Entropy reduction from a detailed fluctuation theorem for a nonequilibrium stochastic system driven under feedback control

We show explicitly the entropy reduction from a detailed fluctuation theorem for the general stochastic system driven by nonequilibrium process under feedback control. The effect of interaction of the feedback controller with the system is to reduce the entropy of the system. We define the entropy reduction for the single trajectory and show that the overall entropy production for the stochastic system with feedback control can be either positive or negative. The negative entropy production has been studied earlier for a simple system with velocity dependent feedback control [K. H. Kim and H. Qian, Phys. Rev. Lett. {\bf 93}, 120602 (2004)]. Our general approach provides the overall positive or negative entropy production irrespective of velocity dependent and position dependent feedback control.