Eigenstates and scattering solutions for billiard problems: A boundary wall approach

It was proposed about a decade ago [M.G.E. da Luz, A.S. Lupu-Sax, E.J. Heller, Phys. Rev. E 56 (1997) 2496] a simple approach for obtaining scattering states for arbitrary disconnected open or closed boundaries C, with different boundary conditions. Since then, the so called boundary wall method has been successfully used to solve different open boundary problems. However, its applicability to closed shapes has not been fully explored. In this contribution we present a complete account of how to use the boundary wall to the case of billiard systems. We review the general ideas and particularize them to single connected closed shapes, assuming Dirichlet boundary conditions for the C’s. We discuss the mathematical aspects that lead to both the inside and outside solutions. We also present a different way to calculate the exterior scattering S matrix. From it, we revisit the important inside-outside duality for billiards. Finally, we give some numerical examples, illustrating the efficiency and flexibility of the method to treat this type of problem.     

一类Marangoni对流边界层方程的近似解析解

利用Adomain解析拆分和Padé逼近方法对由Marangoni对流诱发的层流边界层问题进行了研究, 提供了一种求解边界层方程的解析分析方法. 得到了问题的近似解析解并对相应的流动及传热特性进行了探讨. 本文所提出的思想方法可以用于解决其他科学和工程技术问题. 关键词： Marangoni对流 非线性 Adomain拆分法 近似解析解     

Diffusion approximations of the two dimensional master equation

The effects of rotational energy transfer on unimolecular dissociation are assessed using a two-dimensional master equation approach. The collision operator for such a system can be very large, and approximations are sought that reduce the size of the matrix that has to be stored. The density of states of the vibrational energy is large for most molecules and can be treated as virtually continuous. As a consequence, motion in vibrational energy space can be treated using a diffusion equation approach. This reduces the storage requirements greatly and increases the speed at which diagonalization occurs. Results are presented for the dissociation of ethane and methane.     

Similarity solutions for two-dimensional magnetogasdynamic boundary layer flow in a transverse magnetic field

Summary The conditions for the existence of the similarity solution of a two-dimensional boundary layer flow past a plane wall of an electrically conducting gas, in the presence of a transverse magnetic field at small magnetic Reynolds number, are studied. It is found that, in general, the similarity requirements involve as many as eleven parameters out of which eight are the same as these of nonmagnetic gasdynamic boundary layer flow. The solution is obtained for power law velocity distribution, in Illingworth variables, at the outer edge of the boundary layer and for the corresponding variable magnetic field. The Prandtl number of the fluid is taken as unity and the magnetic-field effects are confined to the boundary layer only.

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Riassunto Si studiano le condizioni per l’esistenza della soluzione di somiglianza di un flusso nello stato di confine bidimensionale oltre una parete piana di un gas che conduce electtricità, in presenza di un campo magnetico trasverso con piccolo numero magnetico di Reynolds. Si trova che, in generale, le esigenze di somiglianza coinvolgono undici parametri, otto dei quali sono identici a quelli del flusso della strato di confine non magnetico e gas dinamico. La soluzione è ottenuta per la distribuzione di velocità della legge di potenza, in variabili di Illingworth, nel bordo esterno dello strato limite e per il corrispondente campo magnetico variabile. Il numero di Prandtl del fluido è preso uguale a uno e gli effetti del campo magnetico sono confinati solo allo strato limite.

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Резюме Исследуются условия существования подобного решения для двумерного течения в пограничиом слое вдоль плосокой стенки газа, проводящего электричество, в присутствии поперечного магнитного поля при малых магнитных числах Рейнольдса. Получено, что в общем случае требования подобия включают одиннадцать параметров, восемь из которых являются соответствующими параметрами для немагнитного газодинамического течения в пограничном слое. Ппределено решение в случае степенного закона распределения по скоростям на внешнем краю пограничного слоя и для соотвтствуюего переменного магнитного поля. Число Прандтля для жидкости считается равным единице и эффекты магнитного поля ограничиваются только пограничным слоем.

