  If a system is at thermodynamic equilibrium, an observer cannot tell whether
a film of it is being played forward or in reverse: any transition will occur
with the same frequency in the forward as in the reverse direction. However, if
expenditure of energy changes the rate of even a single transition to yield a
non-equilibrium steady state, such time-reversal symmetry undergoes a
widespread breakdown, far beyond the point at which the energy is expended. An
explosion of interdependency also arises, with steady-state probabilities of
system states depending in a complicated manner on the rate of every transition
in the system. Nevertheless, in the midst of this global non-equilibrium
complexity, we find that cyclic paths have reversibility properties that remain
local, and which can exhibit symmetry, no matter how far the system is from
thermodynamic equilibrium. Specifically, given any cycle of reversible
transitions, the ratio of the frequencies with which the cycle is traversed in
one direction versus the other is determined, in the long-time limit, only by
the thermodynamic force on the cycle itself, without requiring knowledge of
transition rates elsewhere in the system. In particular, if there is no net
energy expenditure on the cycle, then, over long times, the cycle traversal
frequencies are the same in either direction.
