Transition-path probability depends non-monotonically on memory time for non-Markovian systems

Defining low-dimensional reaction coordinates is crucial for analyzing the dynamics of complex systems and for comparison with experiments. The maximal value of the transition-path probability along the reaction coordinate $x, p(\mathrm {TP}|x)$ , is a common estimator for reaction-coordinate quality by comparing to the theoretical maximal value of 1/2 in the overdamped Markovian limit. We show by analytical arguments and simulations that for non-Markovian dynamics $p(\mathrm {TP}|x)$ is non-monotonic as a function of the memory time and exceeds 1/2 for long memory time. This disqualifies $p(\mathrm {TP}|x)$ as a criterion for reaction coordinate quality.