640 research outputs found
Vlasov equation and -body dynamics - How central is particle dynamics to our understanding of plasmas?
Difficulties in founding microscopically the Vlasov equation for
Coulomb-interacting particles are recalled for both the statistical approach
(BBGKY hierarchy and Liouville equation on phase space) and the dynamical
approach (single empirical measure on one-particle
-space). The role of particle trajectories
(characteristics) in the analysis of the partial differential Vlasov--Poisson
system is stressed. Starting from many-body dynamics, a direct derivation of
both Debye shielding and collective behaviour is sketched.Comment: revTeX, 15 p
The Kinetic and Hydrodynamic Bohm Criterions for Plasma Sheath Formation
The purpose of this paper is to mathematically investigate the formation of a
plasma sheath, and to analyze the Bohm criterions which are required for the
formation. Bohm derived originally the (hydrodynamic) Bohm criterion from the
Euler--Poisson system. Boyd and Thompson proposed the (kinetic) Bohm criterion
from kinetic point of view, and then Riemann derived it from the
Vlasov--Poisson system. In this paper, we prove the solvability of boundary
value problems of the Vlasov--Poisson system. On the process, we see that the
kinetic Bohm criterion is a necessary condition for the solvability. The
argument gives a simpler derivation of the criterion. Furthermore, the
hydrodynamic criterion can be derived from the kinetic criterion. It is of
great interest to find the relation between the solutions of the
Vlasov--Poisson and Euler--Poisson systems. To clarify the relation, we also
study the hydrodynamic limit of solutions of the Vlasov--Poisson system.Comment: 24 pages, 2 figure
Weakly collisional Landau damping and three-dimensional Bernstein-Greene-Kruskal modes: New results on old problems
Landau damping and Bernstein-Greene-Kruskal (BGK) modes are among the most
fundamental concepts in plasma physics. While the former describes the
surprising damping of linear plasma waves in a collisionless plasma, the latter
describes exact undamped nonlinear solutions of the Vlasov equation. There does
exist a relationship between the two: Landau damping can be described as the
phase-mixing of undamped eigenmodes, the so-called Case-Van Kampen modes, which
can be viewed as BGK modes in the linear limit. While these concepts have been
around for a long time, unexpected new results are still being discovered. For
Landau damping, we show that the textbook picture of phase-mixing is altered
profoundly in the presence of collision. In particular, the continuous spectrum
of Case-Van Kampen modes is eliminated and replaced by a discrete spectrum,
even in the limit of zero collision. Furthermore, we show that these discrete
eigenmodes form a complete set of solutions. Landau-damped solutions are then
recovered as true eigenmodes (which they are not in the collisionless theory).
For BGK modes, our interest is motivated by recent discoveries of electrostatic
solitary waves in magnetospheric plasmas. While one-dimensional BGK theory is
quite mature, there appear to be no exact three-dimensional solutions in the
literature (except for the limiting case when the magnetic field is
sufficiently strong so that one can apply the guiding-center approximation). We
show, in fact, that two- and three-dimensional solutions that depend only on
energy do not exist. However, if solutions depend on both energy and angular
momentum, we can construct exact three-dimensional solutions for the
unmagnetized case, and two-dimensional solutions for the case with a finite
magnetic field. The latter are shown to be exact, fully electromagnetic
solutions of the steady-state Vlasov-Poisson-Amp\`ere system
Wigner-Poisson and nonlocal drift-diffusion model equations for semiconductor superlattices
A Wigner-Poisson kinetic equation describing charge transport in doped
semiconductor superlattices is proposed. Electrons are supposed to occupy the
lowest miniband, exchange of lateral momentum is ignored and the
electron-electron interaction is treated in the Hartree approximation. There
are elastic collisions with impurities and inelastic collisions with phonons,
imperfections, etc. The latter are described by a modified BGK
(Bhatnagar-Gross-Krook) collision model that allows for energy dissipation
while yielding charge continuity. In the hyperbolic limit, nonlocal
drift-diffusion equations are derived systematically from the kinetic
Wigner-Poisson-BGK system by means of the Chapman-Enskog method. The
nonlocality of the original quantum kinetic model equations implies that the
derived drift-diffusion equations contain spatial averages over one or more
superlattice periods. Numerical solutions of the latter equations show
self-sustained oscillations of the current through a voltage biased
superlattice, in agreement with known experiments.Comment: 20 pages, 1 figure, published as M3AS 15, 1253 (2005) with
correction
Contributions of plasma physics to chaos and nonlinear dynamics
This topical review focusses on the contributions of plasma physics to chaos
and nonlinear dynamics bringing new methods which are or can be used in other
scientific domains. It starts with the development of the theory of Hamiltonian
chaos, and then deals with order or quasi order, for instance adiabatic and
soliton theories. It ends with a shorter account of dissipative and high
dimensional Hamiltonian dynamics, and of quantum chaos. Most of these
contributions are a spin-off of the research on thermonuclear fusion by
magnetic confinement, which started in the fifties. Their presentation is both
exhaustive and compact. [15 April 2016
A High Order Stochastic Asymptotic Preserving Scheme for Chemotaxis Kinetic Models with Random Inputs
In this paper, we develop a stochastic Asymptotic-Preserving (sAP) scheme for
the kinetic chemotaxis system with random inputs, which will converge to the
modified Keller-Segel model with random inputs in the diffusive regime. Based
on the generalized Polynomial Chaos (gPC) approach, we design a high order
stochastic Galerkin method using implicit-explicit (IMEX) Runge-Kutta (RK) time
discretization with a macroscopic penalty term. The new schemes improve the
parabolic CFL condition to a hyperbolic type when the mean free path is small,
which shows significant efficiency especially in uncertainty quantification
(UQ) with multi-scale problems. The stochastic Asymptotic-Preserving property
will be shown asymptotically and verified numerically in several tests. Many
other numerical tests are conducted to explore the effect of the randomness in
the kinetic system, in the aim of providing more intuitions for the theoretic
study of the chemotaxis models
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