The Child–Langmuir Asymptotics of the Vlasov–Poisson Equation for Cylindrically or Spherically Symmetric Diodes Part 1: Statement of the Problem and Basic Estimates

The Child–Langmuir asymptotics of the Vlasov–Poisson system provides a model for vacuum diodes which operate under large biases. In these conditions the energy of the injected particles at the cathode is very small compared with the applied external bias. From the mathematical view point, this leads to an interesting and non-standard asymptotic problem for the Vlasov–Poisson equation, which has already been investigated in the one-dimensional Cartesian case, in [7]. The purpose of this paper is to extend the analysis to the cylindrically or spherically symmetric case. Surprisingly, the behaviour of the solutions of the model is somehow different than in the Cartesian case. This feature had not been noticed by the physicists before. Furthermore, the mathematical analysis is much more involved than in [7] because of the geometrical effects, and the techniques that are used are quite different. They mainly rely on the use of supersolutions in the spirit of [18, 19]. This work is divided in two parts. In this first part, we state the problem and establish the basic estimates which are needed for the asymptotic analysis.     

Analysis of a bipolar energy-transport model for a metal-oxide-semiconductor diode

A simplified bipolar energy-transport model for a metal-oxide-semiconductor diode (MOS) with nonconstant lattice temperature is considered. The electron and hole current densities vanish in the diode but the particle temperature may be large. The existence of weak solutions to the system of quasilinear elliptic equations with nonlinear boundary conditions is proved using a Stampacchia trunction technique and maximum principle arguments. Further, an asymptotic analysis for the one-dimensional MOS diode is presented, which shows that only the boundary temperature influences the capacitance-voltage characteristics of the device. The analytical results are underlined by numerical experiments.     

Un modèle d'expansion de plasma dans le videA model for plasma expansion in vacuum

In this paper, we propose a model describing the expansion of a plasma in vacuum. Our starting point consists of a 2-fluid Euler system (isentropic case) coupled with the Poisson equation. Since numerical simulations of this model are very expensive, we investigate a quasi-neutral limit of it. We show that electron emission happens at the plasma–vaccum interface. This emission is well modeled by a Child–Langmuir law. The difficulty consists in accounting for the motion of the plasma–vacuum interface. In this paper, we formally and numerically justify why electron emission produces a reaction pressure which slows down the plasma expansion. To cite this article: P. Degond et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 399–404.     

Effects of saline volume on lesion formation during saline-infused radiofrequency ablation

Radiofrequency ablation (RFA) is hindered when tissue temperature reaches 100 °C. At this temperature, tissue vaporization occurs, which prevents electrical currents from flowing into the tissue. Smaller lesions are created as a result. One way to overcome this is to infuse saline into the tissue prior to RFA. Saline can increase the electrical conductivity of the surrounding tissue due to the abundance of ions inside the solution. This permits a greater distribution of electrical currents that can lead to larger lesion sizes. Although this procedure has been shown to produce larger lesions, the technique is risky due to the potential for the saline to extravasate into healthy regions of the tissue; hence, leading to unnecessary ablation. A computational model is developed in this paper to predict the formation of lesion following saline-infused RFA. The model incorporates the transport of saline inside the tissue, the resistive heating during ablation and the formation of lesion through cell death modeling. Results show that there is an optimal infusion volume that leads to the largest increase in the lesion size. With further developments, the model may be used for the planning of saline-infused RFA treatment.     

Reflection and transmission of plasma waves at the interface of crystallites

A linearized problem of the reflection and transmission of a plasma wave at the boundary of a half-space (namely, the plane separating two crystallites) is formulated and analytically solved. The electron distribution function and the electric field inside the half-space of a degenerate plasma are found. The reflection and transmission coefficients are determined as functions of the input parameters of the problem. The longwave limit (i.e., the resonance case when the oscillation frequency of the self-consistent electric field is close to the natural (Langmuir) oscillation frequency) is analyzed.     

Lie group analysis of gravity currents

We consider shallow water theory to study the self-similar gravity currents that describe the motion of a heavy fluid flowing into another lighter ambient fluid. Gratton and Vigo investigated the shallow water theory representing the self-similar gravity currents by using dimensional analysis [J. Gratton, C. Vigo, Self-similarity gravity currents with variable inflow revisited: Plane currents, J. Fluid. Mech. 258 (1994) 77–104]. But in this study, the self-similarity solutions of the one-layer shallow-water equations representing gravity currents are investigated by using Lie group analysis and it is shown that Lie group analysis is the generalization of the dimensional analysis for investigating the self-similarity solutions of the one-layer shallow-water equations. Applying Lie group theory, reduced equations of the shallow water equations are found. Therefore, it becomes possible to obtain the similarity forms depending on the Lie group parameters and also the self-similarity solutions for the special values of these group parameters.     

