Unified model equations for microstructure evolution

Unified model equations hidden in microstructure evolution are discovered in this paper. The governing equations in Lifshitz-Slyozov-Wagner theory, and diffusion screening theory are derived with some approximations from the unified model equations. The governing equations in multiparticle diffusion simulation and phase-field simulation in microstructure evolution are also derived from the unified model equations. The advantages and limitations for different theories and simulations in microstructure evolution are compared in detail. This comparison can guide scientists to select computational tools for their needs in microstructure evolution. The unified model equations can be applied in many new technological fields, such as self-assembly in nanoscience.     

基于Lagrange方法封闭的两相湍流场方程模型

两相湍流场方程模型采用基于Euler方法的一阶矩方程,而二阶矩方程由Lagrange方法得到,新模型的封闭不需要附加其它假设.首先基于概率密度函数给出颗粒运动的连续方程和动量方程,其次由基于平均Langevin方程的Lagrange模型推导得到颗粒二阶矩方程,最终获得封闭的二阶矩模型.将新模型用于气固两相壁面射流的数值模拟,结果表明新模型合理有效.     

Lattice model with power-law spatial dispersion for fractional elasticity

A lattice model with a spatial dispersion corresponding to a power-law type is suggested. This model serves as a microscopic model for elastic continuum with power-law non-locality. We prove that the continuous limit maps of the equations for the lattice with the power-law spatial dispersion into the continuum equations with fractional generalizations of the Laplacian operators. The suggested continuum equations, which are obtained from the lattice model, are fractional generalizations of the integral and gradient elasticity models. These equations of fractional elasticity are solved for two special static cases: fractional integral elasticity and fractional gradient elasticity.     

A stochastic model of Maxwell's equations in 1+1 dimensions

We show that the random walk model due to Mark Kac which underlies the telegraph equations may be modified to produce Maxwell's field equations in 1+1 dimensions. This provides the field equations with a representation in terms of classical particles. It also establishes the Kac model as a strong conceptual link between the diffusion, telegraph, and Maxwell equations, and suggests that recent simulations of the Schrödinger and Dirac equations are analogous to Maxwell's equation in terms of interpretation.     

Dispersion Relations in Binary Gas Mixtures From a Kinetic Theory

Sound propagation and the initial value problems in gas mixtures of two components are investigated. By using the eigen theory of linearized Boltzmann equations, a model equations is formed, with the use of the Fourier-Laplace transform for model equations derived, the dispersion relations for both components are obtained.     

On the Partial Regularity of a 3D Model of the Navier-Stokes Equations

We study the partial regularity of a 3D model of the incompressible Navier-Stokes equations which was recently introduced by the authors in [11]. This model is derived for axisymmetric flows with swirl using a set of new variables. It preserves almost all the properties of the full 3D Euler or Navier-Stokes equations except for the convection term which is neglected in the model. If we add the convection term back to our model, we would recover the full Navier-Stokes equations. In [11], we presented numerical evidence which seems to support that the 3D model develops finite time singularities while the corresponding solution of the 3D Navier-Stokes equations remains smooth. This suggests that the convection term play an essential role in stabilizing the nonlinear vortex stretching term. In this paper, we prove that for any suitable weak solution of the 3D model in an open set in space-time, the one-dimensional Hausdorff measure of the associated singular set is zero. The partial regularity result of this paper is an analogue of the Caffarelli-Kohn-Nirenberg theory for the 3D Navier-Stokes equations.     

