主方程与随机路径积分理论
Master equations and the theory of stochastic path integrals
摘要：
这篇评论提供了一个教学和自我包含的介绍主方程和他们的代表性的路径积分。我们讨论的解析和数值方法的主方程的解决方案，保持我们的重点是方法，即使是适用于随机波动很强。回顾的方法包括生成函数技术和泊松表示，以及新的方式映射到线性偏微分方程（PDE）的向前和向后的主方程。谱方法，WKB近似，和变分方法已被提出的生成函数服从的偏微分方程的分析。在概述了这些方法之后，我们用两个路径积分来求解所导出的偏微分方程。路径积分提供求解主方程的条件概率分布的不同精确表示。在分析基元化学反应时，我们对这两种路径积分进行了修正。在此基础上，我们讨论了稀有事件概率的近似方法，并导出了Fokker-Planck方程的路径积分表示。为了使我们的评论能够被广泛的社区所接受，我们使用了概率论的语言，而不是量子（场）理论。支持各种技术概念的概率结构，如相干态，Doi移位和正常有序的可观测量，从而明确。
英文摘要：
This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. We discuss analytical and numerical methods for the solution of master equations, keeping our focus on methods that are applicable even when stochastic fluctuations are strong. The reviewed methods include the generating function technique and the Poisson representation, as well as novel ways of mapping the forward and backward master equations onto linear partial differential equations (PDEs). Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE obeyed by the generating function. After outlining these methods, we solve the derived PDEs in terms of two path integrals. The path integrals provide distinct exact representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Furthermore, we review a method for the approximation of rare event probabilities and derive path integral representations of Fokker-Planck equations. To make our review accessible to a broad community, we have used the language of probability theory rather than quantum (field) theory. The probabilistic structures underpinning various technical concepts, such as coherent states, the Doi-shift, and normal-ordered observables, are thereby made explicit.
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