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Jacobian transformed and detailed balance approximations for photon induced scattering

Photon emission and scattering are enhanced by the number of photons in the final state, and the photon transport equation reflects this in scattering–emission kernels and source terms. This is often a complication in both theoretical and numerical analyzes, requiring approximations and assumptions about background and material temperatures, incident and exiting photon energies, local thermodynamic equilibrium, plus other related aspects of photon scattering and emission. We review earlier schemes parameterizing photon scattering–emission processes, and suggest two alternative schemes. One links the product of photon and electron distributions in the final state to the product in the initial state by Jacobian transformation of kinematical variables (energy and angle), and the other links integrands of scattering kernels in a detailed balance requirement for overall (integrated) induced effects. Compton and inverse Compton differential scattering cross sections are detailed in appropriate limits, numerical integrations are performed over the induced scattering kernel, and for tabulation induced scattering terms are incorporated into effective cross sections for comparisons and numerical estimates. Relativistic electron distributions are assumed for calculations. Both Wien and Planckian distributions are contrasted for impact on induced scattering as LTE limit points. We find that both transformed and balanced approximations suggest larger induced scattering effects at high photon energies and low electron temperatures, and smaller effects in the opposite limits, compared to previous analyzes, with 10–20% increases in effective cross sections. We also note that both approximations can be simply implemented within existing transport modules or opacity processors as an additional term in the effective scattering cross section. Applications and comparisons include effective cross sections, kernel approximations, and impacts on radiative transport solutions in 1D geometry. The additional computing time for processing opacities (cross sections) within these approximations is negligible as induced terms are merely added (multipliers) to cross sections at the end of the processing cycle.     

Elastic scattering cross section of photons by water: limits for the free atom approximation

Several papers can be found in the literature in which elastic and inelastic experimental scattering cross sections of photons by water with energies in the keV range are measured. Usually, the authors use some kind of normalization with theoretical values, because of the difficulties in obtaining some parameters necessary to obtain absolute cross sections from the measurements. A reasonable range of momentum transfer has been covered, but nevertheless, inconsistencies and gaps still remain. In this work, we address the problem performing absolute cross section measurements in the high momentum transfer region (0.33 Å^?1 ≤ x ≤ 4.5 Å^?1), where no structure effects can be observed. We compare the results with theory to validate it in an intermediary region (0.33 Å^?1 ≤ x ≤ 1.72 Å^?1) and then use the experimental values to normalize data in the low‐energy region (0.078 Å^?1 ≤ x ≤ 1.72 Å^?1) to obtain the experimental cross section in that region. From the comparison, we concluded that for momentum transfers, x between 0.7 and 4.5 Å^?1 the scattering can be considered as being due to free atoms in free molecules. Therefore, the scattering cross sections may be considered as a sum of those from free atoms. Theoretically, the scattering cross sections can be obtained from the form factors as well as from the second‐order perturbation theory (S‐matrix) because there is no significant difference between both. On the other hand, for x ≤ 0.7 Å^?1, interference between the photon‐scattered amplitudes due to liquid structure generates oscillations in the cross section values. Copyright © 2014 John Wiley & Sons, Ltd.     

Single and double scattering approximations for a two-dimensional cylindrical medium

Large spatial frequency expansions for the source function, radiative flux, and intensity are obtained for an isotropically scattering finite two-dimensional medium exposed to collimated radiation. With these expansions, the single and double scattering results are obtained which are valid at small optical distances away from the incident radiation. Results are presented for a circular disk, exponential distribution and a Gaussian distribution of incident radiation.     

Similarity solutions for boundary layer flow and heat transfer of a FENE-P fluid with thermal radiation

This Letter presents a numerical study of the flow and heat transfer of an incompressible FENE-P fluid over a non-isothermal surface. The governing partial differential equations are converted into ordinary differential equations by a similarity transformation. The effects of the thermal radiation are considered in the energy equation, and the variations of dimensionless surface temperature and dimensionless surface temperature gradient, as well as the heat transfer characteristics with various physical parameters are graphed and tabulated. Two cases are studied, namely, (i) the sheet with prescribed surface temperature (PST case) and (ii) the sheet with prescribed heat flux (PHF case). Moreover, the mechanical characteristics of the corresponding flow are also presented.     

Dirac-delta function approximations to the scattering phase function

Dirac-delta function approximations are used to represent the single scattering phase function of large spherical particles or voids. The phase function for a spherical particle or void can be represented by a series of Legendre polynomials; however, as the diameter is increased, forward scattering becomes dominant and the number of terms in the series becomes very large. A Dirac-delta function approximation consists of a Dirac-delta function in the forward direction plus a finite series of Legendre polynomials. The Dirac-delta function accounts for strong forward scattering. Particular attention is given to large ice spheres and spherical voids in ice. The Dirac-delta function is shown effective in reducing the number of terms needed to describe the phase function.     

