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By taking full advantage of the dynamic property imposed by the detailedbalance condition, we derive a new refined unified fluctuation theorem (FT) forgeneral stochastic thermodynamic systems. This FT involves the jointprobability distribution functions of the final phase space point and athermodynamic variable. Jarzynski equality, Crooks fluctuation theorem, and theFTs of heat as well as the trajectory entropy production can be regarded asspecial cases of this refined unified FT, and all of them are generalized toarbitrary initial distributions. We also find that the refined unified FT caneasily reproduce the FTs for processes with the feedback control, due to itsunconventional structure that separates the thermodynamic variable from thechoices of initial distributions. Our result is heuristic for furtherunderstanding of the relations and distinctions between all kinds of FTs, andmight be valuable for studying thermodynamic processes with informationexchange.

Jarzynski Equality, Crooks Fluctuation Theorem and the Fluctuation Theorems of Heat for Arbitrary Initial States