Understanding the accuracy of Nanbu’s numerical Coulomb collision operator

We investigate the accuracy of and assumptions underlying the numerical binary Monte Carlo collision operator due to Nanbu [K. Nanbu, Phys. Rev. E 55 (1997) 4642]. The numerical experiments that resulted in the parameterization of the collision kernel used in Nanbu’s operator are argued to be an approximate realization of the Coulomb–Lorentz pitch-angle scattering process, for which an analytical solution for the collision kernel is available. It is demonstrated empirically that Nanbu’s collision operator quite accurately recovers the effects of Coulomb–Lorentz pitch-angle collisions, or processes that approximate these (such interspecies Coulomb collisions with very small mass ratio) even for very large values of the collisional time step. An investigation of the analytical solution shows that Nanbu’s parameterized kernel is highly accurate for small values of the normalized collision time step, but loses some of its accuracy for larger values of the time step. Careful numerical and analytical investigations are presented, which show that the time dependence of the relaxation of a temperature anisotropy by Coulomb–Lorentz collisions has a richer structure than previously thought, and is not accurately represented by an exponential decay with a single decay rate. Finally, a practical collision algorithm is proposed that for small-mass-ratio interspecies Coulomb collisions improves on the accuracy of Nanbu’s algorithm.     

Properties of the Collision Operators of the Electron Boltzmann Equation for a Weakly Ionized Plasma. II. Compactness Proof and Statements on the Spectrum

The compactness has been proved of the inscattering operator covering elastic, exciting and deexciting processes between electrons and heavy neutral particles. Starting point is the proof of the boundedness of the different inscattering operators for the individual collision processes using certain assumptions on the structure of the corresponding differential cross sections. Then, under somewhat more confined conditions the proff of compactness could be performed with the aid of the square integrability of an eightfold iterated kernel. On the basis of this property it was possible to draw conclusions with regard to the spectrum of the total collision operator. They are related to the continuous as well as the discrete part of the spectrum and are discussed in detail. For elastic collisions additional statements can be deduced from our investigation.     

Calculation of the linear kernel of the collision integral for the hard-sphere potential

The expansion of a distribution function in spherical harmonics transforms the Boltzmann equation into a system of integro-differential equations with kernels depending only of the magnitudes of velocities. The kernels can be expressed in terms of the sums including the matrix elements (MEs) of the collision integral. The kernels are constructed using new results of ME calculations; analysis of errors is carried out with the help of analytic expressions for kernels, which were derived by Hilbert and Hecke for the hard-sphere model. The concept of generalized matrix elements is introduced and their asymptotic representation is constructed for large values of indices. Analytic expressions for the contribution from MEs with large indices to the kernels are constructed. The high accuracy of the construction of a kernel using MEs is demonstrated.     

Calculation of the collision integral linear kernel for Maxwellian molecules in the isotropic case

Application of the method of nonlinear moments to solve the Boltzmann equation generates the need to sum a series that is the expansion of the distribution function in basis functions. This series converged only if the Grad test is fulfilled. Such a limitation can be removed if the expansion of the distribution function is summed over the index related to only the expansion in velocity magnitude. In this case, the distribution function and the collision integral become expanded in only spherical harmonics and the expansion coefficients satisfy integro-differential equations. The kernels of these equations are the sums of the Sonine polynomials in the velocities of colliding and outgoing particles multiplied by matrix elements of the collision integral. For a number of arguments, the direct calculation of the kernels requires that a very large number of terms in the sum be taken into consideration. In this respect, an approach seems to be promising in which the asymptotics of the matrix elements and Sonine polynomials at large indices are used and summation over index is replaced by integration. In this paper, we apply this approach to calculate the linear kernel in the isotropic case, assuming that interaction between particles is described by a pseudopower law. With this approach, the collision integral kernel can be calculated with a high accuracy using as little as a few tens of series terms and the asymptotic estimate of the residue.     

