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Journal articles on the topic 'Model Vlasov-Poisson'

Author: Grafiati

Published: 4 June 2021

Last updated: 10 February 2022

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1

Abdallah, Naoufel Ben, and Florian Méhats. "On a Vlasov–Schrödinger–Poisson Model." Communications in Partial Differential Equations 29, no. 1-2 (January 2005): 173–206. http://dx.doi.org/10.1081/pde-120028849.

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2

Griffin-Pickering, Megan, and Mikaela Iacobelli. "Global strong solutions in $ {\mathbb{R}}^3 $ for ionic Vlasov-Poisson systems." Kinetic & Related Models 14, no. 4 (2021): 571. http://dx.doi.org/10.3934/krm.2021016.

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<p style='text-indent:20px;'>Systems of Vlasov-Poisson type are kinetic models describing dilute plasma. The structure of the model differs according to whether it describes the electrons or positively charged ions in the plasma. In contrast to the electron case, where the well-posedness theory for Vlasov-Poisson systems is well established, the well-posedness theory for ion models has been investigated more recently. In this article, we prove global well-posedness for two Vlasov-Poisson systems for ions, posed on the whole three-dimensional Euclidean space <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^3 $\end{document}</tex-math></inline-formula>, under minimal assumptions on the initial data and the confining potential.</p>

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3

Caprino, S. "A Vlasov-Poisson plasma model with non-L1 data." Mathematical Methods in the Applied Sciences 27, no. 18 (2004): 2211–29. http://dx.doi.org/10.1002/mma.551.

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4

Teichmann, J. "Linear Vlasov stability in one-dimensional double layers." Laser and Particle Beams 5, no. 2 (May 1987): 287–93. http://dx.doi.org/10.1017/s0263034600002779.

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Analytical study of the linear stability of one-dimensional double layers in nonmagnetized plasmas based on the solution of the Vlasov–Poisson system is presented. Electromagnetic effects are not included. A self-consistent equilibrium electrostatic potential Φ0(z) that monotonically increases from a low level at z = − ∞ to a high level at z = + ∞ is assumed. We model this potential as a piecewise continuous function of z and we assume that Φ0(z) has constant values for − ∞ z ≤ 0 and L ≤ z < ∞, L being the thickness of the double layer. The BGK states for the Vlasov–Poisson system provide an explicit expression for the velocity distribution of the reflected electrons required for the particular double layer configuration. The stability of the double layers is studied via the linearized Vlasov and Poisson equations using the WKB approximation.

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5

Hagstrom, George I., and P. J. Morrison. "Caldeira–Leggett model, Landau damping, and the Vlasov–Poisson system." Physica D: Nonlinear Phenomena 240, no. 20 (October 2011): 1652–60. http://dx.doi.org/10.1016/j.physd.2011.02.007.

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6

Tayeb, Mohamed Lazhar. "Homogenized Diffusion Limit of a Vlasov–Poisson–Fokker–Planck Model." Annales Henri Poincaré 17, no. 9 (April 26, 2016): 2529–53. http://dx.doi.org/10.1007/s00023-016-0484-7.

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7

DUAN, RENJUN, TONG YANG, and HUIJIANG ZHAO. "THE VLASOV–POISSON–BOLTZMANN SYSTEM FOR SOFT POTENTIALS." Mathematical Models and Methods in Applied Sciences 23, no. 06 (March 17, 2013): 979–1028. http://dx.doi.org/10.1142/s0218202513500012.

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An important physical model describing the dynamics of dilute weakly ionized plasmas in the collisional kinetic theory is the Vlasov–Poisson–Boltzmann system for which the plasma responds strongly to the self-consistent electrostatic force. This paper is concerned with the electron dynamics of kinetic plasmas in the whole space when the positive charged ion flow provides a spatially uniform background. We establish the global existence and optimal convergence rates of solutions near a global Maxwellian to the Cauchy problem on the Vlasov–Poisson–Boltzmann system for angular cutoff soft potentials with -2 ≤ γ < 0. The main idea is to introduce a time-dependent weight function in the velocity variable to capture the singularity of the cross-section at zero relative velocity.

