库仑碰撞截面在等离子体粒子模拟中的应用

在等离子体粒子模拟中,TA模型和NanBu模型被广泛用于处理库仑碰撞,这两种模型要求每个时间步长内全部粒子参与计算。为了降低参与碰撞的粒子数,提高库仑碰撞的计算效率,提出了一种基于截面的库仑碰撞模拟方法,并给出了库仑碰撞概率的计算公式。采用该方法对不同温度不同密度电子气的弛豫过程进行模拟,分别对比了电子速度分布函数、电子温度以及电子x、y方向上的温度与电子温度之比的模拟值与理论值,验证了该方法的准确性。在相同的小时间步长上,该方法相比TA模型计算效率提升可达40%以上。对于较大的时间步长,该方法仍能得到与理论解近似的模拟结果,相比Nanbu模型,在相同的精度下可取更大的时间步长,计算效率也有所提升。研究表明,该方法同样适用于电子-离子碰撞。因此在提高库仑碰撞计算效率上,该方法具有碰撞粒子数少以及适用于大时间步长的优势。     

Particle simulation of Coulomb collisions: Comparing the methods of Takizuka & Abe and Nanbu

The interactions of charged particles in a plasma are governed by long-range Coulomb collision. We compare two widely used Monte Carlo models for Coulomb collisions. One was developed by Takizuka and Abe in 1977, the other was developed by Nanbu in 1997. We perform deterministic and statistical error analysis with respect to particle number and time step. The two models produce similar stochastic errors, but Nanbu’s model gives smaller time step errors. Error comparisons between these two methods are presented.     

Ta及Nanbu库仑碰撞模型数值对比研究

深入研究库仑碰撞,对两种库仑碰撞模型-Ta模型与Nanbu模型在理论上进行了对比分析,详细阐述了两种模型中散射角大小的区别.在已有的采用Ta模型的全三维粒子模拟/蒙特长罗(PIC/MCC)算法基础上,采用Nanbu模型对电子间库仑碰撞计算模块重新进行了算法设计.分别应用Ta模型、Nanbu模型和无库仑碰撞的全三维PIC/MCC算法对国外热门负氢离子源JAEA 10A中的电子能量沉积进行了模拟分析.模拟结果与实验结果的对比分析发现:1)库仑碰撞使电子能量分布更接近于麦克斯韦分布;2)相对于传统的Ta模型,Nanbu模型的散射角考虑了多体碰撞的累加效果从而具有更高的期望值,按其模拟得到的电子温度具有更高的精度.这些为国内外学者设计相关算法指明了方向.     

A field theoretic approach to the atomic collision problems

It is an endeavour to make field theoretic approach available to the domain of electronic and atomic collision physics. The capacity of QED is demonstrated in explaining atomic collision phenomena in Coulomb gauge and depending on energy, in relativistic Lorentz gauge. Feynman diagrams are used to calculate bound state collision problems in atomic physics.     

Modeling of beam ions loss and slowing down with Coulomb collisions in EAST

This paper uses the implicit Monte-Carlo full-orbit-following parallel program ISSDE to calculate the prompt loss and slowing down process of neutral beam injection (NBI)-generated fast ions due to Coulomb collisions in the equilibrium configuration of Experimental Advanced Superconducting Tokamak (EAST). This program is based on the weak equivalence of the Fokker-Planck equation under Rosenbluth MacDonald Judd (RMJ) potential and Stratonovich stochastic differential equation (SDE). The prompt loss with the LCFS boundary and the first wall (FW) boundary of the two co-current neutral injection beams are studied. Simulation results indicate that the loss behavior of fast ions using the FW boundary is very different from that of the LCFS boundary, especially for fast ions with a large gyration radius. According to our calculations, about 5.11% of fast ions generated by perpendicular injection drift out of the LCFS and then return inside the LCFS to be captured by the magnetic field. The prompt loss ratio of fast ions and the ratio of orbital types depend on the initial distribution of fast ions in the P_ζ-$\varLambda$ space. Under the effect of Coulomb collisions, the pitch-angle scattering and stochastic diffusion happens, which will cause more fast ion loss. For short time scales, among the particles lost due to collisions, the fraction of banana ions reaches 92.31% in the perpendicular beam and 58.65% in the tangential beam when the fraction of banana ions in the tangential beam is 3.4% of the total ions, which means that the effect of Coulomb collisions on banana fast ions is more significant. For long time scales, the additional fast ion loss caused by Coulomb collisions of tangential and perpendicular beams accounted for 16.21% and 25.05% of the total particles, respectively. We have also investigated the slowing down process of NBI fast ions.     

