Derivation of a hydrodynamic equation for Ginzburg-Landau models in an external field

The lattice approximation to a time-dependent Ginzburg-Landau equation is investigated in the presence of a small external field. The evolution law conserves the spin, but is not reversible. A nonlinear diffusion equation of divergence type is obtained in the hydrodynamic limit. The proof extends to certain stochastically perturbed Hamiltonian systems.     

辐射烧蚀界面的单模Rayleigh-Taylor不稳定性分析

首先对惯性约束聚变(ICF)遇到的流体力学界面不稳定性作了简要的分析;其次讨论了Rayleigh-Taylor(RT)不稳定性在ICF中所起的作用;第三,根据一维数值计算结果,按照考虑密度梯度和烧蚀致稳的RT不稳定性增长的定标规律对不稳定性增长作了估计。最后在计算结果的讨论中,对有关实验测量提出某些倾向性的建议。     

Hydrodynamic Limit of Brownian Particles Interacting with Short- and Long-Range Forces

We investigate the time evolution of a model system of interacting particles moving in a d-dimensional torus. The microscopic dynamics is first order in time with velocities set equal to the negative gradient of a potential energy term plus independent Brownian motions: is the sum of pair potentials, V(r)+ ^d J(r); the second term has the form of a Kac potential with inverse range . Using diffusive hydrodynamic scaling (spatial scale ^–1, temporal scale ^–2) we obtain, in the limit 0, a diffusive-type integrodifferential equation describing the time evolution of the macroscopic density profile.     

The hydrodynamic limit of a one-dimensional nearest neighbor gradient system

We study the hydrodynamic behavior of a one-dimensional nearest neighbor gradient system with respect to a positive convex potential . In the hydrodynamic limit the density distribution is shown to evolve according to the nonlinear diffusion equation ,(q)/t= (²/dq²){F([1/₁(q)]), with F= –.     

On the Boundary Condition and Related Instability in the Smoothed Particle Hydrodynamics

In this work, we explore various relevant aspects of the Smoothed Particle Hydrodynamics regarding Burger's equation. The stability, precision, and efficiency of the algorithm are investigated in terms of different implementations.In particular, we argue that the boundary condition plays an essential role in the stability of numerical implementation.Besides, the issue is shown to be closely associated with the initial particle distribution and the interpolation scheme.Among others, we introduce an interpolation scheme termed symmetrized finite particle method. The main advantage of the scheme is that its implementation does not involve any derivative of the kernel function. Concerning the equation of motion, the calculations are carried out using two distinct scenarios, where the particles are chosen to be either stationary or dynamically evolved. The obtained results are compared with those obtained by using the standard finite difference method for spatial derivatives. Our numerical results indicate subtle differences between different schemes regarding the choice of boundary condition. In particular, a novel type of instability is observed where the regular distribution is compromised as the particles start to traverse each other. Implications and further discussions of the present study are also addressed.     

Hydrodynamic limits for the Boltzmann process

We study the behavior of the nonlinear Markov process associated to the Boltzmann equation under both hyperbolic and parabolic space-time scalings. In the first case the limit of the process is the solution of an o.d.e. with vector field given by a solution of the Euler equation, while in the second case the limit of the process, in the incompressible case, turns out to be a diffusion process whose drift is a solution of the incompressible Navier-Stokes equation.     

On the derivation of the incompressible Mavier-Stokes equation for Hamiltonian particle systems

We consider a Hamiltonian paticle system interacting by means of a pair potetial. We look at the behavior of the system on a space scale of order ^-1, times of order ^-2 and mean velocities of order , with a scale parameter. Assuming that the phase space density of the particles is give by a series in (the analog of the Chapman-Enskog expansion), the behavior of the system under this rescaling is described, to the lowest order in , by the incompressible Navier-Stokes equations. The viscosity is given in terms of microscopic correlations, and its expression agrees with the Green-Kubo formula.     

X光背光观测烧蚀面扰动引起内界面扰动的增长

侧向背光照相能直接反映靶表面扰动幅度的变化情况。在神光Ⅱ装置上,实验利用侧向背光照相技术,对烧蚀面扰动引起的内界面扰动增长进行了观测。实验结果表明,观察到的内界面扰动幅度大于期望值。分析认为,造成内界面较大扰动增长的原因主要是2维效应。X光辐照的主要是烧蚀面的中间部分,烧蚀面扰动引起的内界面的扰动就呈现出一幅从中间的扰动区域逐渐过渡到四周的图像。由此,提出了新的靶优化设计方案,应尽可能减小沿背光方向的样品尺寸。     

On the stability of crystal growth

We investigate properties of solid-on-solid models for crystal growth, involving general microscopic rates of capture of atoms by the crystal surface and of escape of atoms. The rates in this Markov process influence the stability of the growing surface. We prove, for various different ranges of the rate parameters, stability (i.e., ergodicity) and instability (i.e., nullity) of the growth process. Symmetry properties of the process, such as reversibility, dynamic reversibility, and reflection invariance, are proved or disproved under various conditions. We give a measure of surface smoothness that distinguishes between stable and unstable growth.     

Navier–Stokes Limit for a Thermal Stochastic Lattice Gas

We study a stochastic particle system on the lattice whose particles move freely according to a simple exclusion process and change velocities during collisions preserving energy and momentum. In the hydrodynamic limit, under diffusive space-time scaling, the local velocity field u satisfies the incompressible Navier–Stokes equation, while the temperature field solves the heat equation with drift u. The results are also extended to include a suitably resealed external force.     

