Reversal symmetries for cyclic paths in systems far from thermodynamic equilibrium

We have examined time-reversal properties of cyclic paths in non-equilibrium systems described by Markov processes. We find that even for systems far from equilibrium, the long-time limit of the ratio of forward to reverse occurrences of a cyclic path is determined only by the thermodynamic force on the cycle itself. In particular, if traversing the cycle does not expend energy, the frequency with which the cyclic path occurs is the same for the forward and reverse directions. This symmetry holds irrespective of the transition rates elsewhere in the system. This is in contrast to time-reversal symmetry for pairs of transitions, which can break down for all transitions in a system if even a single transition rate is disturbed from its equilibrium value, and to steady-state fluxes and probabilities of system states, which in the non-equilibrium case depend in a complex manner on all transition rates. Our results suggest the existence of fluctuation theorems for cycle counts, which is a topic of current research. They also have experimental applications for biological systems, as they can shed light on which processes in a complex system involve expenditure of energy.