Advances in kinetic theory and computing : selected papers

218 Pages·1994·23.672 MB·English

by  Perthame B. (ed.)

ADVANCES IN
r-KlNETIC
1HE0RYAND
QOMPUTING
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Series on Advances in Mathematics for Applied Sciences - Vol. 22
Klli:;t;c
THEORYAND
OMPUTING
Selected Papers
Editor
B. Perthame
Universite Pierre et Marie Curie
France
11»
World Scientific
II
Singapore· New Jersey· London· Hong Kong
Published by
World Scientific Publishing Co. Pie. Lid.
P O Box 128. Fairer Road. Singapore 9128
USA office: Suite IB, 1060 Main Street. River Edge, NJ 07661
UK office: 73 Lymon Mead. Totteridge, London N20 SDH
ADVANCES IN KINETIC THEORY AND COMPUTING:
SELECTED PAPERS
Copyright© 1994 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved This book, or parts thereof, may not be reproduced in any form
orby any means, electronic or mechanical includingphotocopying, recording or any
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the Copyright Clearance Center, Inc., 27 Congress Street, Salem, MA01970.U5A.
ISBN: 981 -02-1671-8
Printed in Singapore by Utopia Press.
V
FOREWORD
This book contains a collection of papers dealing with the kinetic theory
of gases. Computational aspects are discussed as well as the applications and
modelisation or macroscopic properties related to the kinetic structures. The
underlying models are those of Vlasov-Poisson or Boltzmann equations as used
in the modern sciences (plasma physics, semiconductors, hypersonic flows etc.).
The idea of collecting these papers came out after a one-day workshop
organized by J. Mossino for physicists and mathematicians in Orleans. Al
though the book contains a larger number of papers, its main body is that of
the conference and I would like to thank J. Mossino for the time she spent
in organizing this successful conference and M. Feix for the hospitality of the
CNRS in Orleans.
B. Pertkame
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vii
CONTENTS
Foreword v
I. Vlasov-Poisson in Plasma Physics
The Child-Langmuir Law in the Kinetic Theory of Charged-Particles.
Part 1, Electron Flows in Vacuum
P. Degond 3
Eulerian Codes for the Vlasov Equation
M. R. Feix, P Bertrand and A. Ghizzo 45
Mathematical Models of Ion Extraction and Plasma Sheaths
S. Mas-Gallic and P A. Ravtart 82
II. Quantum Mechanics and Semiconductors
Transport Equations for Quantum Plasmas
G. Manfredi and M. R. Feix 109
Mathematical Theory of Kinetic Equations for Transport
Modelling in Semiconductors
F. Poupaud 141
III. Boltzmann Equations and Gas Dynamics
On Zero Pressure Gas Dynamics
F. Bouckut 171
A Remark Concerning the Chapman-Enskog Asymptotics
L. Dtsvillettts and F. Golse 191
Introduction to the Theory of Random Particle Methods
for Boltzmann Equation
B. Perthame 204
3
THE CHTLD-LANGMUIR LAW IN THE KINETIC THEORY OF
CHARGED-PARTICLES. PART 1, ELECTRON FLOWS IN VACUUM
PIERRE DEGOND
Math&natiques pour ('Industrie el la Physique
C.N.R.S. UFR MIC, University Paul Sabalier
US, route de Narbonne 31062 Toulouse Cedex, FYance
This paper is the first part of a series of three papers reviewing the mathematical
theory of the Child-Langmuir law in the kinetic theory of charged-particle beams
and various of its applications-
1. Introduction
The Child-Langmuir law goes back to the early studies of Child, Langmuir and
Compton [19] in the early 30's. It states that the maximal current which can flow
through a plane vacuum diode cannot exceed a limiting value, independent on the
way the electrons are extracted from the cathode (and in particular on how much of
them are extracted), and which only depends on the length of the diode and on the
applied potential. The Child-Langmuir formula for the current J is the following :
(1.1)
where e is the vacuum permittivity, e and m are the electron charge and mass,
u
<fi is the applied potential and L is the length of the diode.
The reason why the current cannot be increased beyond the value J, by simply
increasing the number of extracted electrons originates from the presence of a space-
charge layer which builds up close to the cathode. This space-charge layer generates
a potential barrier which reflects part of the emitted electrons back to the cathode.
This potential barrier adjusts itself so that it maintains the constant value of the