On the dynamics of excitations in disordered systems

A new, time-local (TL) reduced equation of motion for the probability distribution of excitations in a disordered system is developed. ToO(k²) the TL equation results in a Gaussian spatial probability distribution, i.e, P(r, t) = [(2)^1/2]^–dexp(-r²/2²), where = (t) is a correlation length, andr = ¦r¦. The corresponding distribution derived from the Hahn-Zwanzig (HZ) equation is more complicated and assumes the asymptotic (r ) form: P(r, s)(s ^d )^–1exp(–r/) · (r/)^(1-d)/2 where = (s),d is the space dimensionality, ands is the Laplace transform variable conjugate tot. The HZ distribution generalizes the scaling form suggested by Alexanderet al. ford= 1. In the Markov limit (t)t, (s)1/s, and the two distributions are identical (ordinary diffusion).     

Step instability of the surface during Ino.2Gao.sAs （001） annealing

Anisotropic evolution of the step edges on the compressive-strained In0.2Ga0.8As/GaAs（001） surface has been investigated by scanning tunneling microscopy （STM）. The experiments suggest that step edges are indeed sinuous and protrude somewhere a little way along the [110] direction, which is different from the classical waviness predicted by the theoretical model. We consider that the monatomic step edges undergo a morphological instability induced by the anisotropic diffusion of adatoms on the terrace during annealing, and we improve a kinetic model of step edge based on the classical Burton Cabrer-Frank （BCF） model in order to determine the normal velocity of step enlargement. The results show that the normal velocity is proportional to the arc length of the peninsula, which is consistent with the first result of our kinetic model. Additionally, a significant phenomenon is an excess elongation along the [110] direction at the top of the peninsula with a higher aspect ratio, which is attributed to the restriction of diffusion lengths.     

Step instability of the In_0.2Ga_0.8As （001） surface during annealing

Anisotropic evolution of the step edges on the compressive-strained In0.2Ga0.8 As/GaAs(001) surface has been investigated by scanning tunneling microscopy (STM). The experiments suggest that step edges are indeed sinuous and protrude somewhere a little way along the [110] direction, which is different from the classical waviness predicted by the theoretical model. We consider that the monatomic step edges undergo a morphological instability induced by the anisotropic diffusion of adatoms on the terrace during annealing, and we improve a kinetic model of step edge based on the classical Burton–Cabrera–Frank (BCF) model in order to determine the normal velocity of step enlargement. The results show that the normal velocity is proportional to the arc length of the peninsula, which is consistent with the first result of our kinetic model. Additionally, a significant phenomenon is an excess elongation along the [110] direction at the top of the peninsula with a higher aspect ratio, which is attributed to the restriction of diffusion lengths.     

On the exact solutions of nonlinear diffusion-reaction equations with quadratic and cubic nonlinearities

Attempts have been made to look for the soliton content in the solutions of the recently studied nonlinear diffusion-reaction equations [R S Kaushal, J. Phys. 38, 3897 (2005)] involving quadratic or cubic nonlinearities in addition to the convective flux term which renders the system nonconservative and the corresponding Hamiltonian non-Hermitian.     

Step instability of the In0.2Ga0.8As （001） surface during annealing

Anisotropic evolution of the step edges on the compressive-strained In0.2Ga0.8 As/GaAs(001) surface has been investigated by scanning tunneling microscopy (STM). The experiments suggest that step edges are indeed sinuous and protrude somewhere a little way along the [110] direction, which is different from the classical waviness predicted by the theoretical model. We consider that the monatomic step edges undergo a morphological instability induced by the anisotropic diffusion of adatoms on the terrace during annealing, and we improve a kinetic model of step edge based on the classical Burton–Cabrera–Frank (BCF) model in order to determine the normal velocity of step enlargement. The results show that the normal velocity is proportional to the arc length of the peninsula, which is consistent with the first result of our kinetic model. Additionally, a significant phenomenon is an excess elongation along the [110] direction at the top of the peninsula with a higher aspect ratio, which is attributed to the restriction of diffusion lengths.     

On the Milne Problem and the Hydrodynamic Limit for a Steady Boltzmann Equation Model

For a stationary nonlinear Boltzmann equation in a slab with a particular truncation in the collision operator, the Milne problem for the boundary layer together with a weak type of hydrodynamic behavior in the interior of the slab are studied by nonperturbative methods in the small-mean-free-path limit.     

Stationary states and hydrodynamics of FHP cellular automata

The large scale behavior of FHP-type cellular automata is investigated in the presence of some additional random effects. It is shown that every translationinvariant stationary state of the modified model is a superposition of product measures. By means of the entropy argument of Yau and of Olla, Varadhan, and Yau, the macroscopic (Euler-type) equations governing the hydrodynamic behavior of FHP automata are also derived.Dedicated to Oliver Penrose on the occasion of his 65th birthday.     

A class of stochastic evolutions that scale to the porous medium equation

A class of reversible Markov jump processes on a periodic lattice is described and a result about their scaling behavior stated: Under diffusion scaling, the empirical measure converges to a solution of the porous medium equation on thed-dimensional torus. The process can be viewed as a randomly interacting configuration of sticks that evolves through exchanges of stick pieces between nearest neighbors through a zero-range pressure mechanism, with conservation of total stick length.     

The Asymptotic Behavior of the Ergodic Measure of a Glauber + Kawasaki model

We consider an interacting particle system given by the Glauber + Kawasaki dynamics. It is known that this process has a reaction diffusion equation as hydrodynamic limit. The ergodicity of this process in the presence of a metastable state (double well potential) was recently proved by S. Brassesco et al. In this Letter we prove that, in the limit, as ε → 0, the expected value of each spin converges to the global minimizer of the potential. We also prove decay of correlations of the ergodic measure.AMS Subject Classification (2000). 60K35 (82C22, 82C31)This work was partially supported by CNPq