Solving Langevin equation with the bicolour rooted tree method

Stochastic differential equations, especially the one called Langevin equation, play an important role in many fields of modern science. In this paper, we use the bicolour rooted tree method, which is based on the stochastic Taylor expansion, to get the systematic pattern of the high order algorithm for Langevin equation. We propose a popular test problem, which is related to the energy relaxation in the double well, to test the validity of our algorithm and compare our algorithm with other usually used algorithms in simulations. And we also consider the time-dependent Langevin equation with the Ornstein-Uhlenbeck noise as our second example to demonstrate the versatility of our method.     

Stochastic quantization method for spinning particles in a gravitational plane wave field

The exact and analytic Green functions for spinning relativistic particles in interaction with a gravitational plane wave field are obtained within the Stochastic Quantization Method of Parisi and Wu. We have separated the classical calculations from those related to the quantum fluctuations. The problem has been solved by using a perturbative treatment via the Langevin equation relying on phase and configuration spaces formulation.      

Beyond modified mean field: a case for a stochastic grain growth model

ABSTRACT

We address the basic mechanism of grain growth in two and three dimensions in single phase metallic systems. It is argued, a comprehensive grain growth model is best based on a stochastic process described by a Fokker Planck equation. This concept is sufficient to determine almost all features of grain growth in general terms. The existence of the von Neumann law and its counterpart in three dimensions enables us to provide an explicit form of the Fokker Planck equation and thus determine many features of grain growth in great detail. Finally, we illustrate this model by determining the salient features of grain growth in three dimensions in the long time limit, and comparing them with experimental and computational results.     

Closure approximation error in the mean solution of stochastic differential equations by the hierarchy method

The iterative or stochastic Green's function method of solution of stochastic differential equations is used to find the error terms in the solution and mean solution due to truncation in the hierarchy method. A comparison is made of solutions by the iterative and the hierarchy method.Supported by a grant from the Sloan Foundation.     

Lateral diffusion of the topological charge density in stochastic optical fields

Stochastic (i.e. random and quasi-random) optical fields may contain distributions of optical vortices that are represented by non-uniform topological charge densities. Numerical simulations are used to investigate the evolution under free-space propagation of topological charge densities that are inhomogeneous along one dimension. It is shown that this evolution is described by a diffusion process that has a diffusion parameter which depends on the propagation distance.     

A stochastic approach to hopping transport in semiconductors

Generalized charge carrier equations for hopping transport in semiconductors are derived which include also the widely used Van Roosbroeck equations. The approach is based on a microscopic stochastic interacting particle system which models the hopping of electrons on a random set of states.     

Low-frequency breathers in polyethylene crystal

It is shown both analytically and numerically that three-dimensional low-frequency (acoustic) breathers identified as localized coupled bend and twist nonlinear waves can exist in polyethylene (PE) crystal. Their motion along the chain is accompanied by the simultaneous excitation of both (“zig” and “zag”) sub-chains forming PE macromolecule.In the region of a breather one can observe intensive out-of-plane motion of carbon atoms and relatively small displacements in the plane of the zigzag. In spite of this smallness, the “secondary” nonlinear effects turn out to be crucial for the existence of breathers.Both the existence and stability of low-frequency breathers in free motion are confirmed by computer simulation, using the Molecular Dynamics (MD) procedure. The stability of breathers with respect to thermal excitations as well as to mutual collisions and collisions with optical breathers is also discussed. We study also the breathers’ formation in different conditions. It turns out that the frequencies and extensions of the breathers can be varied in very narrow regions predicted by our analytical solution.