三维静电场线性插值边界元中的解析积分方法

提出求解三维静电场的三角形线性插值边界元解析积分方法.针对含1/R和1/R²的积分项,将单元形状函数分解为常数项、含x的线性项和含y的线性项,从而将边界单元积分简化为6个基本积分组合,并导出其解析计算公式,避免了因形状函数改变而导致的重复计算.该方法不仅可以准确计算远离奇异情况下的边界元积分,而且可以准确计算一阶和二阶接近奇异积分以及一阶奇异积分.计算结果表明,在接近奇异积分和奇异积分比较突出的问题中,当数值积分方法不能给出正确结果时,用同样的边界元网格,解析积分方法可以给出正确的结果,提高了三维静电场线性插值边界元法的计算精度.     

Optimized Rayleigh–Schrödinger expansion of the effective potential

An optimized Rayleigh–Schrödinger expansion scheme of solving the functional Schrödinger equation with an external source is proposed to calculate the effective potential beyond the Gaussian approximation. For a scalar field theory whose potential function has a Fourier representation in a sense of tempered distributions, we obtain the effective potential up to the second order, and show that the first-order result is just the Gaussian effective potential. Its application to the λφ⁴ field theory yields the same post-Gaussian effective potential as obtained in the functional integral formalism.     

Low-rank canonical-tensor decomposition of potential energy surfaces: application to grid-based diagrammatic vibrational Green's function theory

ABSTRACT

A new method is proposed for a fast evaluation of high-dimensional integrals of potential energy surfaces (PES) that arise in many areas of quantum dynamics. It decomposes a PES into a canonical low-rank tensor format, reducing its integral into a relatively short sum of products of low-dimensional integrals. The decomposition is achieved by the alternating least squares (ALS) algorithm, requiring only a small number of single-point energy evaluations. Therefore, it eradicates a force-constant evaluation as the hotspot of many quantum dynamics simulations and also possibly lifts the curse of dimensionality. This general method is applied to the anharmonic vibrational zero-point and transition energy calculations of molecules using the second-order diagrammatic vibrational many-body Green's function (XVH2) theory with a harmonic-approximation reference. In this application, high dimensional PES and Green's functions are both subjected to a low-rank decomposition. Evaluating the molecular integrals over a low-rank PES and Green's functions as sums of low-dimensional integrals using the Gauss–Hermite quadrature, this canonical-tensor-decomposition-based XVH2 (CT-XVH2) achieves an accuracy of 0.1 cm^?1 or higher and nearly an order of magnitude speedup as compared with the original algorithm using force constants for water and formaldehyde.     

转动相对论系统动力学的积分理论

建立转动相对论系统动力学方程的积分理论.给出系统运动的第一积分,分别利用系统的循环积分和能量积分降阶运动方程,得到推广的Routh方程和推广的Whittaker方程,建立系统运动的正则方程和变分方程,并由第一积分构造系统的积分不变量.给出系统的Poincaré-Cartan型积分变量关系和积分不变量. 关键词： 转动相对论 运动方程 积分方法     

Gravitational and electrostatic fields of a homogeneous circular cone

The problem of the simplest elements forming the external field of a gravitating (or electrostatically charged) homogeneous circular cone is formulated and solved by the method of contour integrals. It is found that the tip point has a logarithmic singularity and is not included in the equigravitating frame of the figure. On the whole, only one equigravitating rod with a purely imaginary density described by elementary functions exists for a cone. It is proved that this rod satisfies all necessary requirements: its mass and spatial potential are real-valued and equivalent to analogous characteristics of the cone. Independent verification in the asymptotic limit of an inhomogeneous circular disk confirms the main result. The spatial potential of the cone is expressed, using the rod, first in terms of a single integral and then in terms of standard full elliptic integrals, as well as a special series in harmonic functions. A family of equipotential surfaces is obtained.     

