Dynamically regulated energy barriers with violation of symmetry for reaction path

The motion of a particle on a flexible one-dimensional base is investigated numerically. It is regarded as a barrier-overcoming process in a multi-dimensional phase space. Driving the system far from equilibrium, a directional flow of the particle is observed. In order to understand the appearance of the directionality, two kinds of transition states for the barrier-overcoming process are considered particularly: one of which is for the overcoming process from left to right and the other from right to left. System's conformations for these two transitions are found to be identical at equilibrium, but become different at non-equilibrium, which means that the transition states become direction dependent. The energy barrier heights defined for the two transition states are found to account for biased reaction rates for the directional flow. Thus, the barrier heights work as activation energy extended to far from equilibrium. From the viewpoint of phase space, the directional dependence of the transition states is understood as the symmetry violation of the reaction paths, the route of the reaction being selected depending on the overcoming direction. It should be noted that such a selection is brought about by autonomous dynamics of the system itself. These results suggest the possibility to define an effective potential for a class of dynamical systems in non-equilibrium situations.