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8

Califano, F., L. Galeotti, and A. Mangeney. "The Vlasov-Poisson model and the validity of a numerical approach." Physics of Plasmas 13, no. 8 (August 2006): 082102. http://dx.doi.org/10.1063/1.2215596.

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9

Uhlemann, Cora, and Michael Kopp. "Beyond single-stream with the Schrödinger method." Proceedings of the International Astronomical Union 11, S308 (June 2014): 115–18. http://dx.doi.org/10.1017/s1743921316009716.

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AbstractWe investigate large scale structure formation of collisionless dark matter in the phase space description based on the Vlasov-Poisson equation. We present the Schrödinger method, originally proposed by \cite{WK93} as numerical technique based on the Schrödinger Poisson equation, as an analytical tool which is superior to the common standard pressureless fluid model. Whereas the dust model fails and develops singularities at shell crossing the Schrödinger method encompasses multi-streaming and even virialization.

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10

DUAN, RENJUN, MEI ZHANG, and CHANGJIANG ZHU. "L1STABILITY FOR THE VLASOV–POISSON–BOLTZMANN SYSTEM AROUND VACUUM." Mathematical Models and Methods in Applied Sciences 16, no. 09 (September 2006): 1505–26. http://dx.doi.org/10.1142/s0218202506001613.

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Based on the global existence theory of the Vlasov–Poisson–Boltzmann system around vacuum in the N-dimensional phase space, in this paper, we prove the uniform L1stability of classical solutions for small initial data when N ≥ 4. In particular, we show that the stability can be established directly for the soft potentials, while for the hard potentials and hard sphere model it is obtained through the construction of some nonlinear functionals. These functionals thus generalize those constructed by Ha for the case without force to capture the effect of the force term on the time evolution of solutions. In addition, the local-in-time L1stability is also obtained for the case of N = 3.

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11

Ambroso, Annalisa, Xavier Fleury, and Pierre-Arnaud Raviart. "An Existence Result for a Stationary Vlasov–Poisson Model with Source Term." Mathematical Models and Methods in Applied Sciences 13, no. 08 (August 2003): 1119–55. http://dx.doi.org/10.1142/s0218202503002842.

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In the low energy part of an ion accelerator, the ion beam reaches the first accelerating device through a drift tube where a residual neutral gas is present. A process of ionization takes place and secondary electrons and ions are created. We analyze the coupling between these secondary particles, the beam density and the electrical potential. In a previous article,2 we proved that a stationary Vlasov–Poisson problem with ionization source terms, which, at first sight, seems the natural model for this system, does not admit solutions. To overcome this difficulty, we propose here a heuristic model which consists of modifying the electron source term. In this new frame, we prove that the problem can be solved and we give numerical examples.

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12

Perez, Jérôme. "Equilibrium of stellar dynamical systems in the context of the Vlasov–Poisson model." Communications in Nonlinear Science and Numerical Simulation 13, no. 1 (February 2008): 153–57. http://dx.doi.org/10.1016/j.cnsns.2007.05.011.

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13

Krasovsky, V. L., H. Matsumoto, and Y. Omura. "On the three-dimensional configuration of electrostatic solitary waves." Nonlinear Processes in Geophysics 11, no. 3 (July 2, 2004): 313–18. http://dx.doi.org/10.5194/npg-11-313-2004.

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Abstract. The simplest models of the electrostatic solitary waves observed by the Geotail spacecraft in the magnetosphere are developed proceeding from the concept of electron phase space holes. The technique to construct the models is based on an approximate quasi-one-dimensional description of the electron dynamics and three-dimensional analysis of the electrostatic structure of the localized wave perturbations. It is shown that the Vlasov-Poisson set of equations admits a wide diversity of model solutions of different geometry, including spatial configurations of the electrostatic potential similar to those revealed by Geotail and other spacecraft in space plasmas.

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14

Finkelshtein, Dmitri, Yuri Kondratiev, Yuri Kozitsky, and Oleksandr Kutoviy. "The statistical dynamics of a spatial logistic model and the related kinetic equation." Mathematical Models and Methods in Applied Sciences 25, no. 02 (November 24, 2014): 343–70. http://dx.doi.org/10.1142/s0218202515500128.

