The Augmented Jump Chain,Advanced Theory and Simulations

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The Augmented Jump Chain
Advanced Theory and Simulations ( IF 2.9 ) Pub Date : 2021-03-09 , DOI: 10.1002/adts.202000274
Alexander Sikorski ₁ , Marcus Weber ₁ , Christof Schütte _{1,

2}

Affiliation

Modern methods of simulating molecular systems are based on the mathematical theory of Markov operators with a focus on autonomous equilibrated systems. However, non‐autonomous physical systems or non‐autonomous simulation processes are becoming more and more important. A representation of non‐autonomous Markov jump processes is presented as autonomous Markov chains on space‐time. Augmenting the spatial information of the embedded Markov chain by the temporal information of the associated jump times, the so‐called augmented jump chain is derived. The augmented jump chain inherits the sparseness of the infinitesimal generator of the original process and therefore provides a useful tool for studying time‐dependent dynamics even in high dimensions. Furthermore, possible generalizations and applications to the computation of committor functions and coherent sets in the non‐autonomous setting are discussed. After deriving the theoretical foundations, the concepts with a proof‐of‐concept Galerkin discretization of the transfer operator of the augmented jump chain applied to simple examples are illustrated.

中文翻译：

增强跳链

模拟分子系统的现代方法是基于马尔可夫算子的数学理论，重点是自主平衡系统。但是，非自治的物理系统或非自治的仿真过程变得越来越重要。非自治马尔可夫跳跃过程的表示形式是时空上的自治马尔可夫链。通过关联跳跃时间的时间信息来增强嵌入的马尔可夫链的空间信息，可以得出所谓的增强跳跃链。增强的跳链继承了原始过程的无穷小生成器的稀疏性，因此提供了一个有用的工具，即使在高维中，它也可以用于研究时间相关的动力学。此外，讨论了在非自治环境中提交者函数和相干集合的可能通用化方法及其在计算中的应用。在推导出理论基础之后，说明了概念性加勒金离散化的概念，该概念证明了适用于简单示例的增强型跳链的转移算子。

更新日期：2021-04-11

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