Stochastic echo phenomena in nonequilibrium systems

Abstract

A thermodynamic system is driven out of equilibrium by a time-dependent force or a nonconservative force represented with a protocol λ(t). 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.

On the steady-state probability distribution of nonequilibrium stochastic systems

“…In Ref. [30], it was shown that the systems with λ and −λ are dual to each other with respect to time reversal (see also Ref. [31]).…”

“…where c is given in (30). The steady state distribution P st (x) = e −φ(x) is then determined from the relation…”

On a Question Related to Kolmogorov Forward Equation

“…Note that we made choices of signs consistent with statistical physics. For more detailed discussion and application to statistical physics, one may consult [1], and [2]. The following examples show that the solution of the question is in general not unique without the compactness assumption.…”