This thesis will mainly discuss the transition path theory and its extension to the transition state. The framework of transition path theory (TPT) is developed in the context of continuous-time Markov chains on discrete state-spaces. Under the assumption of ergodicity,Transition path theory will first choose any two subsets (mostly metastable states) in the finite state-space based on the equilibrium distribution of the transition probability, and then it analyzes the statistical properties of those associated reactive trajectories, for instance,those trajectories by which the random walker transits from one subset to another. Transition path theory gives properties of these trajectories, such as their probability distribution, their probability current and flux, and their rate of occurrence and finally the dominantreaction pathways. In this thesis, we will first introduce the framework of transition path theory for Markov chains, and then briefly discuss its relation to the electric resistor network theory and Laplacian eigenmaps, and also diffusion maps is discussed as well.Based on Transition Path Theory (TPT) for Markov jump processes, this thesis develops a general approach for identifying and calculating Transition States (TS) of stochastic chemical reacting networks. The thesis first extend the concept of probability current, originally defined on edges connecting different nodes in the configuration space, to each sub-network. To locate sub-networks with maximal probability current on the separatrix between reactive and non-reactive events, which will give the Transition States of the reaction, constraint optimization is conducted. The thesis further introduce an alternative scheme to compute the transition pathways by topological sorting, which is shown to be highly efficient through analysis. Finally, the theory and algorithms are illustrated in several examples.
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