摘要
从概率密度演化的基本思想出发,阐述了概率密度演化方程的历史、进展与应用,文中首先剖析和澄清了概率守恒原理的物理意义,论述了概率守恒原理的随机事件描述和状态空间描述,并由此阐明了概率密度演化与系统物理演化的内在联系,即:系统的物理状态演化构成了概率密度演化的内在机制.在此基础上,结合概率守恒原理的两类描述以及系统状态的物理演化方程,以与历史上不同的方式,重新推导了经典概率密度演化方程,包括Liouville方程、FPK方程和Dostupov-Pugachev方程,进一步阐明了这些方程的物理意义,以及它们不能降阶的原因.结合概率守恒原理的随机事件描述和解耦的系统物理方程,导出了广义概率密度演化方程.分析了广义概率密度演化方程的物理意义.以非线性结构随机反应的概率密度演化分析为例,展示了概率密度演化理论的应用前景.最后,指出了需要进一步研究的问题.
Based on the ideas of probability density evolution,the history,development and applications of the probability density evolution equations are elaborated in this paper.First,the physical meaning of the principle of preservation of probability is clarified,and the principle is then presented in terms of random event description and state space description,respectively.Meanwhile,the intrinsic relationship between the probability density evolution and the physical evolution of the system is elucidated,i.e.the physical state evolution of the system is the inherent mechanism underlying the probability density evolution. By incorporating the two descriptions of the principle of preservation of probability into the physical evolution equations of the stochastic system,the classical probability density evolution equations including the Liouville equation,FPK equation and the Dostupov-Pugachev equation are revisited via methodologies different from the existing ones.The physical meaning of these equations is clarified together with the reason why their dimension cannot be reduced.Moreover,combining the random event description of the principle of preservation of probability with the uncoupled physical equation leads to the generalized density evolution equation with its physical sense exposed.The application of the probability density evolution theory is exemplified by the probability density evolution analysis of the response of nonlinear structures,and the problems in need of further studies are pointed out at the end of paper.
出处
《力学进展》
EI
CSCD
北大核心
2010年第2期170-188,共19页
Advances in Mechanics
基金
国家自然科学基金项目(59825105
50321803
50621062
10402030
10872148
90715033)
科技部863计划项目(2008AA05Z413)资助~~
关键词
随机动力系统
概率守恒原理
随机事件描述
状态空间描述
广义概率密度演化方程
stochastic dynamical system
principle of preservation of probability
random event description
state space description
generalized density evolution equation