适用于中等耦合等离子体的碰撞算子

完全电离等离子体中，当试探粒子分布函数f_α是关于试探粒子速度v_α的偶函数时，导出了一个新的动力学方程的碰撞算子.该碰撞算子同时包括了大角散射(库仑近碰撞)和小角散射(库仑远碰撞)的二体碰撞的贡献,因此，该碰撞算子同时适用于弱耦合(库仑对数lnΛ≥10)和中等耦合(库仑对数2≤lnΛ≤10)等离子体.而且经过修改的碰撞算子和Rosenbluth势有直接的联系，当试探粒子和场粒子满足条件m_αm_β(如电子-离子碰撞或Lor 关键词： 中等耦合等离子体 小角散射 大角散射 碰撞算子     

颗粒碰撞的直接模拟算法

为研究细微颗粒间平均碰撞率,构建了直接模拟三维空间内颗粒碰撞的数值方法.对剪切流内夹带的无惯性颗粒和自由运动颗粒的碰撞过程分别进行了数值模拟,得到的颗粒平均碰撞率的模拟值与理论值的相对误差小于1%.在碰撞颗粒对的搜索过程中,引入网络排除法等优化措施,使计算量降低两个量级以上,而不增大相对误差.采用不同的Δt模拟同一颗粒群的运动,发现较小的时间步长可以避免因为截断误差增大的相对误差.在不遗失颗粒碰撞的条件下,较密的计算网格可明显减少计算量.     

Bulk viscosity of low-temperature strongly interacting matter

We study the bulk viscosity of a pion gas in unitarized Chiral Perturbation Theory at low and moderate temperatures, below any phase transition to a quark-gluon plasma phase.We argue that inelastic processes are irrelevant and exponentially suppressed at low temperatures. Since the system falls out of chemical equilibrium upon expansion, a pion chemical potential must be introduced, so we extend the existing theories that include it. We control the zero modes of the collision operator and Landau?s conditions of fit when solving the Boltzmann equation with the elastic collision kernel.The dependence of the bulk viscosity with temperature is reminiscent of the findings of Fernández-Fraile and Gómez Nicola (2009) [1], while the numerical value is closer to that of Davesne (1996) [2]. In the zero-temperature limit we correctly recover the vanishing viscosity associated to a non-relativistic monoatomic gas.     

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     

On boltzmann equations and fokker—planck asymptotics: Influence of grazing collisions

In this paper, we are interested in the influence of grazing collisions, with deflection angle near π/2, in the space-homogeneous Boltzmann equation. We consider collision kernels given by inverse-sth-power force laws, and we deal with general initial data with bounded mass, energy, and entropy. First, once a suitable weak formulation is defined, we prove the existence of solutions of the spatially homogeneous Boltzmann equation, without angular cutoff assumption on the collision kernel, fors ≥ 7/3. Next, the convergence of these solutions to solutions of the Landau-Fokker-Planck equation is studied when the collision kernel concentrates around the value π/2. For very soft interactions, 2<s<7/3, the existence of weak solutions is discussed concerning the Boltzmann equation perturbed by a diffusion term     

Triple-collision operator for a quantum gas with bound states

The triple-collision operator of dense gas theory is analyzed for a quantum-mechanical gas obeying Boltzmann statistics. The contribution of two-body bound states is extracted by using Faddeev's representation of the three-body resolvent. The result is a binary atom-molecule collision operator which includes the effects of molecular formation and breakup, and inelastic and rearrangement collisions. An additional contribution is a modification of the Boltzmann collision operator due to the binding of one member of the colliding pair to a third particle. The analysis is carried out in the framework of the Green-Kubo formulas so the operators considered are linear and the results are in a form suitable for the evaluation of the transport coefficients.     