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In this paper, we study an infinite system of point entities in ℝd which reproduce themselves and die, also due to competition. The system's states are probability measures on the space of configurations of entities. Their evolution is described by means of a BBGKY-type equation for the corresponding correlation (moment) functions. It is proved that: (a) these functions evolve on a bounded time interval and remain sub-Poissonian due to the competition; (b) in the Vlasov scaling limit they converge to the correlation functions of the time-dependent Poisson point field the density of which solves the kinetic equation obtained in the scaling limit from the equation for the correlation functions. A number of properties of the solutions of the kinetic equation are also esta- blished.

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15

SCHULZE, ACHIM. "Existence and stability of static shells for the Vlasov–Poisson system with a fixed central point mass." Mathematical Proceedings of the Cambridge Philosophical Society 146, no. 2 (March 2009): 489–511. http://dx.doi.org/10.1017/s0305004108001916.

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AbstractWe consider the Vlasov–Poisson system with spherical symmetry and an exterior potential which is induced by a point mass in the center. This system can be used as a simple model for a newtonian galaxy surrounding a black hole. For this system, we establish a global existence result for classical solutions with shell-like initial data, i.e. the support of the density is bounded away from the point mass singularity. We also prove existence and stability of stationary solutions which describe static shells, where we use a variational approach which was established by Y. Guo and G. Rein.

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16

Bochkarev, S. G., V. Yu Bychenkov, and V. T. Tikhonchuk. "Investigation of ion acceleration in an expanding laser plasma by using a hybrid Boltzmann-Vlasov-Poisson model." Plasma Physics Reports 32, no. 3 (March 2006): 205–21. http://dx.doi.org/10.1134/s1063780x06030032.

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17

Mingalev, O. V., G. I. Mingaleva, M. N. Melnik, and V. S. Mingalev. "Numerical Simulation of the Time Evolution of Small-Scale Irregularities in the F-Layer Ionospheric Plasma." International Journal of Geophysics 2011 (2011): 1–8. http://dx.doi.org/10.1155/2011/353640.

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Dynamics of magnetic field-aligned small-scale irregularities in the electron concentration, existing in the F-layer ionospheric plasma, is investigated with the help of a mathematical model. The plasma is assumed to be a rarefied compound consisting of electrons and positive ions and being in a strong, external magnetic field. In the applied model, kinetic processes in the plasma are simulated by using the Vlasov-Poisson system of equations. The system of equations is numerically solved applying a macroparticle method. The time evolution of a plasma irregularity, having initial cross-section dimension commensurable with a Debye length, is simulated during the period sufficient for the irregularity to decay completely. The results of simulation indicate that the small-scale irregularity, created initially in the F-region ionosphere, decays accomplishing periodic damped vibrations, with the process being collisionless.

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18

Malkov, Evgeny A., and Alexey N. Kudryavtsev. "Non-stationary Antonov self-gravitating layer: analytics and numerics." Monthly Notices of the Royal Astronomical Society 491, no. 3 (November 28, 2019): 3952–66. http://dx.doi.org/10.1093/mnras/stz3276.

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ABSTRACT Large-scale instability of gravitating systems plays a key role in collisionless relaxation and in reaching a quasi-stationary state at the early stage of evolution. Advanced high-resolution methods and permanently increasing performance of computational systems allow this phenomenon to be studied by means of computer simulations at a new level. In this paper, an approach to verification and validation of computer codes implementing high-resolution methods is proposed. The approach is based on comparisons of the simulation results with exact non-stationary solutions of the Vlasov–Poisson equations. The evolution of the gravitating layer model is considered as an example of implementation of this approach. A one-parameter family of exact models of a non-stationary gravitating layer is described, and their stability to large-scale disturbances in the linear approximation is analytically studied. Non-linear instability development is computed with the use of the fifth-order conservative semi-Lagrangian WENO scheme.

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19

Turchetti, G., S. Rambaldi, A. Bazzani, M. Comunian, and A. Pisent. "3D solutions of the Poisson-Vlasov equations for a charged plasma and particle-core model in a line of FODO cells." European Physical Journal C 30, no. 2 (September 2003): 279–90. http://dx.doi.org/10.1140/epjc/s2003-01269-2.

