Asymptotic Analyses Of 2-dimension With Dissipation Maytrix Kramers Problem

A noise activated escape of a particle with variational damped coefficient of X axis from a 2-dimentional potential is considered. A Brownian particle activated by noise will escape across potential barrier and enter into a deeper and more stable potential. This model can be used to explain the chemical reactions, which was first found by Kramers. So this kind of problem are called Kramers problem. Though the Kramers problem is originated from chemical-physical field, but now it plays an important role in many fields. Research on the Kramers problem is very valuable on both theory and application.The classification of the critical points and the domain of attraction, first passage time and the exit distribution, stochastic dynamical systems satisfied as a tail trajectory and the asymptotic formulation of exit distribution of a tail trajectory are examined. By means of ordinary differential equation, partial differential equation, stochastic differential equation, nonlinear dynamics, singular perturbations and so on ,some new results are obtained, which provide a solid ground for the practical use of models. The principal work are as follows:Firstly, the dimension of the space is add to two and damped coefficient is changed of X axis. The equation of the Brownian particle activated in two-dimension space is an partial differential equation with two-order of two-dimension. We can consider it as a stochastic differential equation with one order of three-dimension. And the classification of the critical points and the domain of attraction are analyzed. Secondly, first passage times and the exit distribution is formulated and analyzed by means of ordinary differential equation, partial differential equation, stochastic differential equation, nonlinear dynamics, singular perturbations. The formulation of first passage times and the exit distribution are obtained by asymptotic analysis of Fokker-Planck equation. Thirdly, a tail trajectory is analyzed. The tails are the trajectories of the original dynamics conditioned on reaching the boundaryΓbefore returning to the critical energy contourΓC. The tails obey different dynamic with the original trajectories. The modified dynamic of the tails is obtained by Bayes formula.