Reply to comments on "On asymptotic matching and the sheath Edge"

In our recent paper [Sternberg and Godyak (2004)], we have solved the plasma-sheath problem using asymptotic matching techniques. We have shown that the position of the sheath edge obtained by asymptotic matching yields the position of the ion sheath edge determined by Godyak's condition (E=kT/sub e//(e/spl lambda//sub D1/)) [Godyak (1982)]. In addition, we were able to show how the sheath width depends on the voltage applied to the wall. Our results are theoretical, grounded in mathematics. We used numerical computations mainly to illustrate our point. We have written our paper with great care, so that any scientist, even one not familiar with the intricacies of asymptotic matching, could understand our exposition and be convinced in the correctness of our results. Kaganovich (2004) reproduced our theoretical results numerically up to /spl epsi/=10/sup -11/. Franklin (2004) published a review adapting our results and interpretations. However, we were not able to convince Riemann, although he reproduced some of our results numerically [Riemann (2004)]. This paper is a response to Riemann's comments [Riemann (2004)].     

Comments on "Double-Wave Theory"

Some falsehoods of the so-called double-wave theory (DWT) are pointed out. The logic and the origin of the DWT, and three of the four fundamental hypotheses of DWT are criticized. It is expounded that the hypothesis of two wave functions is unreasonable, the hypothesized Hamiltonian differs from the actually used Hamiltonian, and the hypothesized expression of measured values or mean values of mechanical quantities is wrong and does not express the measured values or mean values at all.     

Comments on "Speech Understanding and Aging"

The hypothesis advanced by the CHABA Working Group on Speech Understanding and Aging [J. Acoust. Soc. Am. 83, 859-895 (1986)] that the systematic decline in speech understanding with age might be explained by concomitant decline in extra-auditory cognitive factors was tested by examining speech audiometric findings in patients with dementia. The fact that performance was consistent with normal central auditory function in 12 of 23 such patients, in spite of deficits in immediate memory for spoken material, tolerance of distraction, mental tracking and sequencing, cognitive flexibility, and set shifting argues against the hypothesis that speech understanding deficits in the elderly can be explained as the simple consequence of cognitive decline.     

On the collisional transient sheath

A theory of the transient sheath spanning the entire range of ion collisionality and treating finite rise-time voltage pulses is developed. This theory predicts the maximum ion impact energy, and the temporal dependence of the sheath width, the ion impact energy, and the ion flux. It is found that the ion impact energy is greatly reduced by even small amounts of collisionality, whereas the sheath width and ion flux at the target (i.e., ion current density) are relatively insensitive to collisionality. In the collisional sheath, the maximum ion impact energy is found to decrease with increasing rise time and collisionality. Good agreement is found between the theory and fluid simulations. In particular, the collisional dependence of the ion impact energy is well described by the theory     

Semiclassical anomalies of the quantum mechanical systems and their modifications for the asymptotic matching

JWKB solutions to the Initial Value Problems (IVPs) of the Time Independent Schrodinger’s Equation (TISE) for the Simple Linear Potentials (SLPs) with a turning point parameter have been studied according to the turning points by graphical analysis to test the results of the JWKB solutions and suggested modifications. The anomalies happening in the classically inaccessible region where the SLP function is smaller than zero and the results of the suggested modifications, which are in consistent with the quantum mechanical theories, to remove these anomalies in this region have been presented. The origins of the anomalies and verifications of the suggested modifications showing a great success in the results have also been studied in terms of a suggested matrix elements made up of the JWKB expansion terms, S_i−1,j (where i = 1, 2, 3 and j = 1, 2). The results of the modifications for the IVPs and their application to the Bound State Problems (BSPs) with an example application of the Harmonic Oscillator (HO) have been presented and their generalization for any potential function have been discussed and classified accordingly.     

On the asymptotic nonlinearity in smectics

Isolated sources of deformation in smectics for which the problem of determining the displacement field is significantly nonlinear at arbitrarily long distances even for a small action amplitude are indicated. Nonlinear asymptotic expressions can be determined in some cases.     

On the rate of asymptotic eigenvalue degeneracy

The gap between asymptotically degenerate eigenvalues of one-dimensional Schrödinger operators is estimated. The procedure is illustrated for two examples, one where the solutions of Schrödinger's equation are explicitly known and one where they are not. For the latter case a comparison theorem for ordinary differential equations is required. An incidental result is that a semiclassical (W-K-B) method gives a much better approximation to the logarithmic derivative of a wave-function than to the wave-function itself; explicit error-bounds for the logarithmic derivative are given.     

On the determination of asymptotic normalization coefficients

The problem of determining asymptotic normalization coefficients (ANC) from experimental data is considered. It is shown that the ANC could be extracted from the single partial-wave phase shift if one applies the analyticity requirement. A new method of determining ANCs is suggested, which makes use both of the experimental information on phase shifts and of the analytic properties of scattering amplitudes.     

