Extending transition path theory: periodically driven and finite-time dynamics
Given two distinct subsets \(A\), \(B\) in the state space of some dynamical system, transition path theory (TPT) was successfully used to describe the statistical behavior of transitions from \(A\) to \(B\) in the ergodic limit of the stationary system. In the present paper, generalizations of TPT are derived, that remove the requirements of stationarity and of the ergodic limit. This provides the tool for the analysis of other dynamical scenarios: periodically forced dynamics and time-dependent finite-time systems. (We point out that originally TPT stems form molecular dynamics). The analysis is restricted to Markov processes on discrete state spaces, running in discrete time. The authors focus on the applicability of the general theory, e.g., nonequilibrium molecular, climate, fluid, or social dynamics (agent-based models). On simple model examples, mostly with numerical support, it is shown how the new tools deliver quantitative understanding about the statistical behavior of such systems. Five numerical examples, which refer to Markov chains on a finite-time interval, are discussed in separate subsections of Section 5. Section 6 is devoted to four numerical examples in the triple well landscape. These refer to several scenarios: infinite-time, stationary dynamics, that with an added periodic forcing, and finite-time dynamics. Technical proofs are relegated to appendices.
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