Sub-iteration leads to accuracy and stability enhancements of semi-implicit schemes for the Navier–Stokes equations

We present an iterative semi-implicit scheme for the incompressible Navier–Stokes equations, which is stable at CFL numbers well above the nominal limit. We have implemented this scheme in conjunction with spectral discretizations, which suffer from serious time step limitations at very high resolution. However, the approach we present is general and can be adopted with finite element and finite difference discretizations as well. Specifically, at each time level, the nonlinear convective term and the pressure boundary condition – both of which are treated explicitly in time – are updated using fixed-point iteration and Aitken relaxation. Eigenvalue analysis shows that this scheme is unconditionally stable for Stokes flows while numerical results suggest that the same is true for steady Navier–Stokes flows as well. This finding is also supported by error analysis that leads to the proper value of the relaxation parameter as a function of the flow parameters. In unsteady flows, second- and third-order temporal accuracy is obtained for the velocity field at CFL number 5–14 using analytical solutions. Systematic accuracy, stability, and cost comparisons are presented against the standard semi-implicit method and a recently proposed fully-implicit scheme that does not require Newton’s iterations. In addition to its enhanced accuracy and stability, the proposed method requires the solution of symmetric only linear systems for which very effective preconditioners exist unlike the fully-implicit schemes.     

Time domain numerical modeling of wave propagation in 2D heterogeneous porous media

This paper deals with the numerical modeling of wave propagation in porous media described by Biot’s theory. The viscous efforts between the fluid and the elastic skeleton are assumed to be a linear function of the relative velocity, which is valid in the low-frequency range. The coexistence of propagating fast compressional wave and shear wave, and of a diffusive slow compressional wave, makes numerical modeling tricky. To avoid restrictions on the time step, the Biot’s system is splitted into two parts: the propagative part is discretized by a fourth-order ADER scheme, while the diffusive part is solved analytically. Near the material interfaces, a space–time mesh refinement is implemented to capture the small spatial scales related to the slow compressional wave. The jump conditions along the interfaces are discretized by an immersed interface method. Numerical experiments and comparisons with exact solutions confirm the accuracy of the numerical modeling. The efficiency of the approach is illustrated by simulations of multiple scattering.     

Scattering of low-energy electrons by positive ions in a magnetic field. II. Resonances,transition probabilities,and transport frequencies

We analyze quantum-mechanically electron-ion collisions in a magnetic field at a low temperature, for which the electron's thermal energy is less than the energy gap between two Landau levels and the electron's Larmor radius is less than the characteristic impact parameter of close collisions without the magnetic field. To calculate transition probabilities, we use the analytical procedure proposed in the first part of our paper. We calculate the energy and lifetime of the resonant (autoionization) states of an electron embedded in the Coulomb electric field of an ion and in a uniform magnetic field. The obtained values coincide in order of magnitude with the known exact numerical values. We find that the electron backward scattering probability irregularly (chaotically) depends on the particle energy and the magnetic field. We propose analytical approximations for the collision transport frequencies, one of which describes the electron braking along the magnetic field and another, equalizing of the temperatures corresponding to the electron motion along and across the magnetic field. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 51, No. 8, pp. 682–699, August 2008.     

Interactions between two-dimensional solitons in the diffractive-diffusive Ginzburg-Landau equation with the cubic-quintic nonlinearity

We report the results of systematic numerical analysis of collisions between two and three stable dissipative solitons in the two-dimensional (2D) complex Ginzburg-Landau equation (CGLE) with the cubic-quintic (CQ) combination of gain and loss terms. The equation may be realized as a model of a laser cavity which includes the spatial diffraction, together with the anomalous group-velocity dispersion (GVD) and spectral filtering acting in the temporal direction. Collisions between solitons are possible due to the Galilean invariance along the spatial axis. Outcomes of the collisions are identified by varying the GVD coefficient, β, and the collision “velocity” (actually, it is the spatial slope of the soliton’s trajectory). At small velocities, two or three in-phase solitons merge into a single standing one. At larger velocities, both in-phase soliton pairs and pairs of solitons with opposite signs suffer a transition into a delocalized chaotic state. At still larger velocities, all collisions become quasi-elastic. A new outcome is revealed by collisions between slow solitons with opposite signs: they self-trap into persistent wobbling dipoles, which are found in two modifications — horizontal at smaller β, and vertical if β is larger (the horizontal ones resemble “zigzag” bound states of two solitons known in the 1D CGL equation of the CQ type). Collisions between solitons with a finite mismatch between their trajectories are studied too.     

