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Time arrow in open-boundary one-dimensional stochastic dynamics

We consider the finite-timestep Brownian dynamics of a single particle confined in one dimension, with a nonuniform temperature profile. In such an open-boundary scenario, one cannot observe any net probability current in the nonequilibrium steady state (NESS). On the other hand, the nonequilibrium nature of this system is exhibited through the asymmetry in forward and backward transition probabilities, as is reported in this work through the stochastic simulation analysis and theoretical arguments. The irreversibility becomes prominent nearby the temperature interface. We propose that the observed irreversibility can be accounted for via a virtual-gyration scenario, while the collapse of virtual gyrations upon the one-dimensional coordinate leads to the absence of probability current.