GEAR算法在随机轨道模型计算中的应用

本文对随机轨道模型中颗粒相常微分方程组的刚性问题进行了分析，结果表明：当采用常规算法如四阶Runge－kutta法求解方程组时，方程组的刚性是导致某些情况下计算发散或计算时间过长的原因。为此，本文将适用于求解刚性方程组的Gear算法应用于随机轨道模型的计算中，取得了良好的效果．     

Jerky flow model as a basis for research in deformation instabilities

A phenomenological jerky flow model was developed in which macroscale plastic strain rates are defined by dislocation kinetics. The model takes into account destructive processes governed by shear and bulk defect accumulation. At the heart of the model lie equations of solid mechanics and relaxation-type constitutive equations. A loaded elastoplastic solid is treated as a nonlinear dynamic system whose evolution, according to synergetic laws, is much contributed by negative and positive feedbacks expressed, respectively, through constitutive equations of the first group (relaxation equations) and constitutive equations of the second group (kinetic equations for deformation defect and damage accumulation rates). The negative feedback stabilizes deformation by relaxation, bringing the process to some local dynamic equilibrium. The positive feedback destabilizes deformation, driving the system to a critical state. Numerical experiment was performed in 2D and 3D statements. Statistical analysis of stress fluctuations about the average trend shows that the jerky flow model of an elastoplastic medium demonstrates evolution characteristic of nonlinear dynamic systems: through states of dynamic chaos and self-organized criticality to a global catastrophe.     

New amplitude equation of single—mode laser

The white-gain model and the white-loss model of a single-mode laser are investigated in the presence of cross-correlations between the real and imaginary parts of quantum noise as well as pump noise. It was found that, like the white cubic model (2001 Chin. Phys. Lett. 18 370), the amplitude equations are all decoupled from the phase equations for the two models, and the same novel term appears in the amplitude equations of the two models. So we can put the amplitude equations of all the models into a general form, that is, the new amplitude equation. We further use this new amplitude equation to study specifically the stationary properties of the laser intensity for the white-gain model.     

Collective motion of active Brownian particles in one dimension

We analyze a model of active Brownian particles with non-linear friction and velocity coupling in one spatial dimension. The model exhibits two modes of motion observed in biological swarms: A disordered phase with vanishing mean velocity and an ordered phase with finite mean velocity. Starting from the microscopic Langevin equations, we derive mean-field equations of the collective dynamics. We identify the fixed points of the mean-field equations corresponding to the two modes and analyze their stability with respect to the model parameters. Finally, we compare our analytical findings with numerical simulations of the microscopic model.     

Large-scale turbulence modelling via α-regularisation for atmospheric simulations

We study the possibility of obtaining a computational turbulence model by means of non-dissipative regularisation of the compressible atmospheric equations for climate-type applications. We use an α-regularisation (Lagrangian averaging) of the atmospheric equations. For the hydrostatic and compressible atmospheric equations discretised using a finite volume method on unstructured grids, deterministic and non-deterministic numerical experiments are conducted to compare the individual solutions and the statistics of the regularised equations to those of the original model. The impact of the regularisation parameter is investigated. Our results confirm the principal compatibility of α-regularisation with atmospheric dynamics and encourage further investigations within atmospheric model including complex physical parametrisations.     

汽车空调模拟计算

汽车空调模拟计算张克忠，韩礼钟，朱明善（清华大学热能工程系北京１０００８４）关键词：计算机模型，汽车空调一、引言目前全球大约有２亿６千万辆汽车装有汽车空调系统，年消耗ＣＦＣ－１２达１０．２万吨，约占ＣＦＣ－１２总产量的４０％。欧美等发达国家１９９４年...     

Transients and basins of attraction in neutral network models

Approximate dynamic mean field equations for a generalized Hopfield model are derived, which allow to calculate transient properties of this model. These equations are exact for short times and yield the replica symmetric solution as a stationary solution. They allow reliable computation of retrieval trajectories and basins of attraction of retrieval states, as demonstrated by comparison with simulations. The equations are derived for networks with arbitrary mean activity and results are given for the standard model and for low activities.     

The concept of “incompatibility” for a simple model of a crystal lattice

A system of equations is derived for a simple model of a crystal lattice disturbed by a dislocation. The model is represented by two rows of atoms. The system of equations is analogous to that describing a singular dislocation in the theory of the continuous distribution of crystal imperfections. The method of solving the derived system of equations is given.     

