Motion by mean curvature by scaling a nonlocal equation: Convergence at all times in the two-dimensional case

The convergence to a motion by mean curvature by diffusively scaling a nonlocal evolution equation, describing the macroscopic behavior of a ferromagnetic spin system with Kac interaction and Glauber dynamics has recently been proved. The convergence is proven up to the times when the motion by curvature is regular. Here we show the convergence at all times in the two-dimensional case. Since, in this case, the only singularity is the shrinking to a point of a closed curve, we verify that the curve actually disappears past the singularity.     

Sharp interface tracking using the phase-field equation

A general interface tracking method based on the phase-field equation is presented. The zero phase-field contour is used to implicitly track the sharp interface on a fixed grid. The phase-field propagation equation is derived from an interface advection equation by expressing the interface normal and curvature in terms of a hyperbolic tangent phase-field profile across the interface. In addition to normal interface motion driven by a given interface speed or by interface curvature, interface advection by an arbitrary external velocity field is also considered. In the absence of curvature-driven interface motion, a previously developed counter term is used in the phase-field equation to cancel out such motion. Various modifications of the phase-field equation, including nonlinear preconditioning, are also investigated. The accuracy of the present method is demonstrated in several numerical examples for a variety of interface motions and shapes that include singularities, such as sharp corners and topology changes. Good convergence with respect to the grid spacing is obtained. Mass conservation is achieved without the use of separate re-initialization schemes or Lagrangian marker particles. Similarities with and differences to other interface tracking approaches are emphasized.     

Distribution functions for the Brownian motion of particles in a periodic potential driven by an external force

The stationary distribution functions for the Brownian motion of particles driven by an external force are calculated by expanding the velocity part into Hermite functions and the space part into a Fourier series. Insertion into the Fokker-Planck equation leads to a matrix continued fraction for the lowest two coefficients of the Hermite functions. Higher order terms are found by reverse iteration. Results are shown for a cosine potential. The good convergence allows the calculation in the full range of damping constants. For small friction the distribution function is in good agreement with previous results and the maxima are given by the solutions without noise.     

Thermostatting the atomic spin dynamics from controlled demons

Deterministic dynamics in extended phase space of a constant temperature interacting spin system is formulated. The spin temperature is recovered through the constrained equation of motion and is in agreement with Rugh’s geometrical approach to temperature for classical Heisenberg spin systems. Detailed comparisons are investigated between state-of-the-art stochastic spin dynamics and deterministic dynamics using a chain of thermostats, for which an accelerated convergence structure is found.     

Steady state self-diffusion at low density

We prove that the motion of a test particle in a hard sphere fluid in thermal equilibrium converges, in the Boltzmann-Grad limit, to the stochastic process governed by the linear Boltzmann equation. The convergence is in the sense of weak convergence of the path measures. We use this result to study the steady state of a binary mixture of hard spheres of different colors (but equal masses and diameters) induced by color-changing boundary conditions. In the Boltzmann-Grad limit the steady state is determined by the stationary solution of the linear Boltzmann equation under appropriate boundary conditions.Supported in part by NSF Grant No. PHY 78-15920-02.Supported by a Heisenberg Fellowship of the Deutsche Forschungsgemeinschaft.     

Tail shortening by discrete hydrodynamics

A discrete formulation of hydrodynamics was recently introduced, whose most important feature is that it is exactly renormalizable. Previous numerical work has found that it provides a more efficient and rapidly convergent method for calculating transport coefficients than the usual Green-Kubo method. The latter's convergence difficulties are due to the well-known long-time tail of the time correlation function which must be integrated over time. The purpose of the present paper is to present additional evidence that these difficulties are really absent in the discrete equation of motion approach. The memory terms in the equation of motion are calculated accurately, and shown to decay much more rapidly with time than the equilibrium time correlations do.     

Nuclear rotations studied by the time-dependent variational method

A general nuclear rotation including precession and wobbling motion is studied by the time-dependent variational method and a classical equation of motion is derived. The intrinsic wave function associated with the general rotational motion is constructed by making use of the constrained Hartree-Fock method and variables necessary in solving the equation are calculated. The method developed here is applied to a schematic extension of the Nilsson model.     

