Journal
MULTISCALE MODELING & SIMULATION
Volume 7, Issue 3, Pages 1192-1219Publisher
SIAM PUBLICATIONS
DOI: 10.1137/070699500
Keywords
transition path theory; Markov jump process; committor function; network; graph theory; reactive trajectories; probability current; transition rate
Funding
- DFG Research Center MATHEON [FZT86]
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The framework of transition path theory (TPT) is developed in the context of continuous-time Markov chains on discrete state-spaces. Under assumption of ergodicity, TPT singles out any two subsets in the state-space and analyzes the statistical properties of the associated reactive trajectories, i.e., those trajectories by which the random walker transits from one subset to another. TPT gives properties such as the probability distribution of the reactive trajectories, their probability current and flux, and their rate of occurrence and the dominant reaction pathways. In this paper the framework of TPT for Markov chains is developed in detail, and the relation of the theory to electric resistor network theory and data analysis tools such as Laplacian eigenmaps and diffusion maps is discussed as well. Various algorithms for the numerical calculation of the various objects in TPT are also introduced. Finally, the theory and the algorithms are illustrated in several examples.
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