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20

Touil, B., A. Bendib, K. Bendib-Kalache, and C. Deutsch. "Dispersion relation of low-frequency electrostatic waves in plasmas with relativistic electrons." Laser and Particle Beams 34, no. 1 (January 11, 2016): 178–86. http://dx.doi.org/10.1017/s026303461500107x.

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AbstractThe dispersion relation of electrostatic waves with phase velocities smaller than the electron thermal velocity is investigated in relativistic temperature plasmas. The model equations are the electron relativistic collisionless hydrodynamic equations and the ion non-relativistic Vlasov equation, coupled to the Poisson equation. The complex frequency of electrostatic modes are calculated numerically as a function of the relevant parameters kλDe and ZTe/Ti where k is the wavenumber, λDe, the electron Debye length, Te and Ti the electron and ion temperature, and Z, the ion charge number. Useful analytic expressions of the real and imaginary parts of frequency are also proposed. The non-relativistic results established in the literature from the kinetic theory are recovered and the role of the relativistic effects on the dispersion and the damping rate of electrostatic modes is discussed. In particular, it is shown that in highly relativistic regime the electrostatic waves are strongly damped.

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21

Chen, L. J., and G. K. Parks. "BGK electron solitary waves: 1D and 3D." Nonlinear Processes in Geophysics 9, no. 2 (April 30, 2002): 111–19. http://dx.doi.org/10.5194/npg-9-111-2002.

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Abstract. This paper presents new results for 1D BGK electron solitary wave (phase-space electron hole) solutions and, based on the new results, extends the solutions to include the 3D electrical interaction (E ~ 1/r 2) of charged particles. Our approach for extending to 3D is to solve the nonlinear 3D Poisson and 1D Vlasov equations based on a key feature of 1D electron hole (EH) solutions; the positive core of an EH is screened by electrons trapped inside the potential energy trough. This feature has not been considered in previous studies. We illustrate this key feature using an analytical model and argue that the feature is independent of any specific model. We then construct azimuthally symmetric EH solutions under conditions where electrons are highly field-aligned and ions form a uniform background along the magnetic field. Our results indicate that, for a single humped electric potential, the parallel cut of the perpendicular component of the electric field (E⊥) is unipolar and that of the parallel component (E||) bipolar, reproducing the multi-dimensional features of the solitary waves observed by the FAST satellite. Our analytical solutions presented in this article capture the 3D electric interaction and the observed features of (E|| ) and E⊥. The solutions predict a dependence of the parallel width-amplitude relation on the perpendicular size of EHs. This dependence can be used in conjunction with experimental data to yield an estimate of the typical perpendicular size of observed EHs; this provides important information on the perpendicular span of the source region as well as on how much electrostatic energy is transported by the solitary waves.

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22

DOLBEAULT, J., and G. REIN. "TIME-DEPENDENT RESCALINGS AND LYAPUNOV FUNCTIONALS FOR THE VLASOV–POISSON AND EULER–POISSON SYSTEMS, AND FOR RELATED MODELS OF KINETIC EQUATIONS, FLUID DYNAMICS AND QUANTUM PHYSICS." Mathematical Models and Methods in Applied Sciences 11, no. 03 (April 2001): 407–32. http://dx.doi.org/10.1142/s021820250100091x.

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We investigate rescaling transformations for the Vlasov–Poisson and Euler–Poisson systems and derive in the plasma physics case Lyapunov functionals which can be used to analyze dispersion effects. The method is also used for studying the long time behavior of the solutions and can be applied to other models in kinetic theory (two-dimensional symmetric Vlasov–Poisson system with an external magnetic field), in fluid dynamics (Euler system for gases) and in quantum physics (Schrödinger–Poisson system, nonlinear Schrödinger equation).

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23

Palmer, P. L., and J. Papaloizou. "Instability Through Anisotropy in Spherical Stellar Systems." Symposium - International Astronomical Union 127 (1987): 515–16. http://dx.doi.org/10.1017/s0074180900186012.