A Front-Tracking Method for Motion by Mean Curvature with Surfactant

In this paper, we present a finite difference method to track a network of curves whose motion is determined by mean curvature. To study the effect of inhomogeneous surface tension on the evolution of the network of curves, we include surfactant which can diffuse along the curves. The governing equations consist of one parabolic equation for the curve motion coupled with a convection-diffusion equation for the surfactant concentration along each curve. Our numerical method is based on a direct discretization of the governing equations which conserves the total surfactant mass in the curve network. Numerical experiments are carried out to examine the effects of inhomogeneous surface tension on the motion of the network, including the von Neumann law for cell growth in two space dimensions.     

Asymptotic Analysis for a Vlasov-Fokker-Planck/ Compressible Navier-Stokes System of Equations

This article is devoted to the asymptotic analysis of a system of coupled kinetic and fluid equations, namely the Vlasov-Fokker-Planck equation and a compressible Navier-Stokes equation. Such a system is used, for example, to model fluid-particle interactions arising in sprays, aerosols or sedimentation problems. The asymptotic regime corresponding to a strong drag force and a strong Brownian motion is studied and the convergence toward a two phase macroscopic model is proved. The proof relies on a relative entropy method. A. Mellet was partially supported by NSF grant DMS-0456647.     

Mapping the Monte Carlo scheme to Langevin dynamics: a Fokker-Planck approach

We propose a general method of using the Fokker-Planck equation (FPE) to link the Monte Carlo (MC) and the Langevin micromagnetic schemes. We derive the drift and diffusion FPE terms corresponding to the MC method and show that it is analytically equivalent to the stochastic Landau-Lifshitz-Gilbert (LLG) equation of Langevin-based micromagnetics. Subsequent results such as the time-quantification factor for the Metropolis MC method can be rigorously derived from this mapping equivalence. The validity of the mapping is shown by the close numerical convergence between the MC method and the LLG equation for the case of a single magnetic particle as well as interacting arrays of particles. We also find that our Metropolis MC method is accurate for a large range of damping factors alpha, unlike previous time-quantified MC methods which break down at low alpha, where precessional motion dominates.     

The dynamic natures of microscopic particles described by nonlinear Schrödinger equation

There are a lot of difficulties and troubles in quantum mechanics, when the linear Schrödinger equation is used to describe microscopic particles. Thus, we here replace it by a nonlinear Schrödinger equation to investigate the properties and rule of microscopic particles. In such a case we find that the motion of microscopic particle satisfies classical rule and obeys the Hamiltonian principle, Lagrangian and Hamilton equations. We verify further the correctness of these conclusions by the results of nonlinear Schrödinger equation under actions of different externally applied potential. From these studies, we see clearly that rules and features of motion of microscopic particle described by nonlinear Schrödinger equation are greatly different from those in the linear Schrödinger equation, they have many classical properties, which are consistent with concept of corpuscles. Thus, we should use the nonlinear Schrödinger equation to describe microscopic particles.     

时间无卷积广义主方程中核函数的精确解和高阶微扰展开:在自旋-玻色模型和激发态能量转移中的应用

The time-convolutionless (TCL) quantum master equation provides a powerful tool to simulate reduced dynamics of a quantum system coupled to a bath. The key quantity in the TCL master equation is the so-called kernel or generator, which describes effects of the bath degrees of freedom. Since the exact TCL generators are usually hard to calculate analytically, most applications of the TCL generalized master equation have relied on approximate generators using second and fourth order perturbative expansions. By using the hierarchical equation of motion (HEOM) and extended HEOM methods, we present a new approach to calculating the exact TCL generator and its high order perturbative expansions. The new approach is applied to the spin-boson model with different sets of parameters, to investigate the convergence of the high order expansions of the TCL generator. We also discuss circumstances where the exact TCL generator becomes singular for the spin-boson model, and a model of excitation energy transfer in the Fenna-Matthews-Olson complex.     

Asymptotic of Grazing Collisions and Particle Approximation for the Kac Equation without Cutoff

The subject of this article is the Kac equation without cutoff. We first show that in the asymptotic of grazing collisions, the Kac equation can be approximated by a Fokker-Planck equation. The convergence is uniform in time and we give an explicit rate of convergence. Next, we replace the small collisions by a small diffusion term in order to approximate the solution of the Kac equation and study the resulting error. We finally build a system of stochastic particles undergoing collisions and diffusion, that we can easily simulate, which approximates the solution of the Kac equation without cutoff. We give some estimates on the rate of convergence.     

