The Augmented Jump Chain. Issue 4 (9th March 2021) - Sikorski, Alexander; Weber, Marcus; Schütte, Christof
Abstract: 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. Abstract : This article shows how Markov models can be extended to non‐equilibrium systems. The augmented jump chain allows to represent non‐autonomous Markov jump processes as autonomous Markov chains on space‐time. By this, it inherits the sparseness of the infinitesimal generator of the original process and provides a useful tool for studying time‐dependent dynamics in high dimensions.