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Generalized detailed Fluctuation Theorem under Nonequilibrium Feedback control

It has been shown recently that the Jarzynski equality is generalized under nonequilibrium feedback control [T. Sagawa and M. Ueda, Phys. Rev. Lett. {\bf 104}, 090602 (2010)]. The presence of feedback control in physical systems should modify both Jarzynski equality and detailed fluctuation theorem [K. H. Kim and H. Qian, Phys. Rev. E {\bf 75}, 022102 (2007)]. However, the generalized Jarzynski equality under forward feedback control has been proved by consider that the physical systems under feedback control should locally satisfies the detailed fluctuation theorem. We use the same formalism and derive the generalized detailed fluctuation theorem under nonequilibrium feedback control. It is well known that the exponential average in one direction limits the calculation of precise free energy differences. The knowledge of measurements from both directions usually gives improved results. In this aspect, the generalized detailed fluctuation theorem can be very useful in free energy calculations for system driven under nonequilibrium feedback control.