Matrix continued fraction solutions of the Kramers equation and their inverse friction expansions

The distribution function in position and velocity space for the Brownian motion of particles in an external field is determined by the Kramers equation, i.e., by a two variable Fokker-Planck equation. By expanding the distribution function in Hermite functions (velocity part) and in another complete set satisfying boundary conditions (position part) the Laplace transform of the initial value problem is obtained in terms of matrix continued fractions. An inverse friction expansion of the matrix continued fractions is used to show that the first Hermite expansion coefficient may be determined by a generalized Smoluchowski equation. The first terms of the inverse friction expansion of this generalized Smoluchowski operator and of the memory kernel are given explicitly. The inverse friction expansion of the equation determining the eigenvalues and eigenfunctions is also given and the connection with the result of Titulaer is discussed.     

Three-dimensional finite-difference lattice Boltzmann model and its application to inviscid compressible flows with shock waves

In this paper, a three-dimensional (3D) finite-difference lattice Boltzmann model for simulating compressible flows with shock waves is developed in the framework of the double-distribution-function approach. In the model, a density distribution function is adopted to model the flow field, while a total energy distribution function is adopted to model the temperature field. The discrete equilibrium density and total energy distribution functions are derived from the Hermite expansions of the continuous equilibrium distribution functions. The discrete velocity set is obtained by choosing the abscissae of a suitable Gauss–Hermite quadrature with sufficient accuracy. In order to capture the shock waves in compressible flows and improve the numerical accuracy and stability, an implicit–explicit finite-difference numerical technique based on the total variation diminishing flux limitation is introduced to solve the discrete kinetic equations. The model is tested by numerical simulations of some typical compressible flows with shock waves ranging from 1D to 3D. The numerical results are found to be in good agreement with the analytical solutions and/or other numerical results reported in the literature.     

Iterative calculation, manufacture and investigation of DOE forming unimodal complex distribution

We present results of iterative calculation, manufacture and experimental as well as theoretical investigations of a novel diffractive optical element (DOE) which transforms a Gaussian TEM₀₀ input beam into a unimodal Gauss–Hermite (1, 0) complex distribution. The iterative calculation procedure is based on the application of the method of generalized projections. The projection operator onto a set of modal functions is implemented through partition of the focal plane into a ‘useful’ and an ‘auxiliary’ domain. This element is manufactured as a 16 level surface profile by (variable dose) electron-beam direct-writing into a PMMA resist film, and a subsequent development procedure of the resist. The final element consists of a fused-silica substrate coated with the structured PMMA film. Both computational and experimental results are presented and demonstrate a good conformity with each other. The achieved results show good prospects of such an approach for the formation of unimodal distributions.     

The Calogero-Sutherland Model and Generalized Classical Polynomials

Multivariable generalizations of the classical Hermite, Laguerre and Jacobi polynomials occur as the polynomial part of the eigenfunctions of certain Schr?dinger operators for Calogero-Sutherland-type quantum systems. For the generalized Hermite and Laguerre polynomials the multidimensional analogues of many classical results regarding generating functions, differentiation and integration formulas, recurrence relations and summation theorems are obtained. We use this and related theory to evaluate the global limit of the ground state density, obtaining in the Hermite case the Wigner semi-circle law, and to give an explicit solution for an initial value problem in the Hermite and Laguerre case. Received: 16 August 1996 / Accepted: 21 January 1997     

Three-dimensional coupled double-distribution-function lattice Boltzmann models for compressible Navier–Stokes equations

Two three-dimensional (3D) lattice Boltzmann models in the framework of coupled double-distribution-function approach for compressible flows, in which specific-heat ratio and Prandtl number can be adjustable, are developed in this paper. The main differences between the two models are discrete equilibrium density and total energy distribution function. One is the D3Q25 model obtained from spherical function, and the other is the D3Q27 standard lattice model obtained from Hermite expansions of the corresponding continuous equilibrium distribution functions. The two models are tested by numerical simulations of some typical compressible flows, and their numerical stability and precision are also analysed. The results indicate that the two models are capable for supersonic flows, while the one from Hermite expansions is not suitable for compressible flows with shock waves.     

Zur mikrophysikalischen Beschreibung des schwachionisierten Säulenplasmas von Glimmentladungen in Stickstoff-Neon-Gemischen. I. Vergleich zwischen gemessenen und berechneten Geschwindigkeitsverteilungsfunktionen der Elektronen

The isotropic part of the velocity distribution function of electrons is determined experimentally and theoretically for the plasma of the positive column of glow discharges in N₂-Ne mixtures. The distribution functions are measured with electrical probes using the Druyvesteyn method. The calculation of the isotropic part of the distribution functions for homogeneous and stationary conditions from the Boltzmann equation takes into consideration the essential elastical and unelastical collision processes. The character of the obtained distribution functions is fixed by the reduced field strength E/p₀ and the mixture ratio x = N₂/N₂ + Ne), which is varied in this paper in the range between pure nitrogen and pure neon. The distribution functions are found to be influenced by elastic and unelastic collisions of electrons with nitrogen molecules even at smal admixtures of N₂. Comparing the experimental results for the distribution functions with those received by the calculations a good agreement is observed.     