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We consider the linear stability of spherical stellar systems by solving the Vlasov and Poisson equations which yield a matrix eigenvalue problem to determine the growth rate. We consider this for purely growing modes in the limit of vanishing growth rate. We show that a large class of anisotropic models are unstable and derive growth rates for the particular example of generalized polytropic models. We present a simple method for testing the stability of general anisotropic models. Our anlysis shows that instability occurs even when the degree of anisotropy is very slight.

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24

Serre, Denis. "Compensated integrability. Applications to the Vlasov–Poisson equation and other models in mathematical physics." Journal de Mathématiques Pures et Appliquées 127 (July 2019): 67–88. http://dx.doi.org/10.1016/j.matpur.2018.06.025.

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25

Bahcall, John N. "Dark Matter in the Galactic Disk." Symposium - International Astronomical Union 117 (1987): 17–31. http://dx.doi.org/10.1017/s0074180900149794.

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The Poisson and Vlasov equations are solved self-consistently for realistic Galaxy models which include multiple disk components, a Population II spheroid, and an unseen massive halo. The total amount of matter in the vicinity of the Sun is determined by comparing the observed distributions of tracer stars, samples of F dwarfs and of K giants, with the predictions of the Galaxy models. Results are obtained for a number of different assumed distributions of the unseen disk mass. The major uncertainties, observational and theoretical, are estimated. For all the observed samples, typical models imply that about half of the mass in the solar vicinity must be in the form of unobserved matter. The volume density of unobserved material near the Sun is about 0.1M⊙pc−3; the corresponding column density is about 30M⊙pc−2. This so far unseen material must be in a disk with an exponential scale height of less than 0.7 kpc. If the unseen material is in the form of stars with masses less than 0.1M⊙, then the nearest such object is about 1 pc away and has a proper motion of more than 1 arcsecond per year.

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26

Lorenzon, Denis, Sergio A. Elaskar, and Andrés M. Cimino. "Numerical Simulations Using Eulerian Schemes for the Vlasov–Poisson Model." International Journal of Computational Methods, March 30, 2021, 2150031. http://dx.doi.org/10.1142/s0219876221500316.

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The Vlasov equation describes the temporal evolution of the distribution function of particles in a collisionless plasma and, if magnetic fields are negligible, the mean electric field is prescribed by Poisson equation. Eulerian numerical methods discretize and directly solve the Vlasov equation on a mesh in phase space and can provide high accuracy with low numerical noise. In this paper, we present a comprehensive analysis and comparison between the most used Eulerian methods for the two-dimensional Vlasov–Poisson system, including finite-differences, finite-volumes and semi-Lagrangian ones. The schemes are evaluated and compared through classical problems and conclusions are drawn regarding their accuracy and performance.

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27

Miloshevich, George, and Joshua W. Burby. "Hamiltonian reduction of Vlasov–Maxwell to a dark slow manifold." Journal of Plasma Physics 87, no. 3 (June 2021). http://dx.doi.org/10.1017/s0022377821000556.

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We show that non-relativistic scaling of the collisionless Vlasov–Maxwell system implies the existence of a formal invariant slow manifold in the infinite-dimensional Vlasov–Maxwell phase space. Vlasov–Maxwell dynamics restricted to the slow manifold recovers the Vlasov–Poisson and Vlasov–Darwin models as low-order approximations, and provides higher-order corrections to the Vlasov–Darwin model more generally. The slow manifold may be interpreted to all orders in perturbation theory as a collection of formal Vlasov–Maxwell solutions that do not excite light waves, and are therefore ‘dark’. We provide a heuristic lower bound for the time interval over which Vlasov–Maxwell solutions initialized optimally near the slow manifold remain dark. We also show how the dynamics on the slow manifold naturally inherits a Hamiltonian structure from the underlying system. After expressing this structure in a simple form, we use it to identify a manifestly Hamiltonian correction to the Vlasov–Darwin model. The derivation of higher-order terms is reduced to computing the corrections of the system Hamiltonian restricted to the slow manifold.

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28

Crestetto, A., F. Deluzet, and D. Doyen. "Bridging kinetic plasma descriptions and single-fluid models." Journal of Plasma Physics 86, no. 5 (September 15, 2020). http://dx.doi.org/10.1017/s0022377820000884.