Schwinger Boson Formulation and Solution of the Crow-Kimura and Eigen Models of Quasispecies Theory

We express the Crow-Kimura and Eigen models of quasispecies theory in a functional integral representation. We formulate the spin coherent state functional integrals using the Schwinger Boson method. In this formulation, we are able to deduce the long-time behavior of these models for arbitrary replication and degradation functions. We discuss the phase transitions that occur in these models as a function of mutation rate. We derive for these models the leading order corrections to the infinite genome length limit.     

The radiation impedance of a rectangular piston

The radiation impedance of a rectangular piston is expressed as the Fourier transform of its impulse response, which is obtained from the recent work of Lindermann [1]. The analytical evaluation of the transform is performed and new integral expressions are presented for both the radiation resistance and reactance. The integrals are readily evaluated in terms of elementary functions at both the low and high frequency limits. The integrals are also expressed as series of Bessel functions which are valid for all frequencies and aspect ratios. Numerical results are presented to illustrate the behavior of the radiation resistance and reactance as a function of the aspect ratio of the piston and a normalized frequency parameter. Additional numerical results are then presented to illustrate the accuracy of the analytical expressions for the radiation resistance and reactance at low and high frequencies. Finally, numerical results are presented to illustrate the application and accuracy of using standard FFT algorithms to evaluate the radiation resistance and reactance directly from the impulse responses.     

Integrable cosmological potentials

The problem of classification of the Einstein–Friedman cosmological Hamiltonians H with a single scalar inflaton field \(\varphi \), which possess an additional integral of motion polynomial in momenta on the shell of the Friedman constraint \(H=0\), is considered. Necessary and sufficient conditions for the existence of the first-, second- and third-degree integrals are derived. These conditions have the form of ODEs for the cosmological potential \(V(\varphi )\). In the case of linear and quadratic integrals we find general solutions of the ODEs and construct the corresponding integrals explicitly. A new wide class of Hamiltonians that possess a cubic integral is derived. The corresponding potentials are represented in parametric form in terms of the associated Legendre functions. Six families of special elementary solutions are described, and sporadic superintegrable cases are discussed.     

The singularities of the integrals in Mayer's ionic solution theory

The terms of the expression previously obtained by Friedman for the cluster integral sum in Mayer's theory are examined as to order in total ion concentration, c, at the limit, c = 0. The order of the singularity at c = 0 is calculated for several of those terms of the irreducible cluster integrals that correspond to graphs of low connectivity but with an arbitrary number of vertices. On the basis of these calculations and two postulates concerning the relation of the singularities of these integrals to others, it is concluded that the terms of the cluster integral sum, when arranged in order of increasing index, are also in increasing order of concentration, and that the first term, κ ³/12π, has a lower order than any other. The order in concentration of the higher terms depends on whether the third moment of the concentration of charge types, σc₈z₈ ³, vanishes, as it does in solutions of electrolytes of symmetrical charge type.     

A physically important class of integrals expressed as a parameter derivative

The paper deals with a class of integrals, the integrands of which contain the square of a solution of a second-order linear, ordinary differential equation. Such integrals often arise in quantum mechanics as normalization integrals or expectation values. A generalized, unified procedure for rewriting such an integral, associated with a differential equation of the Sturm-Liouville type with unspecified boundary conditions, as a parameter derivative is presented. The formula thus obtained can be used for the evaluation of various integrals of physical interest. As an application we present a simplified derivation of a formula given by de Alfaro and Regge, in which the quantal normalization integral is expressed in terms of the Jost function. Other applications to integrals involving special functions and to integrals associated with the one-dimensional Schrödinger equation are also presented. Furthermore, it is explained why an approximate formula for expectation values is much more accurate than one can expect from the usual, crude derivation of it, and why certain attempts to improve that derivation have failed.     

