  By taking full advantage of the dynamic property imposed by the detailed
balance condition, we derive a new refined unified fluctuation theorem (FT) for
general stochastic thermodynamic systems. This FT involves the joint
probability distribution functions of the final phase space point and a
thermodynamic variable. Jarzynski equality, Crooks fluctuation theorem, and the
FTs of heat as well as the trajectory entropy production can be regarded as
special cases of this refined unified FT, and all of them are generalized to
arbitrary initial distributions. We also find that the refined unified FT can
easily reproduce the FTs for processes with the feedback control, due to its
unconventional structure that separates the thermodynamic variable from the
choices of initial distributions. Our result is heuristic for further
understanding of the relations and distinctions between all kinds of FTs, and
might be valuable for studying thermodynamic processes with information
exchange.