Linear and nonlinear response of pipeline suspension bridge due to moving fluid

Basing on the nonlinear dynamic model of flexible pipeline suspended by spatial system of cables, described in Ref. [1], the linear and nonlinear vibrations are investigated in order to estimate the nonlinear effects. The model is based on substructure technique and formulated including features specific to analyzed structure, for example large displacements and time dependent parameters appearing in equations of motion due to fluid flowing inside the pipeline. Due to the fact that modelling problem for the analyzed structure is one's own complicated, a simple case when the conveying fluid is idealized simply as a ballast moving inside the pipe is considered. This paper presents a short numerical analysis of linear and nonlinear, static and dynamic response of exemplary structure for three different cases: during filling the pipe with fluid, when the pipeline is completely filled and during emptying the pipe. Moreover, for the linear problem, the influence of a speed of the fluid on the stability of the pipeline suspension bridge is investigated. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vanishing viscosity and Debye-length limit to rarefaction wave with vacuum for the 1D bipolar Navier–Stokes–Poisson equation

In this paper, we consider the one-dimensional (1D) compressible bipolar Navier–Stokes–Poisson equations. We know that when the viscosity coefficient and Debye length are zero in the compressible bipolar Navier–Stokes–Poisson equations, we have the compressible Euler equations. Under the case that the compressible Euler equations have a rarefaction wave with one-side vacuum state, we can construct a sequence of the approximation solution to the one-dimensional bipolar Navier–Stokes–Poisson equations with well-prepared initial data, which converges to the above rarefaction wave with vacuum as the viscosity and the Debye length tend to zero. Moreover, we also obtain the uniform convergence rate. The results are proved by a scaling argument and elaborate energy estimate.     

Modeling the interaction of a wave front with a forest

We investigate the processes that arise when a wave front hits a natural obstacle in the form of a forest. The modeling is carried out in the framework of a single methodological approach that uses the Euler equation to describe the motion of the air mass both over an open area and inside the forest. In the latter case the equations include mass forces associated with the vegetation. The numerical solution is obtained by Godunov’s method using parallel programming techniques. Two types of incident wave front are investigated: a plane shockwave and a nonlinear acoustic impulse modeling a spherical explosion wave at a large distance from the source. The specific features of the interaction process, including penetration of the wave front into the forest, partial reflection from the near boundary, and diffraction above the top boundary, are investigated for different types of vegetation (coniferous and deciduous forests). The numerical results reveal the formation of a pair of ascending and descending currents in the upper part of the forest (inside the tree crowns). The existence of this structure is confirmed by experimental findings. __________ Translated from Prikladnaya Matematika i Informatika, No. 21, pp. 48–71, 2005.     

On the Measurement of Information in the Field of Criminal Detection

A method for measuring the value of information in those fields where the meaning of messages is important is discussed. At present there exists no accepted measure of information in such problem areas, the classical unit of bits flowing per second being unacceptable. It is shown that where two sets of phenomena are associated in some way with a given set of probabilities, e.g. a population and the crimes which are committed within it, the problem reduces to how much “choice” exists. This enables the concept of entropy to be used to advantage.After developing a measure of the value of information in the general case, the paper applies the method to the investigation of a case of simple larceny. This case illustrates several interesting features. Perhaps the most important feature is that some pieces of information, although they are very important to the police, only change the level of uncertainty very slightly. In such cases the piece of information generally demands that the police perform an action which either may provide useful information or lead to further action. To overcome this problem a potential entropy change is defined which takes this factor into account.It is hoped that the work may lead to a fuller understanding of how information flows in the police network. Thus it may be possible to see if the right information is getting into the police system, getting lost inside it or whether it is being used most efficiently.     

Computing the Particle Paths in an Open-Trap Sharp-Point Geometry

The article presents simulation results for the motion of electrons in two different regions: a high-current vacuum diode and an open trap. The entire system is immersed in an external magnetic field with sharp-point geometry. The electrons are continuously injected from a part of the cathode into the diode region, where they are accelerated to relativistic velocities. In this region, the self-consistent problem of particle motion in electromagnetic fields is solved by the large particle method. In the magnetic trap region, the particles experience only the external magnetic field, and their paths are computed by the test particle method. The simulation efficiency is improved by partitioning the problem into two: separate simulation of the high-voltage diode and integration of the particle paths in the trap. Calculations show that the bulk of the particles leave the trap through the side walls, moving along the magnetic forcelines, and only a small part of the particles undergo multiple reflections and remain longer inside the trap.     

Image of the braid groups inside the finite Temperley–Lieb algebras

We determine the image of the braid groups inside the Temperley–Lieb algebras, defined over finite field, in the semisimple case, and for suitably large (but controlable) order of the defining (quantum) parameter. We also prove that, under natural conditions on this parameter, the representations of the Hecke algebras over a finite field are unitary for the action of the braid groups.     

The nonlinear current behaviour of a driven R–L-Varactor in the low frequency range

The nonlinear behaviour of an R–L-Varactor circuit, simulated by Multisim 7.0 at a driving frequency that is below the circuit resonance frequency, is reported and evaluated. A new high amplitude oscillation is observed and attributed to the emergence of a large diode junction capacitance. Increasing the driving signal amplitude, the circuit is led to a non-periodic mode of operation, producing trajectories of increasing chaotic content in a four-dimensional phase space. At specific amplitudes a two-dimensional tori was monitored, where trajectories are periodic, like in a two oscillator system with commensurate frequencies. Poincaré cross-sections, FFT spectra and correlation dimension calculations, suggest the quasi-periodicity route to chaos.     