On the approach to kinetic theory due to Gross

A classical kinetic theory introduced by Gross is explored in further detail. The theory consists of a sequence of approximations to the Liouville distribution function, with each approximation leading to a truncation of the BBGKY hierarchy at successively higher order. We formulate the truncation scheme at general order in terms of a set of time-dependent equilibrium correlation functions. It has the correct symmetries and, as is implied by the work of Gross with the first two approximations, is such that the interparticle potential appears only implicitly via static equilibrium correlation functions. We arrange the theory as a sequence of linear kinetic equations for the phase-space density correlation function, and solve for the collision kernels which result in each order. The collision kernel of the second approximation, which involves only binary dynamics, is shown to be a mean-field generalization of the known low-density kernel. The third approximation gives a similar generalization of the triple-collision kernel. The nth approximation also reproduces the frequency moments of S(kω) through order ω²ⁿ. More generally, the approximations are shown to give a continued-fraction expansion of the collision kernel, with the levels governed by the dynamics of successively larger numbers of particles. This is a renormalized kinetic theory in the sense that the potential is eliminated and clusters of particles are never isolated.     

等直径微液滴碰撞过程的改进光滑粒子动力学模拟

运用一种改进光滑粒子动力学(SPH)方法模拟了相溶和不相溶两种情况下的等直径微液滴碰撞动力学过程. 为提高传统SPH方法的数值精度和稳定性, 采用一种不涉及核导数计算的核梯度改进形式; 为处理液滴界面张力采用修正的van der Waals表面张力模型. 通过模拟牛顿液滴碰撞聚并变形过程并与相关文献或试验结果进行对比, 验证了改进SPH 方法模拟微液滴碰撞过程的可靠性. 随后, 研究了基于van der Waals模型相溶聚合物微液滴碰撞聚并变形过程及不相溶微液滴碰撞后的反弹、分离过程, 讨论了碰撞过程中碰撞速度、碰撞角度、密度比等参数对碰撞变形过程的影响, 分析了流体桥、旋转角度等因素的变化情况. 关键词： 光滑粒子动力学 微液滴 聚合物液滴 碰撞     

The interaction potential of the He+-Au system determined from Rutherford backscattering data

The potential-internuclear-distance dependences for the He⁺-Au system are obtained by processing measurements of particle scattering cross sections at various collision energies. The potential is shown to be independent of the collision velocity. The values obtained made it possible to introduce a correction for the difference of the above potential from the Coulomb one involved in the Rutherford backscattering method, and thus to improve the measurement accuracy. The obtained data on the screening constants enable estimation of the screening effect on the increase in the nucleosynthesis cross sections.     

Entropy production estimates for Boltzmann equations with physically realistic collision kernels

We establish strict entropy production bounds for the Boltzmann equation with the hard-sphere collision kernel. Using these entropy production bounds, we prove results asserting that the rate at which strongL ¹ convergence to equilibrium occurs is uniform in wide classes of initial data. This extends our previous results in this direction, which applied only to a very special collision kernel. Moreover, the present results provide computable lower bounds; compactness arguments are entirely avoided. The uniformity is an important ingredient in our study of scaling limits of solutions of the non-spatially homogeneous Boltzmann equation, and is the main focus of this paper. However, the results obtained here provide the only framework known to us in which one can obtain computable estimates on the time it takes a solution of the spatially homogeneous Boltzmann equation with initial data far from equilibrium to reach any given small strongL ¹ neighborhood of equilibrium.     

A Weak Formulation of the Boltzmann Equation Based on the Fourier Transform

In this article we present an alternative formulation of the spatially homogeneous Boltzmann equation. Rewriting the weak form of the equation with shifted test functions and using Fourier techniques, it turns out that the transformed problem contains only a three-fold integral. Explicit formulas for the transformed collision kernel are presented in the case of VHS models for hard and soft potentials. For isotropic Maxwellian molecules, a classical result by Bobylev is recovered, too.     

