Stochastic echo phenomena in nonequilibrium systems

A thermodynamic system is driven out of equilibrium by a time-dependent force or a nonconservative force represented with a protocol λ(t). The dynamics of such a system is irreversible so that the ensemble of trajectories under a time-reversed protocol λ(-t) is not equivalent to that of timereversed trajectories under λ(t). We raise a question whether one can find a suitable protocol under which the system exhibits time-reversed motions of the original system. Such a phenomenon is referred to as a stochastic echo phenomenon. We derive a condition for the optimal protocol that leads to the stochastic echo phenomenon in Langevin systems. We find that any system driven by timeindependent nonconservative forces has a dual system exhibiting the stochastic echo phenomenon perfectly. The stochastic echo phenomena are also demonstrated for harmonic oscillator systems driven by time-dependent forces. Our study provides a novel perspective on the time-irreversibility of nonequilibrium systems.