On the forward/backward symmetry of transition path time distributions in nonequilibrium systems

Recent single-molecule studies have been able to observe molecular transition paths, i.e., short and rare excursions where the molecule is caught in transit from one stable molecular conformation to another, motivating a flurry of theoretical work. Under equilibrium conditions, the temporal duration of a transition path, or the transition path time, exhibits a fundamental property that is a consequence of the time reversal symmetry: the distribution of the transition path time is independent of the transition direction (forward or backward). Many conformational changes occurring in living systems, however, take place away from equilibrium. Molecular motors, for example, make more steps in the forward than in the backward direction, resulting in overall unidirectional motion. Is the symmetry between the transition path times for individual steps in the forward and backward directions preserved in this case? Here, we show that this symmetry is still preserved if the transition occurs between (precisely) known microscopic states. However, since most experiments can only monitor some low-dimensional property of the system (e.g., the position of the molecular motor along its track), this symmetry, when applied to experimental observations, is generally broken, except in equilibrium. In a different language, the forward/backward symmetry violation occurs only if two conditions are met simultaneously: (1) the dynamics of the low-dimensional experimental observables is non-Markovian and (2) the system is not in equilibrium.