Inert fluorinated gas T1 calculator

The physics of spin-rotation interaction in roughly spherical perfluorinated gas molecules has been studied extensively. But, it is difficult to calculate a spin-lattice relaxation time constant T1 for any given temperature and pressure using the published literature. We give a unified parameterization that makes use of the Clausius equation of state, Lennard-Jones collision dynamics, and a formulaic temperature dependence for collision cross section for rotational change. The model fits T1s for SF6, CF4, C2F6, and c-C4F8 for temperatures from 180 to 360 K and pressures from 2 to 210 kPa and in mixtures with other common gases to within our limits of measurement. It also fits previous data tabulated according to known number densities. Given a pressure, temperature, and mixture composition, one can now calculate T1s for common laboratory conditions with a known accuracy, typically 0.5%. Given the success of the model's formulaic structure, it is likely to apply to even broader ranges of physical conditions and to other gases that relax by spin-rotation interaction.     

Multiregion, multigroup collision probability method with white boundary condition for light water reactor thermalization calculations

A multiregion, multigroup collision probability method with white boundary condition is developed for thermalization calculations of light water moderated reactors. Hydrogen scatterings are treated by Nelkin's kernel while scatterings from other nuclei are assumed to obey the free-gas scattering kernel. The isotropic return (white) boundary condition is applied directly by using the appropriate collision probabilities. Comparisons with alternate numerical methods show the validity of the present formulation. Comparisons with some experimental results indicate that the present formulation is capable of calculating disadvantage factors which are closer to the experimental results than alternative methods.     

Electroproduction of two light vector mesons in next-to-leading BFKL: study of systematic effects

The forward electroproduction of two light vector mesons is the first example of a collision process between strongly interacting colorless particles for which the amplitude can be written completely within perturbative QCD in the Regge limit with next-to-leading accuracy. In a previous paper we have given a numerical determination of the amplitude in the case of equal photon virtualities by using a definite representation for the amplitude and a definite optimization method for the perturbative series. Here we estimate the systematic uncertainty of our previous determination, by considering a different representation of the amplitude and different optimization methods of the perturbative series. Moreover, we compare our result for the differential cross section at the minimum |t| with a different approach, based on collinear kernel improvement.     

模拟颗粒凝并过程的快速Monte Carlo方法

对常规异权值Monte Carlo(MC)方法进行改进,基于强核函数思想,通过颗粒群单重遍历即可求得强核函数最大值,采用接受-拒绝法随机搜寻凝并对,并利用搜寻过程中拒绝和接受的所有凝并对的信息来估计凝并事件的等待时间(时间步长),从而避免颗粒群的双重遍历,以提高MC的效率.对典型工况的模拟结果显示该快速方法计算代价仅为O(N_s),能够显著提高计算效率,同时保持足够的计算精度,较好地协调计算代价与计算精度之间的矛盾.     

Cercignani's Conjecture is Sometimes True and Always Almost True

 We establish several new functional inequalities comparing Boltzmann's entropy production functional with the relative H functional. First we prove a longstanding conjecture by Cercignani under the nonphysical assumption that the Boltzmann collision kernel is superquadratic at infinity. The proof rests on the method introduced in [39] combined with a novel use of the Blachman-Stam inequality. If the superquadraticity assumption is not satisfied, then it is known that Cercignani's conjecture is not true; however we establish a slightly weakened version of it for all physically relevant collision kernels, thus extending previous results from [39]. Finally, we consider the entropy-entropy production version of Kac's spectral gap problem and obtain estimates about the dependence of the constants with respect to the dimension. The first two results are sharp in some sense, and the third one is likely to be, too; they contain all previously known entropy estimates as particular cases. This gives a first coherent picture of the study of entropy production, according to a program started by Carlen and Carvalho [12] ten years ago. These entropy inequalities are one step in our study of the trend to equilibrium for the Boltzmann equation, in both its spatially homogeneous and spatially inhomogeneous versions. Received: 25 July 2002 / Accepted: 13 September 2002 Published online: 10 February 2003 Communicated by J. L. Lebowitz     