current (1.1) whatever the emission conditions are.
This phenomenon can be mathematically modelled by a suitable perturbation
analysis applied to the kinetic model of electron transport: the Vlasov equation.
Indeed, the plane vacuum diode can be described by a boundary value problem
4
for the Vlasov-Poisson equation, in which the typical energy associated with the
boundary value of the distribution function is small compared with the applied
potential. In scaled variables, this leads to a perturbation problem which cannot be
reduced to any known singular perturbation problem. The purpose of this paper
is to review a series of works which have been done on this problem. A certain
number of its extensions, including application to plasmas, magnetized flws and
semiconductors will be reviewed in two forthcoming papers [5] and [6], following
the works done in [7], [16], [lj, [3], [4], [2], [8], [14] and [13],
The paper is organized as follows : in section 2., the simplest case of the plane
vacuum diode will be analyzed and the general features of the Child-Langmuir
asymptotics will be outlined, following [15]. In section 3., the analysis of the bound
ary layer close to the cathode will be performed (see [15)). In section 4., the first step
towards the multidimensional extension of this analysis will be done : it consists
in the investigation of the cylindrical or spherical diode, and follows [10] and [11].
Section 5. is devoted to a formal extension of this work to the fully multidimensional
case (see [12]} Section 6. concludes the paper. In the whole paper, attention will
be paid at the same time to the rigorous mathematical theory and to the physical
relevence of the models. Physical references on the Child-Langmuir law are [19]
or [20], Most of the devices relying on electron beam propagation operate in the
Child-Langmuir regime. Numerical computations of such devices can be found in,
among other references, [18] or [22].
2. The plane vacuum diode
2.1. Setting of the problem
We consider a plane diode which consists of a pair of electrodes the cathode is
located at X = 0, and the anode is located at X = L. Electrons are emitted at the
cathode only (none are emitted at the anode). We only consider elctrons flowing in
the vacuum, and we neglect binary interactions. The electric held which accelerates
the electrons is due to the charge of the electrons themselves (self-consistent field)
and to the imposed voltage between the electrodes. We set the potential 5>(A)to
zero at the cathode and to the applied potential $/, at the anode. We denote the
electron distribution function by F{X, V),X g [0, LJ, V € IR.
The flow of electrons between the electrodes is governed by the Vlasov-Poisson
system :
Z^.= ±N(X), A-e[0,L], (2.2)
F(X,V)dV, X E [O.LJ. (2.3)
-oo
5
F(0,V) = G(V), V>0, (2.4)
F{L,V) = 0, V <0, (2.5)
${0) = 0, *(Z) = * > 0. (2.6)
L
where N{X) is the electron density and G(V), V > 0, is the distribution of in
jected electrons at the cathode. The existence of solutions of the non-linear problem
(2.1)-(2.6) is proved in [17], An extension to the multidimensional case can be found
in 121],
The typical energy associated with the boundary value G[V) is the thermal
energy \mV£ where
V = ( V2G{V)dV I / G(V)dV)i (2.7)
C
Jo Jo
Throughout this paper, we shall investigate what happens when this thermal
energy is small compared with the applied potential energy
|mVg
< eft (2.8)
We thus introduce a small parameter e by •
and we shall be concerned with the limit s —> 0.
In physical terms, V is a typical value of the velocity of the injected electrons
G
inside the domain. Formula (2.8) implies that such a velocity is very small compared
with the typical velocity Vr, that the elctrons can acquire due to their acceleration
by the electric field, and which can be measured by
1 ;2e4>i
-mVl = e*i., V = yj . (2.10)
L
If velocities are scaled by V/,, this means that V is small of the order of e. If
G
simultaneously the density associated with the boundary value G :
N = I G{V)dV (2.11)
G
is left of order 1, the injected current in the diode NVq = O(e) is very small,
G
and very few interesting phenomena are likely to happen. Therefore, in our asymp
totics, we'd better impose the injected current :
J = / VG{V)dV = 0(1) (2.12)
G
Jo
to be of order 1, and the density to be large of order l/e.