螺旋线行波管注波互作用时域理论

研究了螺旋线行波管中电子注与高频场互作用的时域理论.电子对场的作用由高频场方程和空间电荷场方程模拟,场对电子注的作用由运动方程模拟.在螺旋导电面模型下利用安培环路定理和法拉第电磁感应定律得到了时域高频场方程.利用空间电荷波模型处理空间电荷场,得到了空间电荷场方程.将高频场和空间电荷场代入洛伦兹力方程,得到了运动方程.利用耦合阻抗处理高频场方程的激励源,使得高频场方程的求解能够借助诸如HFSS或HFCS等高频模拟软件来实现,增强了时域理论的灵活性.基于上述理论,编写软件数值模拟某螺旋线行波管,验证了时域理论的可行性.     

Macroscopic description of a subdiffusion-controlled bimolecular reaction

The random trap model is used to derive equations describing reaction-subdiffusion systems with diffusion-controlled (infinitely fast) bimolecular reaction. A hierarchy of equations in terms of distribution functions is closed by using a quasi-equilibrium condition in the equation for the two-particle distribution function. The reaction terms in the resulting equations contain products between concentrations and diffusion jump rates, rather than products of concentrations as dictated by the law of mass action. The same equations are also derived in the framework of a nonlinear continuous-time random walk model. The equations are used to show that inhomogeneity of the medium may manifest itself by fractional-order reaction terms.     

蒙特卡罗方法在解微分方程边值问题中的应用

介绍了蒙特卡罗方法的基本原理以及随机数的产生方法。基于蒙特卡罗方法的思想,结合有限差分方法,建立了求解微分方程边值问题的随机概率模型,并以第一类边界条件的拉普拉斯方程和一个给定初值及边界条件的非稳态热传导方程为数值算例,研究了蒙特卡罗方法在求解微分方程边值问题中的应用。结果表明:利用蒙特卡罗方法,不仅可以有效解决给定边界条件的微分方程,对于给定初值条件的微分方程,也可以从时域有限差分方程出发,采用蒙特卡罗方法进行求解。数值模拟和对误差的理论分析均表明,增加蒙特卡罗试验中的模拟粒子点数,可以提高计算结果的精度。     

剪切流动条件下液滴变形和断裂的数值模拟

本文采用扩散界面法研究了剪切流动条件下悬浮液滴变形和断裂的动力学机制。控制方程采用考虑表面张力影响的Navier-Stokes-Cahn-Hilliard方程描述。计算网格采用均匀矩形交错网格。采用基于压力增量的近似投影法计算 Navier-Stokes方程,采用完全近似多重网格法计算Cahn-Hilliard方程。稳态液滴变形规律及液滴拉伸断裂过程的计算结果与试验结果符合较好,表明本文模型能够很好的研究液滴变形及断裂机理。     

Bifurcation and Relaxation Oscillations in Divertor Plasmas

This paper presents a contribution to bifurcation phenomena in scrape-off layer modelling. The bifurcations are caused by the non-monotonic temperature behaviour of the radiation characteristics of plasma impurities. Low-frequency relaxation oscillations observed in divertor plasmas of the ASDEX Upgrade tokamak are analyzed and computed using a simple model for the dynamics of the impurity density and the electron temperature. The corresponding model equations are a special case of a general system of transport equations whose stability is analyzed. Bifurcations are calculated by solving autonomous and parametrically driven model equations.     

Star-triangle relations in the exactly solvable statistical models

A regular method for analysis of lattice spin models with a nearest neighbour interaction is proposed. Star-triangle relations in the form of functional equations are used. Parametric families of transfer matrices commuting due to star-triangle relations are constructed. The eigenvalues of transfer matrices as functions of the spectral parameter are shown to obey two functional equations. The solution of these equations for the maximal eigenvalue yields the partition function of the model. The method is applied for evaluation of the partition function of the critical Potts models, the Ising model, the Ashkin-Teller model equivalent to the eight-vertex model.