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If a system is at thermodynamic equilibrium, an observer cannot tell whethera film of it is being played forward or in reverse: any transition will occurwith the same frequency in the forward as in the reverse direction. However, ifexpenditure of energy changes the rate of even a single transition to yield anon-equilibrium steady state, such time-reversal symmetry undergoes awidespread breakdown, far beyond the point at which the energy is expended. Anexplosion of interdependency also arises, with steady-state probabilities ofsystem states depending in a complicated manner on the rate of every transitionin the system. Nevertheless, in the midst of this global non-equilibriumcomplexity, we find that cyclic paths have reversibility properties that remainlocal, and which can exhibit symmetry, no matter how far the system is fromthermodynamic equilibrium. Specifically, given any cycle of reversibletransitions, the ratio of the frequencies with which the cycle is traversed inone direction versus the other is determined, in the long-time limit, only bythe thermodynamic force on the cycle itself, without requiring knowledge oftransition rates elsewhere in the system. In particular, if there is no netenergy expenditure on the cycle, then, over long times, the cycle traversalfrequencies are the same in either direction.

Reversal symmetries for cyclic paths away from thermodynamic equilibrium