Shear Viscosity and Temperature Inequalities for Multi-Species Plasmas

The first coefficients in the orthogonal expansions of the velocity distribution functions with respect to Grad's tensorial (three-dimensional) Hermite polynomials are shown to be proportional to ther-modynamic fluxes, if the weight functions are local Maxwellians centered at the mean mass velocity and widened with a mean temperature. Balance equations for the stress tensors are established and reduced to linear algebraic systems under certain restrictions. Explicit solutions for the traces and the tracefree parts of the stress tensors are given.     

Generalized Hermite Polynomials and the Heat Equation for Dunkl Operators

Based on the theory of Dunkl operators, this paper presents a general concept of multivariable Hermite polynomials and Hermite functions which are associated with finite reflection groups on ℝ ^N . The definition and properties of these generalized Hermite systems extend naturally those of their classical counterparts; partial derivatives and the usual exponential kernel are here replaced by Dunkl operators and the generalized exponential kernel K of the Dunkl transform. In the case of the symmetric group S _N , our setting includes the polynomial eigenfunctions of certain Calogero-Sutherland type operators. The second part of this paper is devoted to the heat equation associated with Dunkl's Laplacian. As in the classical case, the corresponding Cauchy problem is governed by a positive one-parameter semigroup; this is assured by a maximum principle for the generalized Laplacian. The explicit solution to the Cauchy problem involves again the kernel K, which is, on the way, proven to be nonnegative for real arguments. Received: 10 March 1997 / Accepted: 7 July 1997     

Nonclassicality of photon-modulated spin coherent states in the Holstein—Primakoff realization

Two new photon-modulated spin coherent states (SCSs) are introduced by operating the spin ladder operators J_± on the ordinary SCS in the Holstein-Primakoff realization and the nonclassicality is exhibited via their photon number distribution, second-order correlation function, photocount distribution and negativity of Wigner distribution. Analytical results show that the photocount distribution is a Bernoulli distribution and the Wigner functions are only associated with two-variable Hermite polynomials. Compared with the ordinary SCS, the photon-modulated SCSs exhibit more stronger nonclassicality in certain regions of the photon modulated number k and spin number j, which means that the nonclassicality can be enhanced by selecting suitable parameters.     

Two-particle Wigner functions in a one-dimensional Calogero-Sutherland potential

We calculate the Wigner distribution function for the Calogero-Sutherland system which consists of harmonic and inverse-square interactions. The Wigner distribution function is separated out into two parts corresponding to the relative and center-of-mass motions. A general expression for the relative Wigner function is obtained in terms of the Laguerre polynomials by introducing a new identity between Hermite and Laguerre polynomials.     

Classical Skew Orthogonal Polynomials and Random Matrices

Skew orthogonal polynomials arise in the calculation of the n-point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry. In particular, the distribution functions are completely determined by a certain sum involving the skew orthogonal polynomials. In the case that the eigenvalue probability density function involves a classical weight function, explicit formulas for the skew orthogonal polynomials are given in terms of related orthogonal polynomials, and the structure is used to give a closed-form expression for the sum. This theory treates all classical cases on an equal footing, giving formulas applicable at once to the Hermite, Laguerre, and Jacobi cases.     

Two-Variable Hermite Function as Quantum Entanglement of Harmonic Oscillator＇s Wave Functions

We reveal that the two-variable Hermite function hm,n, which is the generalized Bargmann representation of the two-mode Fock state, involves quantum entanglement of harmonic oscillator＇s wave functions. The Schmidt decomposition of hm,n is derived. It also turns out that hm,n can be generated by windowed Fourier transform of the single-variable Hermite functions. As an application, the wave function of the two-variable Hermite polynomial state S（γ）Hm,n （μa1^＋, μa2^＋│00〉, which is the minimum uncertainty state for sum squeezing, in （ η│representation is calculated.     

Generating functions for Hermite polynomials of arbitrary order

We propose a systematic method for the construction of generating functions for Hermite polynomials of arbitrary order. The procedure is based on a suitable formula for the Hermite polynomials and our results contain ones obtained earlier by Nieto and Truax [Phys. Lett. A 208 (1995) 8] as particular cases.     