On the asymptotic decay of pair correlations

Asymptotic decay of pair correlations is analyzed on the basis of the analytical solution of the Percus-Yevick equation for a square-well system. It is shown that in the — T plane there is a closed region containing both the triple and the critical point where the decay is monotonic. In the high-density region along the line of solidification and in the very high temperature region for all densities the decay has the form of damped oscillations.     

On the asymptotic equivalence between the Enskog and the Boltzmann equations

An asymptotic equivalence theorem is proven between the solutions of the initial value problem in all space for the Boltzmann and Enskog equations for initial data which assure global existence for the solutions to the initial value problem for one of the two equations. The proof is given starting from the solution of the Boltzmann equation, then the proof line is simply indicated when one starts from the Enskog equation. The proof holds for Knudsen numbers of the order of unity and equivalence is proven when the scale of the dimensions of the gas particles characterizing the Enskog equation tends to zero.On leave from Department of Mathematics, University of Warsaw, Poland.     

评"相对论的质心运动定理"

在相对论情况下,质点系统的质心是一个不确定的概念,在相对论力学中,质心系（质量中心参照系）亦被有确切定义的动量中心系（系统总动量为零的参照系）所代替,本文对《相对论的质心运动定理与质量亏损》一文的论证和结果作出了评论。     

On the asymptotic behaviour of infinite gradient systems

We consider gradient systems of infinitely many particles in one-dimensional space interacting via a positive invariant pair potential with a hard core. The main assumption is that is strictly convex within the rangeR of (whereR is a fixed number ). Under some technical conditions we prove the following theorems: Let the initial distribution be given by a translation invariant point process onR ¹. Then there exists only one extreme equilibrium state with a given intensityI() satisfyingI()R ^–1, and all ergodic initial distributions with an intensityI()R ^–1 converge weakly ast to the extreme equilibrium state with the same intensity.     

On the asymptotic behavior of an earthquake model

We present evidence showing that there are important differences in the relaxation times associated with different aspects of the model of Olami, Feder and Christensen. The asymptotic behavior is characterized by a critical exponent that is independent of α, the degree of conservation of the model. The analysis of temporal series of avalanches gives helpful information to study the approach to self-organized criticality, and to put in evidence any quasi-periodic evolution.     

On the asymptotic behavior of random walks on an anisotropic lattice

A random walk on a two-dimensional lattice with homogeneous rows and inhomogeneous columns, which could serve as a model for the study of some transport phemonema, is discussed. Subject to an asymptotic density condition on the columns it is shown that the horizontal motion of the walk is asymptotically like that of rescaled Brownian motion. Various consequences of this are derived including central limit, iterated logarithm, and mean square displacement results for the horizontal component of the walk.     

On matching conditions in the WKB method

The modified matching conditions for quasiclassical wave functions on both sides of a turning point for the radial Schrödinger equation have been obtained. They differ significantly from the usual Kramers condition which holds for the one-dimensional case. Namely, the ratio C₂/C₁ in the subbarrier and the classical allowed regions is not a universal constant ( , as usual), but depends on the values of the orbital angular momentum l, energy E and on the behaviour of the potential V(r) at r → 0. The comparison with exact and numerical solutions of the Schrödinger equation shows that the modified matching conditions not only make the quasiclassical approximation in the subbarrier region asymptotically exact within the n → ∞ limit, but also considerably enhances its accuracy even in the case of small quantum numbers, n 1. The power-law, funnel and short-range potentials are considered in detail.     

Comments on the Bohm Criterion and on the Determination of the Ion Velocity at the Sheath Edge in Non‐Maxwellian Plasma

This work is devoted to the study of the Bohm criterion in the general case of the electron energy distribution function (EEDF). Investigations are performed by means of a Monte Carlo integration method. We resolve the cold fluid equation system describing the ion motion within the sheath, assuming collisionless conditions, singly charged and mono kinetic incoming ions (BOHM model). Results confirm that the limit ion velocity at the sheath edge to assure a monotone electric field with a positive charge over the entire sheath is v_i ≥ (kT_e/M_i) or ε_i ≥ 1/3 <ε_e> in the case of Maxwellian electrons. We show that in the case of a Druyvesteyn electron energy distribution, this limit is larger, it is ε_i ≥ 0.6 <ε_e>. The study is also extended to other distributions functions. Because of the large controversy in recent publications, concerning the boundary conditions at the sheath entrance, we discuss the collisionless conditions at the sheath edge according to the plasma parameters. It is shown that in a collisionless sheath, the condition n_i(χ) ≥ n_e(χ) can be used to determine the limit ion velocity at the sheath edge of the negatively biased collector (Langmuir probe for instance) (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the asymptotic eigenvalues of an inhomogeneous circular waveguide

An asymptotic formula is presented by means of which WKB's eigenvalues of a circular waveguide can be corrected in analytic form. Maslov's formula derived in 1960 for the determination of Schrödinger's eigenvalues is given for comparison. Both results are shown to be in complete agreement in a specific case.