A variational approach to second-order multisymplectic field theory

This paper presents a geometric-variational approach to continuous and discrete second-order field theories following the methodology of [Marsden, Patrick, Shkoller, Comm. Math. Phys. 199 (1998) 351–395]. Staying entirely in the Lagrangian framework and letting Y denote the configuration fiber bundle, we show that both the multisymplectic structure on J³Y as well as the Noether theorem arise from the first variation of the action function. We generalize the multisymplectic form formula derived for first-order field theories in [Marsden, Patrick, Shkoller, Comm. Math. Phys. 199 (1998) 351–395], to the case of second-order field theories, and we apply our theory to the Camassa–Holm (CH) equation in both the continuous and discrete settings. Our discretization produces a multisymplectic-momentum integrator, a generalization of the Moser–Veselov rigid body algorithm to the setting of nonlinear PDEs with second-order Lagrangians.     

A variational approach to stochastic nonlinear problems

A variational principle is formulated which enables the mean value and higher moments of the solution of a stochastic nonlinear differential equation to be expressed as stationary values of certain quantities. Approximations are generated by using suitable trial functions in this variational principle and some of these are investigated numerically for the case of a Bernoulli oscillator driven by white noise. Comparison with exact data available for this system shows that the variational approach to such problems can be quite effective.     

Structure of some liquid transition metals using integral equation theory

We present the results of calculations of the structure factorS(q) of some liquid 3d transition metals using the self consistent hybridized mean spherical approximation (HMSA) integral equation. The local pseudopotential used is composed of the empty core model and a part that takes care of s-d mixing through an inverse scattering approach to model the interionic pair potential. The results presented are in very good agreement with experiment for most of the systems investigated near freezing, as well as for the noble metals Cu, Ag and Au, thus, confirming the reliability of the pseudopotential in the present integral equation scheme.     

基于随机表面微面元理论的二向反射分布函数几何衰减因子修正

现有几何光学方法的二向反射分布函数BRDF (bidirectional reflection distribution function)模型在计算阴影遮蔽效应时普遍应用Blinn几何衰减效应假设, 其等倾角V形槽近似得出的分段折线形式的几何衰减因子导致BRDF曲线存在较大的误差. 基于倾斜角随机高斯分布的微面元理论提出了一种新的几何衰减模型, 得出了积分形式的几何衰减因子表达式, 数值模拟比较了Blinn几何衰减因子与修正后的积分型衰减因子以及对应的BRDF模型曲线. 结果表明: 提出的几何衰减因子在物理合理性以及模拟精度方面都有明显提升, 使BRDF模型曲线与已有BRDF 数据之间的标准误差由0.0636减小到0.0084.     

有限精度函数理论与Einstein-Maxwell方程的光子解

提出了有限精度函数理论,并基于该理论求出了有限精度的Einstein-Maxwell方程的光子解,揭示了经典光子的电磁、引力与时空几何的重要性质. 关键词： 有限精度函数理论 Einstein-Maxwell方程 光子解     

Diffusion in a symmetric bistable potential: A variational approach

An approximation procedure for the solution of stochastic nonlinear equations, which was derived from a variational principle in a previous paper, is applied to the problem of a particle that diffuses in a symmetric bistable potential starting from the point of unstable equilibrium. The second moment and variance for the particle's position are calculated as functions of the timet. Good agreement is found with results recently obtained by Baibuzet al. from an approximate evaluation of a path integral expression for the probability density.     

A variational approach to bound states in quantum field theory

We consider here in a toy model an approach to bound state problem in a nonperturbative manner using equal time algebra for the interacting field operators. The potential is replaced by offshell bosonic quanta inside the bound state of nonrelativistic particles. The bosonic dressing is determined through energy minimisation, and mass renormalisation is carried out in a nonperturbative manner. Since the interaction is through a scalar field, it does not include spin effects. The model however nicely incorporates an intuitive picture of hadronic bound states in which the gluon fields dress the quarks providing the binding between them and also simulate the gluonic content of hadrons in deep inelastic collisions.     

