Escape Rate Of Brownian Particle Across The Barrier In The Harmonic Noise

A large number of problems in nature can be modeled as Brownian motion.The establishment of Langevin equation and Fokker-Planck equation provide basis for people to study Brownian motion. The system plus heat bath model provides a good microscopic model to study Brownian motion.The resolution of the Fokker-Planck equation is very difficult.At present,only a few systems, such as free field,linear potential and harmonic potential can be solved exactly.In most cases,we have to get the approximate analytic or numerical solution.Stochastic linearization method is an effective way for the solution of nonlinear systems which are in steady-state and driven by colored noise.The study how random force of colored noise impact the probability density function, in particular, the movement of Brownian particles driven by a colored noise, is an important issue in the theory of stochastic nonlinear systems.This paper studies the escape of a Brownian particle driven by a internal harmonic noise in a metastable potential field. This system can be described by a high dimensional Langevin equations which is Markovian type, then we can get the corresponding Fokker-Planck equation. By the exact solution of the probability density, we study the characteristics and physical mechanism of escape rate(Kramers rate).We resolve the Fokker-Planck equation approximately using the mothod of stochastic linearization and investigate the effect of the barrier shape on the Kramers rate and the physical mechanisms.