Accurate models of collisions in glow discharge simulations

Very detailed, self-consistent kinetic glow discharge simulations are used to examine the effect of various models of collisional processes. The effects of allowing anisotropy in elastic electron collisions with neutral atoms instead of using the momentum transfer cross-section, the effects of using an isotropic distribution in inelastic electron-atom collisions, and the effects of including a Coulomb electron-electron collision operator are all described. It is shown that changes in any of the collisional models, especially the second and third described above, can make a profound difference in the simulation results. This confirms that many discharge simulations have great sensitivity to the physical and numerical approximations used. Our results reinforce the importance of using a kinetic theory approach with highly realistic models of various collisional processes     

A unified gas-kinetic scheme for continuum and rarefied flows

With discretized particle velocity space, a multiscale unified gas-kinetic scheme for entire Knudsen number flows is constructed based on the BGK model. The current scheme couples closely the update of macroscopic conservative variables with the update of microscopic gas distribution function within a time step. In comparison with many existing kinetic schemes for the Boltzmann equation, the current method has no difficulty to get accurate Navier–Stokes (NS) solutions in the continuum flow regime with a time step being much larger than the particle collision time. At the same time, the rarefied flow solution, even in the free molecule limit, can be captured accurately. The unified scheme is an extension of the gas-kinetic BGK-NS scheme from the continuum flow to the rarefied regime with the discretization of particle velocity space. The success of the method is due to the un-splitting treatment of the particle transport and collision in the evaluation of local solution of the gas distribution function. For these methods which use operator splitting technique to solve the transport and collision separately, it is usually required that the time step is less than the particle collision time. This constraint basically makes these methods useless in the continuum flow regime, especially in the high Reynolds number flow simulations. Theoretically, once the physical process of particle transport and collision is modeled statistically by the kinetic Boltzmann equation, the transport and collision become continuous operators in space and time, and their numerical discretization should be done consistently. Due to its multiscale nature of the unified scheme, in the update of macroscopic flow variables, the corresponding heat flux can be modified according to any realistic Prandtl number. Subsequently, this modification effects the equilibrium state in the next time level and the update of microscopic distribution function. Therefore, instead of modifying the collision term of the BGK model, such as ES-BGK and BGK–Shakhov, the unified scheme can achieve the same goal on the numerical level directly. Many numerical tests will be used to validate the unified method.     

An action function for a higher step Grushin operator

The purpose of this paper is to discuss how we can construct the heat kernel for (sub)-Laplacian in an explicit (integral) form in terms of a certain class of special functions. Of course, such cases will be highly limited. Here we only treat a typical operator, called Grushin operator. So, first we explain two methods to construct the heat kernel of a “step 2” Grushin operator. One is the eigenfunction expansion which leads to an integral form for the heat kernel, then we treat the formula by a method called, complex Hamilton–Jacobi method invented by Beals–Gaveau–Greiner. One of the main result in this paper is to construct an action function for a higher order oscillator. Until now, no explicit expression of the heat kernel for higher order cases have been given in an explicit form and we show a phenomenon that our action function will play a role toward the construction of the heat kernel of higher step Grushin operators.     

Boltzmann equation for a dissociating gas

The incorporation of three-body collisions for dissociation/recombination into the Boltzmann equation is discussed. Conditions are assumed such that collisions are completed in the sense of scattering theory, so the collision operator is determined by scattering and reaction cross sections. The resulting equation has anH-theorem, and the equilibrium solution requires the law of mass action in addition to the Maxwellian dependence on momentum. A brief discussion is given of the normal solution and the transport coefficients.This paper is dedicated to Prof. E. G. D. Cohen on the occasion of his 65th birthday.