Approximate formulae for the overlap integral of two harmonic oscillator wave functions

The use of second-order perturbation theory to derive approximate formulae for the overlap integral of two harmonic oscillator wave functions is discussed, and the results applied to the theory of intensity distributions in vibrational progressions in electronic spectra. For the vibrational progression m←0 an approximate formula is given which, when the vibrational frequencies of the initial and final states differ by less than 10%, reproduces to an accuracy of 1% or less the intensity profile calculated using the exact formulae for the overlap integrals.     

Topological Invariants in Braid Theory

Many invariants of knots and links have their counterparts in braid theory. Often, these invariants are most easily calculated using braids. A braid is a set of n strings stretching between two parallel planes. This review demonstrates how integrals over the braid path can yield topological invariants. The simplest such invariant is the winding number – the net number of times two strings in a braid wrap about each other. But other, higher-order invariants exist. The mathematical literature on these invariants usually employs techniques from algebraic topology that may be unfamiliar to physicists and mathematicians in other disciplines. The primary goal of this paper is to introduce higher-order invariants using only elementary differential geometry.Some of the higher-order quantities can be found directly by searching for closed one-forms. However, the Kontsevich integral provides a more general route. This integral gives a formal sum of all finite order topological invariants. We describe the Kontsevich integral, and prove that it is invariant to deformations of the braid.Some of the higher-order invariants can be used to generate Hamiltonian dynamics of n particles in the plane. The invariants are expressed as complex numbers; but only the real part gives interesting topological information. Rather than ignoring the imaginary part, we can use it as a Hamiltonian. For n = 2, this will be the Hamiltonian for point vortex motion in the plane. The Hamiltonian for n = 3 generates more complicated motions.     

Master Equation in the General Gauge: On the Problem of Infinite Reducibility

The master equation is quantized. This is an example of quantization of a gauge theory with nilpotent generators. No ghosts are needed for the generation of a gauge algebra. The point about nilpotent generators is that one can not write down a single functional integral for this theory. Instead, one has to write down a product of two coupled functional integrals and take a square root. In a special gauge where the gauge conditions are commuting, the functional integrals decouple and one recovers the known result.     

Nonlinear transport in the Boltzmann limit

Formal expressions for the irreversible fluxes of a simple fluid are obtained as functionals of the thermodynamic forces and local equilibrium time correlation functions. The Boltzmann limit of the correlation functions is shown to yield expressions for the irreversible fluxes equivalent to those obtained from the nonlinear Boltzmann kinetic equation. Specifically, for states near equilibrium, the fluxes may be formally expanded in powers of the thermodynamic gradients and the associated transport coefficients identified as integrals of time correlation functions. It is proved explicitly through nonlinear Burnett order that the time correlation function expressions for these transport coefficients agree with those of the Chapman-Enskog expansion of the nonlinear Boltzmann equation. For states far from equilibrium the local equilibrium time correlation functions are determined in the Boltzmann limit and a similar equivalence to the Boltzmann equation solution is established. Other formal representations of the fluxes are indicated; in particular, a projection operator form and its Boltzmann limit are discussed. As an example, the nonequilibrium correlation functions for steady shear flow are calculated exactly in the Boltzmann limit for Maxwell molecules.Research supported in part by NSF grant PHY 76-21453.     

Asymptotics of Tracy-Widom Distributions and the Total Integral of a Painlevé II Function

The Tracy-Widom distribution functions involve integrals of a Painlevé II function starting from positive infinity. In this paper, we express the Tracy-Widom distribution functions in terms of integrals starting from minus infinity. There are two consequences of these new representations. The first is the evaluation of the total integral of the Hastings-McLeod solution of the Painlevé II equation. The second is the evaluation of the constant term of the asymptotic expansions of the Tracy-Widom distribution functions as the distribution parameter approaches minus infinity. For the GUE Tracy-Widom distribution function, this gives an alternative proof of the recent work of Deift, Its, and Krasovsky. The constant terms for the GOE and GSE Tracy-Widom distribution functions are new.     