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The purpose of this paper is to bridge kinetic plasma descriptions and low-frequency single-fluid models. More specifically, the asymptotics leading to magnetohydrodynamic regimes starting from the Vlasov–Maxwell system are investigated. The analogy with the derivation, from the Vlasov–Poisson system, of a fluid representation for ions coupled to the Boltzmann relation for electrons is also outlined. The aim is to identify asymptotic parameters explaining the transitions from one microscopic description to a macroscopic low-frequency model. These investigations provide groundwork for the derivation of multi-scale numerical methods, model coupling or physics-based preconditioning.

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29

Vedenyapin, V. V., N. N. Fimin, and V. M. Chechetkin. "The generalized Friedmann model as a self-similar solution of Vlasov–Poisson equation system." European Physical Journal Plus 136, no. 6 (June 2021). http://dx.doi.org/10.1140/epjp/s13360-021-01659-7.

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30

Yong Lee, Jae, Jin Woo Jang, and Hyung Ju Hwang. "The model reduction of the Vlasov-Poisson-Fokker-Planck system to the Poisson-Nernst-Planck system via the Deep Neural Network Approach." ESAIM: Mathematical Modelling and Numerical Analysis, August 3, 2021. http://dx.doi.org/10.1051/m2an/2021038.

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The model reduction of a mesoscopic kinetic dynamics to a macroscopic continuum dynamics has been one of the fundamental questions in mathematical physics since Hilbert's time. In this paper, we consider a diagram of the diffusion limit from the Vlasov-Poisson-Fokker-Planck (VPFP) system on a bounded interval with the specular reflection boundary condition to the Poisson-Nernst–Planck (PNP) system with the no-flux boundary condition. We provide a Deep Learning algorithm to simulate the VPFP system and the PNP system by computing the time-asymptotic behaviors of the solution and the physical quantities. We analyze the convergence of the neural network solution of the VPFP system to that of the PNP system via the Asymptotic-Preserving (AP) scheme. Also, we provide several theoretical evidence that the Deep Neural Network (DNN) solutions to the VPFP and the PNP systems converge to the a priori classical solutions of each system if the total loss function vanishes.

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31

Cavenago, Marco. "Integrodifferential models of electron transport for negative ion sources." Journal of Plasma Physics 81, no. 6 (October 8, 2015). http://dx.doi.org/10.1017/s0022377815001130.

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Thanks to the presence of a transverse magnetic flux density ($B_{x}$ and $B_{y}$ where $z$ is the extraction axis), the undesired extraction of electrons from a negative ion source is reduced and it is due to collisions. The electron transport is studied with a kinetic model, including Vlasov–Poisson effects and atomic collisions. The integrodifferential equations (IDE) resulting from a reduction to a one-dimensional problem (1-D) by integration on characteristic orbits are strongly affected by the trapped orbits, as here evaluated; a kernel calculation with a partial wave approximation is introduced. Dependencies from the local drift velocity $v_{d}$ and effective Larmor radius $L_{e}$ are found. Solutions are investigated in simple cases with a constant electron current (no additional electron production). Equilibrium solution and electron conductivity are analytically obtained. Presheath solutions are discussed; the approximated conversion to differential equations that are adequate for presheath only (with moderated electric field gradient $E_{z,z}>-eB_{x}^{2}/m$) and their numeric solutions coupled to Poisson equation are reported, and compared to iterative IDE solutions. Examples with different values of $L_{e}$ and mean free path (mfp) ratio are described.

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32

Bukhari, S., S. Ali, S. A. Khan, and J. T. Mendonca. "Twisted waves and instabilities in a permeating dusty plasma." Journal of Plasma Physics 84, no. 2 (March 13, 2018). http://dx.doi.org/10.1017/s0022377818000223.

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New features of the twisted dusty plasma modes and associated instabilities are investigated in permeating plasmas. Using the Vlasov–Poisson model equations, a generalized dispersion relation is obtained for a Maxwellian distributed plasma to analyse the dust-acoustic and dust-ion-acoustic waves with finite orbital angular momentum (OAM) states. Existence conditions for damping/growth rates are discussed and showed significant modifications in twisted dusty modes as compared to straight propagating dusty modes. Numerically, the instability growth rate, which depends on particle streaming and twist effects in the wave potential, is significantly modified due to the Laguerre–Gaussian profiles. Relevance of the study to wave excitations due to penetration of solar wind into cometary clouds or interstellar dusty plasmas is discussed.

