  We discuss the consequences of a variant of the Hatano-Sasa relation in which
a non-stationary distribution is used in place of the usual stationary one. We
first show that this non-stationary distribution is related to a difference of
traffic between the direct and dual dynamics. With this formalism, we extend
the definition of the adiabatic and non-adiabatic entropies introduced by M.
Esposito and C. Van den Broeck in Phys. Rev. Lett. 104, 090601 (2010) for the
stationary case. We also obtain interesting second-law like inequalities for
transitions between non-stationary states.