On the theory of brownian motion. III. Two-body distribution function

The equation of evolution governing the probability density of a pair of heavy particles in a fluid of lighter particles is derived. The derivation starts from the Liouville equation and proceeds by expansion in the ratio of light to heavy masses, using the technique previously applied successfully to the singlet distribution.This work is part of research supported by NSF Grant GP-8497.     

A variational approach to connecting orbits in nonlinear dynamical systems

We propose a variational method for determining homoclinic and heteroclinic orbits including spiral-shaped ones in nonlinear dynamical systems. Starting from a suitable initial curve, a homotopy evolution equation is used to approach a true connecting orbit. The procedure is an extension of a variational method that has been used previously for locating cycles, and avoids the need for linearization in search of simple connecting orbits. Examples of homoclinic and heteroclinic orbits for typical dynamical systems are presented. In particular, several heteroclinic orbits of the steady-state Kuramoto–Sivashinsky equation are found, which display interesting topological structures, closely related to those of the corresponding periodic orbits.     

Improved but low-x approximated relation betweenF 2 p scaling violation and gluon momentum distribution function

We present a brief analysis on the approximate methods for the determination of gluon distribution from the scaling violation of proton structureF ₂ ^p in the low-x limit. In the leading order, a general low-x approximated relation is presented having more accuracy than the previous methods. Next-to-leading order correction is presented incorporating double-log-approximation. The proposed method is found to give good agreement with data. It may also be used to discriminate between the sets of gluon distributions in the low-x.     

毛细作用的变分法理论

以圆柱形毛细管内壁、毛细管内的液体和气体为研究对象.设毛细管液体的表面为旋转面,写出了系统的自由能(表面能及重力势能)的泛函.使自由能泛函取极值的欧拉方程就是关于附加压强的拉普拉斯公式.泛函取极值时在可动边界(毛细管内壁)上的边界条件就可导出关于接触角的杨氏方程.此二方程是自由能取极值的密不可分的两个必要条件.     

Conditional distribution function approach to the theory of anchoring transitions

The interfacial structure in the nematic phase near a wall is studied by means of a method of conditional distribution functions. It was found that with decreasing strength of the wall anchoring potential, the theory predicts a first-order oblique to oblique anchoring transition observed experimentally by Ryschenkow and Kleman. In the case of increasing strength of the solid wall anchoring potential the model predicts homogeneous anchoring.     

Renormalized theory of the time-dependent pair distribution function. I. General formulation

A classical renormalized theory of a time-dependent pair-distribution function (TDPDF), previously introduced by Oppenheim and Bloom, is presented. An equation of motion for the TDPDF is derived in which the memory function of the system appears. This is then split into a part which contains only static correlation functions and a part which describes the dynamics. The mean field approximation is discussed in some detail and contact is made witn the theory of Oppenheim and Bloom.Work supported in part by a National Research Council of Canada operating grant.     

A variational approach to superfluidity

A variational approach to problems in quantum statistical mechanics is described and it is shown how to determine the best quasi-free approximation to the equilibrium state. The relation between this approximation and the Bogoliubov approximation in superfluidity is discussed.     

Solution of Ornstein-Zernike equation for wall-particle distribution function

The Ornstein-Zernike (OZ) equation is considered for the wall-particle distribution functiong ₀(x) in the case of a flat, impenetrable wall atx = 0 and a fluid of hard-core particles whose centers are constrained by the wall to occupy the semiinfinite spacex >/2, where is the particle diameter. A solution is given in terms of the wall-particle direct correlation function c₀(x) forx >/2, the bulk-fluid direct correlation function c_B (t), and p_B, the average bulk density. Explicit formulas for the contact surface density, total excess surface density, and the Laplace transform of the fluid density near the wall are given. For mean spherical type approximations, c₀ (x) forx >/2 and c_B (t) are both prescribed functions; for this case, a closed-form solution is obtained. An example is discussed and additional equations that enable one to go beyond the approximations considered above are introduced.Report #270, February 1976.The observation of this paper that the wall-particle problem can be treated using standard Wiener-Hopf techniques was independently made by Percus in his work, which came to our attention too late to be compared to, or incorporated into, our own results here.