Second-law violating events are not rare in nonequilibrium nonsteady states

For nonequilibrium systems in which fluctuations are important, it is well established that there can be events that violate the second law of thermodynamics on the trajectory level. For nonequilibrium steady-states, these second-law violating events are rare, but this may not be true for far-from-equilibrium nonsteady processes. In the paradigm system of a Brownian particle trapped under a time-dependent compressing potential, we demonstrate that second-law violating trajectories can outnumber the second-law obeying ones. In particular, for abrupt compressing harmonic and anharmonic potentials, analytic expressions for the total entropy production distribution and the fraction of second-law violating events are derived to show explicitly that second-law violating events can be of significant majority. These results are further confirmed in experiments.