizers by Aki, Dolbeault and Sparber. In this paper, we are concerned with the correspondence between quantum and classical isotropic states. Precisely, we prove that as the Planck constant $\hbar$ goes to zero, free energy minimizers for the Hartree equation converge to those for the Vlasov-Poisson equation in terms of potential functions as well as via the Husimi transform and the T\"oplitz quantization.

Quantum corrections to plasma kinetic equations: A deformation approach.

Plasmas, as well as several other many-body systems of technological interest, have been studied mostly as a purely classical subject. However, in dense plasmas, and in some semiconductor devices, metallic nanostructures and thin metal films, when the de Broglie wavelength of the charge carriers is comparable to the interparticle distance, quantum effects come into play. Because the classical kinetic equations are phase-space equations with positions and momenta as variables, which variables are noncommuting in quantum mechanics, kinetic equations are not directly applicable to quantum plasmas. Therefore, most treatments consider a full quantum many-body problem in Hilbert space and then, by reduction, obtain the quantum version of the kinetic equations. However, quantum mechanics may also be directly formulated in phase space by modifying the Poisson algebra into a new deformed algebra, hence the classical kinetic equations may also be deformed into their corresponding quantum versions. This is the approach followed here and applied to derive the quantum corrections to the Vlasov-Poisson, Vlasov-Maxwell, and Boltzmann equations (in the latter case also within the relaxation-time approximation).

Solutions of a Class of Degenerate Kinetic Equations Using Steepest Descent in Wasserstein Space

We use the steepest descent method in an Orlicz–Wasserstein space to study the existence of solutions for a very broad class of kinetic equations, which include the Boltzmann equation, the Vlasov–Poisson equation, the porous medium equation, and the parabolic p-Laplacian equation, among others. We combine a splitting technique along with an iterative variational scheme to build a discrete solution which converges to a weak solution of our problem.