A mechanical model of Brownian motion in half-space

We consider the motion of a heavy mass in an ideal gas in a semi-infinite system, with elastic collisions at the boundary. The motion is determined by elastic collisions. We prove in the Brownian motion limit the convergence of the position and velocity process of the heavy particle to a diffusion process in which velocity and position remain coupled.     

Modeling the light scattering by nanoparticles of complex shape using the generalized source method

The results of modeling the scattering of light by nanoparticles of complex shape using the generalized source method are presented. It is shown that the approach implementing direct solution of the implicit equation, which is derived by the generalized source method using the three-dimensional Green’s function, in conjunction with the fast Fourier transform significantly improves convergence of the method of volume integral equation and considerably extends the range of its applicability.     

Accelerating a FFT-based solver for numerical homogenization of periodic media by conjugate gradients

In this short note, we present a new technique to accelerate the convergence of a FFT-based solver for numerical homogenization of complex periodic media proposed by Moulinec and Suquet [1]. The approach proceeds from discretization of the governing integral equation by the trigonometric collocation method due to Vainikko [2], to give a linear system which can be efficiently solved by conjugate gradient methods. Computational experiments confirm robustness of the algorithm with respect to its internal parameters and demonstrate significant increase of the convergence rate for problems with high-contrast coefficients at a low overhead per iteration.     

Approximation of the Boltzmann equation by discrete velocity models

Two convergence results related to the approximation of the Boltzmann equation by discrete velocity models are presented. First we construct a sequence of deterministic discrete velocity models and prove convergence (as the number of discrete velocities tends to infinity) of their solutions to the solution of a spatially homogeneous Boltzmann equation. Second we introduce a sequence of Markov jump processes (interpreted as random discrete velocity models) and prove convergence (as the intensity of jumps tends to infinity) of these processes to the solution of a deterministic discrete velocity model.     

费米型级联运动方程的绝热截断方案

研究开放量子系统的量子耗散动力学对于理解许多新奇量子现象背后的机制和实现量子器件的精确量子态控制具有重要意义. 级联运动方程方法已成为研究这类量子耗散动力学最常用的数值方法之一. 然而，在处理强电子关联系统时，准确描述强关联效应需要高的级联截断层数. 这导致级联运动方程方法需要耗费大量物理内存和计算时间. 为了解决该问题，将具有最快耗散速率的耗散模式与其他较慢的耗散模式分离，提出了一种级联运动方程的绝热截断方案. 在单杂质安德森模型上进行的数值测试表明，与传统的方案相比，该截断方案显著地降低了级联运动方程收敛需要的截断层数. 此外，该截断方案缓解了长时间耗散动力学中的数值不稳定性.     

Effective one-dimensional equation of motion for nuclear fission

An approach for describing the dynamics of nuclear fission in the framework of generalized quantum mechanics is discussed. The collective kinetic energy is assumed to be two dimensional, and the reduced mass is allowed to vary with the coordinates. The generalized calculus of variation is employed for minimizing the action after being properly quantized as required by Hamilton's principle, employing a curvilinear coordinate system. The corresponding Euler Lagrange equation is identified as the required generalized equation of motion. The proposed generalized two-dimensional equation of motion is separated into a vibrational eigenvalue equation and a set of coupled-channel one-dimensional equations which describe the translational motion, by exploiting the completeness of the vibrational eigenfunctions. Such a system of coupled equations can be decoupled by replacing the coupling matrix elements by a nonlocal interaction, which can be rendered local after employing the effective mass approximation. As a consequence this differential equation is provided with an effective mass, an effective potential barrier, and a differential boundary term which is responsible for restoring the self-adjointness of the kinetic energy differential operator.     

A perturbation technique for the prediction of the displacement of a line-driven plate with discontinuities

A novel technique is presented for obtaining approximate analytic expressions for an inhomogeneous line-driven plate. The equation of motion for the inhomogeneous plate is transformed, and the transform of the total displacement is written as a sum of the solution for a homogeneous line-driven plate plus a term due to the inhomogeneity. The result is an integral equation for the transform of the inhomogeneous contribution. This expression may in general be solved numerically. However, by introducing a small parameter into the problem, it may be solved approximately using perturbation techniques. This series may not be convergent, but its convergence may be improved using Pade approximation. Results are presented for the case of a single mass discontinuity, and a distribution of mass discontinuities.