基于大偏差理论的随机动力学研究

The researches on the stochastic dynamics based on the large deviation theory

本文介绍了大偏差理论的基本思想、基本概念以及大偏差理论在离出问题研究中的应用.本文评述了有关离出问题的三个重要指标:平均首次离出时间、离出位置分布和最优离出路径相关研究的思路和方法,而其中对最优离出路径的刻化是结构性的难题.针对平均首次离出时间,本文介绍了它与拟势的关系,并应用平均首次离出时间的结论分析了随机共振以及自诱导随机共振中的时间匹配机制.对于离出位置分布,本文介绍了提高蒙特卡罗模拟速度的相关算法,并重点评述了其中的概率演化算法和相关的算例.最后,对于最优离出路径的研究,本文讨论了几类计算方法,分析了最优路径满足的辅助哈密尔顿系统轨线由于非线性多值性形成的拉格朗日流形拓扑结构的奇异性及其动力学含义,并进一步给出了有限噪声强度激励条件下的作用量修正方法.最后,给出了大偏差理论应用发展的一些开放性问题的展望.

This paper introduces the basic ideas and concepts of large deviation theory and its application in the study of exit problems.Three critical indicators of exit problems are reviewed:mean first passage time,exit location distribution and most probable escape path.Among them,the characterization of the most probable escape path is a structural conundrum.For the mean first passage time,its relationship with quasi-potential is introduced,and it is applied to analyze the time matching mechanism in stochastic resonance and self-induced stochastic resonance.For the exit location distribution,the relevant algorithms to accelerate the Monte Carlo numerical simulation are discussed,and the probability evolution method is specially clarified with some interesting examples.For the study of the most probable escape path,several calculation methods are discussed,and the singularity of the topological structure and its dynamical implications of the Lagrangian manifold formed by the auxiliary Hamilton system trajectories are analyzed.Furthermore,the corrected action method under the condition of finite noise intensity is given.Finally,the prospect of some open problems for the application and development of large deviation theory is discussed.