Evaluation of multidimensional canonical integrals in semiclassical collision theory

The evaluation of the multidimensional canonical integrals that occur in the uniform asymptotic representations of the S matrix in the semiclassical theory of inelastic and reactive molecular collisions is considered. For the non-separable two-dimensional canonical integral considered earlier, an exact series expansion is obtained with the help of convergence factors. This method avoids the complex variable techniques used previously. The uniform asymptotic formulae derived by Miller and Marcus are discussed, and compared with the approach adopted in the present paper.     

Invariant surfaces and orbital behaviour in dynamical systems of 3 degrees of freedom. II

Two formal integrals of motion besides the Hamiltonian are found in a system of three degrees of freedom with three unperturbed frequencies ω₁, ω₂, ω₃ which are close to, or exactly at, the double resonance 6:4:3. The integrals are truncated at various orders and they are used to find theoretical invariant surfaces on a surface of section.The theoretical invariant surfaces are then compared with the empirical results. The agreement is improved as the order of the integrals increases from the 5th up to the 11th degree. This is surprising because theoretical considerations indicate that one formal integral breaks down beyond order 6, because of the second resonance 4:3. (However it is possible to construct one further integral by a different method). As expected the agreement is worse when the perturbation becomes large. A check of the constancy of the truncated integrals along various orbits shows that the relative variations become smaller as the order increases; they become larger as the perturbation increases.     

Routh Order Reduction Method of Relativistic Birkhoffian Systems

Routh order reduction method of the relativistic Birkhoffian equations is studied.For a relativistic Birkhoffian system,the cyclic integrals can be found by using the perfect differential method.Through these cyclic integrals,the order of the system can be reduced.If the relativistic Birkhoffian system has a cyclic integral,then the Birkhoffian equations can be reduced at least by two degrees and the Birkhoffian form can be kept.The relations among the relativistic Birkhoffian mechanics,the relativistic Hamiltonian mechanics,and the relativistic Lagrangian mechanics are discussed,and the Routh order reduction method of the relativistic Lagrangian system is obtained.And an example is given to illustrate the application of the result.     

First Integrals and Integral Invariants of Relativistic Birkhoffian Systems

For a relativistic Birkhoflan system, the first integrals and the construction of integral invariants are studied. Firstly, the cyclic integrals and the generalized energy integral of the system are found by using the perfect differential method. Secondly, the equations of nonsimultaneous variation of the system are established by using the relation between the simultaneous variation and the nonsimultaneous variation. Thirdly, the relation between the first integral and the integral invariant of the system is studied, and it is proved that, using a t~rst integral, we can construct an integral invarlant of the system. Finally, the relation between the relativistic Birkhoflan dynamics and the relativistic Hamilton;an dynamics is discussed, and the first integrals and the integral invariants of the relativistic Hamiltonian system are obtained. Two examples are given to illustrate the application of the results.     

Exact form factors in integrable quantum field theories: the sine-Gordon model (II)

A general model independent approach using the ‘off-shell Bethe Ansatz’ is presented to obtain an integral representation of generalized form factors. The general techniques are applied to the quantum sine-Gordon model alias the massive Thirring model. Exact expressions of all matrix elements are obtained for several local operators. In particular soliton form factors of charge-less operators as for example all higher currents are investigated. It turns out that the various local operators correspond to specific scalar functions called p-functions. The identification of the local operators is performed. In particular the exact results are checked with Feynman graph expansion and full agreement is found. Furthermore all eigenvalues of the infinitely many conserved charges are calculated and the results agree with what is expected from the classical case. Within the frame work of integrable quantum field theories a general model independent ‘crossing’ formula is derived. Furthermore the ‘bound state intertwiners’ are introduced and the bound state form factors are investigated. The general results are again applied to the sine-Gordon model. The integrations are performed and in particular for the lowest breathers a simple formula for generalized form factors is obtained.