On the Boltzmann Equation for 2D Bose-Einstein Particles

This paper considers the spatially homogeneous Boltzmann equation for 2D Bose-Einstein particles. Suppose the collision kernel satisfies some assumptions that include the hard disk model and other possible physical models. We prove the existence of global in time conservative measure solutions of the equation for isotropic initial data, and that for any initial datum which is not totally singular and has positive energy, the solution always converges strongly to the Bose-Einstein distribution as time goes to infinity. This implies that for the present 2D model there is no Bose-Einstein condensation in the sense of long-time limit.     

Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equation

We first consider the Boltzmann equation with a collision kernel such that all kinematically possible collisions are run at equal rates. This is the simplest Boltzmann equation having the compressible Euler equations as a scaling limit. For it we prove a stability result for theH-theorem which says that when the entropy production is small, the solution of the spatially homogeneous Boltzmann equation is necessarily close to equilibrium in the entropie sense, and therefore strongL ¹ sense. We use this to prove that solutions to the spatially homogeneous Boltzmann equation converge to equilibrium in the entropie sense with a rate of convergence which is uniform in the initial condition for all initial conditions belonging to certain natural regularity classes. Every initial condition with finite entropy andp ^th velocity moment for some p>2 belongs to such a class. We then extend these results by a simple monotonicity argument to the case where the collision rate is uniformly bounded below, which covers a wide class of slightly modified physical collision kernels. These results are the basis of a study of the relation between scaling limits of solutions of the Boltzmann equation and hydrodynamics which will be developed in subsequent papers; the program is described here.On leave from School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332.On leave from C.F.M.C. and Departamento de Matemática da Faculdade de Ciencias de Lisboa, 1700 Lisboa codex, Portugal.     

Relations between nonlinear kernels of the collision integral

Previously, to solve the Boltzmann equation by the moments method with expansion of the distribution function by spherical Hermit polynomials, a new computational method was suggested which allowed to construct nonlinear matrix elements of the collision integral with very large indices. This made it possible to substantially advance in construction of the distribution function. Limitations to convergence of the distribution function that appear in moment method are eliminated if we come to expansion by spherical harmonics from expansion by spherical Hermit polynomials. In this case, a complex five-fold collision integral is replaced by a set of comparatively simple integral operators, and kernels G _l1,l2 ^l (c, c ₁, c ₂) of these operators become the analog of matrix elements. We found the relations between expansions of the distribution function in the reference frames with various velocities of motion along marked axis. Starting from the invariance condition of the collision integral with respect to selection of such reference frames, we derived recurrent relations between the kernels with various indices. These relations allow us to construct any nonlinear kernel G _{l1, l2} ^l (c, c ₁, c ₂), if the kernel G _0,0⁰(c, c ₁, c ₂) is known.     