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33

Adkins, T., and A. A. Schekochihin. "A solvable model of Vlasov-kinetic plasma turbulence in Fourier–Hermite phase space." Journal of Plasma Physics 84, no. 1 (January 25, 2018). http://dx.doi.org/10.1017/s0022377818000089.

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A class of simple kinetic systems is considered, described by the one-dimensional Vlasov–Landau equation with Poisson or Boltzmann electrostatic response and an energy source. Assuming a stochastic electric field, a solvable model is constructed for the phase-space turbulence of the particle distribution. The model is a kinetic analogue of the Kraichnan–Batchelor model of chaotic advection. The solution of the model is found in Fourier–Hermite space and shows that the free-energy flux from low to high Hermite moments is suppressed, with phase mixing cancelled on average by anti-phase-mixing (stochastic plasma echo). This implies that Landau damping is an ineffective route to dissipation (i.e. to thermalisation of electric energy via velocity space). The full Fourier–Hermite spectrum is derived. Its asymptotics are $m^{-3/2}$ at low wavenumbers and high Hermite moments ($m$) and $m^{-1/2}k^{-2}$ at low Hermite moments and high wavenumbers ($k$). These conclusions hold at wavenumbers below a certain cutoff (analogue of Kolmogorov scale), which increases with the amplitude of the stochastic electric field and scales as inverse square of the collision rate. The energy distribution and flows in phase space are a simple and, therefore, useful example of competition between phase mixing and nonlinear dynamics in kinetic turbulence, reminiscent of more realistic but more complicated multi-dimensional systems that have not so far been amenable to complete analytical solution.

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34

Friedland, L., and A. G. Shagalov. "Standing autoresonant plasma waves." Journal of Plasma Physics 86, no. 3 (May 11, 2020). http://dx.doi.org/10.1017/s0022377820000380.

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The formation and control of strongly nonlinear standing plasma waves (SPWs) from a trivial equilibrium by a chirped frequency drive are discussed. If the drive amplitude exceeds a threshold, after passage through the linear resonance in this system, the excited wave preserves the phase locking with the drive, yielding a controlled growth of the wave amplitude. We illustrate these autoresonant waves via Vlasov–Poisson simulations, showing the formation of sharply peaked excitations with local electron density maxima significantly exceeding the unperturbed plasma density. The Whitham averaged variational approach applied to a simplified water bag model yields the weakly nonlinear evolution of the autoresonant SPWs and the autoresonance threshold. If the chirped driving frequency approaches some constant level, the driven SPW saturates at a target amplitude, avoiding the kinetic wave breaking.

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35

Fiřt, Roman, and Gerhard Rein. "Stability of disk-like galaxies – Part I: Stability via reduction." Analysis 26, no. 4 (January 1, 2006). http://dx.doi.org/10.1524/anly.2006.26.4.507.

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We prove the existence and stability of flat steady states of the Vlasov–Poisson system, which in astrophysics are used as models of disk-like galaxies. We follow the variational approach developed by GUO and REIN [5, 6, 7] for this type of problems and extend previous results of REIN [11]. In particular, we employ a reduction procedure which relates the stability problem for the Vlasov–Poisson system to the analogous question for the Euler–Poisson system.

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Haas, Fernando. "Modelling of relativistic ion-acoustic waves in ultra-degenerate plasmas." Journal of Plasma Physics 82, no. 6 (November 23, 2016). http://dx.doi.org/10.1017/s0022377816001100.

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Abstract:

We consider the relativistic ion-acoustic mode in a plasma composed by cold ions and an ultra-degenerate electron gas, described the relativistic Vlasov–Poisson system. A critical examination of popular fluid models for relativistic ion-acoustic waves is provided, comparing kinetic and hydrodynamic results. The kinetic linear dispersion relation is shown to be reproduced by the rigorous relativistic hydrodynamic equations with Chandrasekhar’s equation of state.

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