Vanishing electron mass limit in the bipolar Euler–Poisson system

The bipolar Euler–Poisson system in physics consists of the conservation laws for the electron and ion densities and their current densities, coupled with the Poisson equation for the electrostatic potential. The limit of vanishing ratio of the electron mass to the ion mass in the $n$-dimensional flat torus is proved in the case of well prepared initial data. The limiting system is composed of two separated equations, where the equation for electron is the incompressible Euler equation with damping, which means physically that the evolution for electrons and ions can be treated as separated motions in the small ratio case.     

Global existence and asymptotic behavior of smooth solutions to a bipolar Euler–Poisson equation in a bound domain

In this paper, we present a bipolar hydrodynamic model from semiconductor devices and plasmas, which takes the form of bipolar isentropic Euler–Poisson with electric field and frictional damping added to the momentum equations. We firstly prove the existence of the stationary solutions. Next, we present the global existence and the asymptotic behavior of smooth solutions to the initial boundary value problem for a one-dimensional case in a bounded domain. The result is shown by an elementary energy method. Compared with the corresponding initial data case, we find that the asymptotic state is the stationary solution.     

The effects of the no-touch gap on the no-touch bipolar radiofrequency ablation treatment of liver cancer: A numerical study using a two compartment model

The no-touch bipolar radiofrequency ablation (RFA) for cancer treatment is advantageous primarily because of its capability to prevent tumour track seeding (TTS). In this technique, the RF probes are placed at a distance (no-touch gap) away from the tumour boundary. Ideally, the RF probes should be placed sufficiently far from the tumour in order to avoid TTS. However, having a gap that is too large can lead to ineffective ablation. This paper investigates how the selection of the no-touch gap can affect the tissue electrical and thermal responses during the no-touch bipolar RFA treatment. Simulations were carried out on a two compartment model using the finite element method. Results obtained indicated that a gap that is too large may lead to incomplete ablation and failure to achieve significant ablation margin. However, keeping the gap to be too small may not be clinically practical. It was suggested that the incomplete ablation and the insufficient ablation margin observed in some of the cases may require the placement of additional probes around the tumour. The present study stresses on the importance of identifying the optimal no-touch gap that can avoid TTS without compromising the treatment outcome.     

Identification of an inclusion in multifrequency electric impedance tomography

The multifrequency electrical impedance tomography is considered to image a conductivity inclusion inside a homogeneous background medium by injecting one current. An original spectral decomposition of the solution of the forward conductivity problem is used to retrieve the Cauchy data corresponding to the extreme case of perfect conductor. Using results based on the unique continuation, we then prove the uniqueness of multifrequency electrical impedance tomography and obtain rigorous stability estimates. Our results in this paper are quite surprising in inverse conductivity problem since in general infinitely many input currents are needed to obtain the uniqueness in the determination of the conductivity.     

A study on the method of fundamental solutions using an image concept

In this paper, both analytical and semi-analytical solutions for Green’s functions are obtained by using the image method which can be seen as a special case of method of fundamental solutions (MFS). The image method is employed to solve the Green’s function for the annular, eccentric and half-plane Laplace problems. In addition, an analytical solution is derived for the fixed-free annular case. For the half-plane problem with a circular hole and an eccentric annulus, semi-analytical solutions are both obtained by using the image concept after determining the strengths of two frozen image points and a free constant by matching boundary conditions. It is found that two frozen images terminated at the two focuses in the bipolar coordinates for the problems with two circular boundaries. A boundary value problem of an eccentric annulus without sources is also considered. Error distribution is plotted after comparing with the analytical solution derived by Lebedev et al. using the bipolar coordinates. The optimal locations for the source distribution in the MFS are also examined by using the image concept. It is observed that we should locate singularities on the two focuses to obtain better results in the MFS. Besides, whether the free constant is required or not in the MFS is also studied. The results are compared well with the analytical solutions.     

Modelling of displacement washing of pulp fibers using orthogonal collocation on finite elements

Flow of fluid through packed bed of porous particles is modelled with the help of Peclet number (Pe) and Biot number (Bi). Packed bed is divided into three zones, flowing liquor, intrapore solute present in pores of particles and solute adsorbed on particle surface. Langmuir isotherm is used to describe the relationship between intrapore solute concentration and concentration of solute adsorbed on particle surface, whereas the bulk fluid concentration and the intrapore solute concentration are related by linear adsorption isotherm. Model predicted values are also compared with the experimental values. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Equilibrium State of Inhomogeneous Plasma

We consider the problem of a self-consistent determination of an essentially inhomogeneous equilibrium state of classical plasma. The solutions of the stationary Vlasov–Poisson equations are constructed in the form of a localized transition layer that separates the domains of homogeneous plasmas with different equilibrium parameters. The layer can also transform into a local perturbation inside a homogeneous plasma. In both cases, the solution contains neither mass currents nor electric currents, and all electrodynamic and hydrodynamic quantities and their derivatives are continuous. The parameters of the adjacent domains uniquely determine the transition layer structure.