高光谱的刺五加黑斑病的早期检测研究

对感染黑斑病的刺五加叶片进行光谱特性研究,能为药用植物病害的早期筛选与精准治疗提供重要研究资料。实验目的,运用高光谱成像技术实现植物病害的自动监督分类与识别。实验过程,首先使用高光谱成像系统在可见光波段（380～960 nm）内采集刺五加黑斑病的叶片样本,光谱数据经过去除亮暗噪声和平滑预处理后,再经过主成分分析实现数据降维,继而运用基于不同核函数的支持向量机法建立分类模型,最后利用总体分类精度、Kappa系数等因子评价不同核函数对分类器性能的影响。根据叶片表面的特征将其分为四类样本：健康亮部、健康暗部、轻度病害和重度病害等。对比各类样本的光谱可知,刺五加的健康样本在540 nm波长存在一个明显峰值,在620～680 nm光谱曲线急剧上升;而病害样本的光谱反射率呈现缓慢且平稳的上升趋势,上述特征能够将图像空间上反射强度接近的健康亮部和严重病害完全区分开。经对比发现前四个主成分（PC1,PC2,PC3,PC4）在分类表达上存在差异,主要表现为PC1含有的信息多,能够较好地区分各类样本;PC2则出现健康亮部和严重病害的交叉混淆;PC3是对于PC2的补充,能基本完整地表达轻微病害;PC4的贡献率仅有0.19%,依然能够准确地识别严重病害。不同主成分分量在表达各类样本特征中存在的差异能够作为复杂样本分类的参考依据。对比四种核函数对支持向量机分类器性能的影响,结果显示线性核函数的识别过程受光强反射的影响较大,Sigmoid核函数的训练精度易受数据集大小的影响,在识别健康亮或暗,以及轻微病害上均存在一定的误差,多项式核函数与径向基核函数的效果较好,其中,多项式核函数的精度更高,为92.77%。研究表明,利用高光谱成像技术能够准确地识别刺五加的健康叶片和患病叶片,为实现自动诊断药用植物叶片病害提供新方法。     

基于近红外高光谱成像技术的小麦不完善粒检测方法研究

小麦作为主要的粮食作物在我国农业生产、运输、食品加工等方面占有重要地位。不完善籽粒严重影响了小麦质量与粮食安全。不完善籽粒主要在生产、存储、包装等过程中产生，目前我国小麦质量检测多以人工分选为主，但存在人主观性较强，肉眼易疲劳，且费时费力等问题，因此，如何快速准确鉴别小麦不完善粒是现阶段提高生产率和保证粮食安全的重要问题。运用高光谱成像技术和特征波段选取方法提出一种快速有效的小麦不完善粒鉴别方法。利用近红外高光谱成像系统获得1 000粒小麦样本在862.9～1 704.2 nm共256个波段的高光谱反射图像，其中包括健康粒、生芽粒、霉变粒和赤霉粒各250粒，提取每个样本感兴趣区域的平均反射率光谱作为分类特征。本文首先对提取的全波段光谱信息进行窗口平滑、一阶导数差分、矢量归一化等数据预处理，将原始光谱数据的隐藏信号放大并消除随机误差；在预处理的基础上运用伪偏最小二乘(DPLS)和正交化线性判别分析(OLDA)对光谱进行特征提取，降低数据的冗余度；最后采用仿生模式识别(BPR)建立四类小麦的鉴别模型。实验结果表明，采用全波段光谱信息建立的小麦不完善粒鉴别模型的平均识别精度达到97.8%，分析结果可知，利用近红外高光谱成像技术的全波段光谱信息对小麦不完善粒鉴别是可行的。尽管全波段光谱信息取得了较好的鉴别效果，但高光谱成像设备较为昂贵，获取高光谱全波段光谱信息数据量较大，无法满足对现场设备运算速度的高要求，因此，采用连续投影算法(SPA)对全波段光谱数据进行特征波段的选择，使波段数量由256维降低到10维，从而提高系统的可行性和运算速度。采用选取的10个特征波段建立小麦不完善粒鉴别模型，实验结果表明10个特征波段的平均识别精度仅为83.2%，分析结果可知，尽管采用10个特征波段提高了系统实时性，但鉴别准确性较差。为达到与全波段特征基本相当的鉴别效果，利用光谱特征与图像特征结合的方法建立小麦不完善粒鉴别模型，将上述选取的10个特征波段的形态信息、纹理信息和光谱信息进行结合，实验结果表明，10个特征波段的光谱信息与图像信息结合使鉴别的平均识别精度达到94.2%，此识别效果与利用全波段光谱数据的识别效果基本相当。利用高光谱成像系统探索了小麦不完善粒鉴别的可行性，通过分析以上实验可知，基于近红外高光谱成像技术对小麦不完善粒检测具有良好的效果，在有效的提高运算速度的同时也保证了系统的鉴别精度，为后期小麦不完善粒快速检测设备的开发提供了有效的研究方向。