The kinetic boundary layer for the Fokker-Planck equation with absorbing boundary

The Fokker-Planck equation for the distribution of position and velocity of a Brownian particle is a particularly simple linear transport equation. Its normal solutions and an apparently complete set of stationary boundary layer solutions can be determined explicitly. By a numerical algorithm we select linear combinations of them that approximately fulfill the boundary condition for a completely absorbing plane wall, and that approach a linearly increasing position space density far from the wall. Various aspects of these approximate solutions are discussed. In particular we find that the extrapolated asymptotic density reaches zero at a distance x_M beyond the wall. We find x_M=1.46 in units of the velocity persistence length of the Brownian particle. This study was motivated by certain problems in the theory of diffusion-controlled reactions, and the results might be used to test approximate theories employed in that field.     

Measurement of the hadronic cross-section for the scattering of two virtual photons at LEP

The interaction of virtual photons is investigated using the reaction hadrons based on data taken by the OPAL experiment at centre-of-mass energies GeV, for GeV and at an average of 17.9 GeV. The measured cross-sections are compared to predictions of the Quark Parton Model (QPM), to the Leading Order QCD Monte Carlo model PHOJET, to the NLO prediction for the reaction , and to BFKL calculations. PHOJET, NLO , and QPM describe the data reasonably well, whereas the cross-section predicted by a Leading Order BFKL calculation is too large. Received: 3 September 2001 / Published online: 5 April 2002     

Anomalous elastic scattering of X-ray photons by the molybdenum atom

The differential cross section for elastic scattering of an X-ray photon by the molybdenum atom is calculated in the region of the ionization threshold of the 1s shell. The calculation is performed in the nonrelativistic approximation taking into account the rearrangement of electron shells of the atomic core in the field of the virtual 1 s vacancy and processes of one-photon virtual excitation (ionization) of the ground state. Good agreement with experimental results is obtained     

On the validity of some new acoustic scattering approximations

Abstract

Two new approximations for predicting the elastic scattering of plane acoustic waves by a weak scatterer are proposed. The approximations have been obtained by drawing an analogy between acoustic and light scattering problems. The validity of these approximations has been examined numerically for the exactly soluble case of scattering by a homogeneous sphere. Results show that for small angle scattering the proposed approximations have a considerably larger domain of validity in comparison to the extensively used Born approximation.     

On the validity of some new acoustic scattering approximations

Two new approximations for predicting the elastic scattering of plane acoustic waves by a weak scatterer are proposed. The approximations have been obtained by drawing an analogy between acoustic and light scattering problems. The validity of these approximations has been examined numerically for the exactly soluble case of scattering by a homogeneous sphere. Results show that for small angle scattering the proposed approximations have a considerably larger domain of validity in comparison to the extensively used Born approximation.     

Interference between nuclear resonance and nuclear Thomson scattering of photons

The elastic differential scattering cross section from ⁵⁵Mn has been measured at θ = 140° with E_γ = 5.5–11.4 MeV photons. Evidence for destructive interference between nuclear resonance (NR) scattering and nuclear Thomson (NT) scattering has been obtained.     

Multiple scattering by cylinders immersed in fluid: high order approximations for the effective wavenumbers

Acoustic wave propagation in a fluid with a random assortment of identical cylindrical scatterers is considered. While the leading order correction to the effective wavenumber of the coherent wave is well established at dilute areal density (n0) of scatterers, in this paper the higher order dependence of the coherent wavenumber on n0 is developed in several directions. Starting from the quasi-crystalline approximation (QCA) a consistent method is described for continuing the Linton and Martin formula, which is second order in n0, to higher orders. Explicit formulas are provided for corrections to the effective wavenumber up to O (n0(4)). Then, using the QCA theory as a basis, generalized self-consistent schemes are developed and compared with self-consistent schemes using other dynamic effective medium theories. It is shown that the Linton and Martin formula provides a closed self-consistent scheme, unlike other approaches.     

HF+He密耦散射的半经典研究

拟合在对称性匹配微扰理论(SAPT)水平下精确计算的HF分子在3个不同核间距时的相互作用能,获得了He-HF复合物的解析振转势能面;这个势能面与现有的理论势符合得很好.在此势能面上,使用密耦近似计算了He-HF碰撞能量从160 cm~(-1)到694 cm~(-1)范围内的分波截面,并采用半经典方法详细讨论了长程吸引和短程各向异性相互作用对弹性和非弹性分波截面的影响.