On Extrapolation of Electromagnetic Responses in Time and Frequency Domains

The complete electromagnetic responses from conducting objects in time and frequency domains are generated by using their early time and low frequency information. Utilizing two kinds of Hermite polynomials and their Fourier transform, the time-domain signal and its corresponding frequency response can be expressed as a weighted sum of these quantities in an efficient way. The general properties of these two families of Hermite functions are studied, which greatly affect the performance of the proposed method. Due to the performance of the algorithm being sensitive to the choice of the origin and the scaling factor, how to properly choose the initial values of these parameters is considered. An optimal algorithm is also developed to find the above parameters so as to achieve the best performance. A criterion is also provided to assess the sensitivity of the performance. The excellent agreement between the computed results by the proposed method and those obtained by earlier approaches is demonstrated in each case. This work was supported in part by natinal nature science foundatin of China (NO. 60432040).     

Orr-Sommerfeld方程的数值解

本文讨论用有限元求解Orr-Sommerfeld方程的方法。由于选取7阶Hermite多项式为元素的基函数和按照流体在各个区域中的不同物理特性选取元素的网格分布,保证了函数在元素节点处C³连续及计算误差的较好控制,所得的结果比以前较为准确。应用这种方法讨论Plane Poi-seuille流的稳定性问题,求得临界雷诺数R_c=5772.2218,在雷诺数从R_c到R=10¹⁰范围内较精确地计算了该流体的中性曲线。     

二维非均匀介质中大地电磁响应的计算

在二维埋均匀介质中,将大地电磁问题可看作电磁波在有耗开放波导中的传播问题,借用研究波导问题的数值模式匹配法予以解决,在计算Ex型问题时,使用Hermite基函数和非均匀无素;在计算Hx型问题时,使用变型Hermite基函数,并与有限元法进行了比较。     

Quasiprobability Distribution Functions of Squeezed Pair Coherent States

Using the entangled state representation of Wigner operator and some formulae related to the two-variable Hermite polynomials, the Wigner function of the squeezed pair coherent state (SPCS) and its two marginal distributions are derived. Based on the entangled Husimi operator introduced by Fan et al. (Phys. Lett. A 358:203, 2006) and the Weyl ordering invariance under similar transformations, we also obtain the Husimi function of the SPCS and its marginal distribution functions. The comparison between the two quasibability functions shows that, for the same amount of information included in two functions, the solving process of the Husimi function is simpler than that of the Wigner function. Work supported by the Natural Science Foundation of Shandong Province of China under Grant Y2008A23 and the Natural Science Foundation of Liaocheng University under Grant X071049.     

Analytical relationship for the cranking inertia

The wave functions of a spheroidal harmonic oscillator without spin-orbit interaction are expressed in terms of associated Laguerre and Hermite polynomials. The pairing gap and Fermi energy are found by solving the BCS system of two equations. Analytical relationships for the matrix elements of inertia are obtained as a function of the main quantum numbers and potential derivative. They may be used to test complex computer codes developed in a realistic approach of the fission dynamics. Results given for the ²⁴⁰Pu nucleus are compared with a hydrodynamical model. The importance of taking into account the correction term due to the variation of the occupation number is stressed.     

Nonclassical Properties of Multiple-Photon-Added Two-Mode Squeezed Coherent States

We investigate how the photon addition operation affects the nonclassical properties of the non-Gaussian squeezed state generated by adding photons to each mode of the two-mode squeezed coherent state (TMSCS). By the generating function of two-variable Hermite polynomials, the compact expression of normalization factor is derived. We show that the fields in such states exhibit remarkable sub-Poissonian photon statistics. The photon addition operation can enhance the cross-correlation for appropriate combinations of several parameters involved in the TMSCS. Compared with that of TMSCS, the Wigner function of the photon–added TMSCS (PA-TMSCS) is modulated by a factor which is also related with two-variable Hermite polynomials. Such Wigner functions have some negativity regions and show a strong quantum mechanical interference. In addition, the normalization factor, Mandel’s Q parameter, cross-correlation function and Wigner functions are all sensitive to the compound phase involved in TMSCS.     

Prediction of the frequency spectrum of AT-cut contoured quartz resonators by means of X-ray diffraction topography

This paper presents some important relationships relating to frequencies in contoured AT-cut quartz resonators. It is shown that frequency interval relations are not affected by the piezoelement geometry but are functionally related solely to the indices of the Hermite functions. According to the analysis of trapped-energy resonators, an X-ray technique for predicting the frequency spectrum can be derived. It is based on the use of X-ray patterns of two wave motions and enables one to determine the whole frequency spectrum in the vicinity of any odd harmonic overtone of vibration. Two contoured resonator analyses show that the X-ray topography predictions are in good agreement with experimental data.