  Integrating out fast degrees of freedom is known to yield, to a good
approximation, memory-less, i.e. Markovian, dynamics. In the presence of such a
time-scale separation local detailed balance is believed to emerge and to
guarantee thermodynamic consistency arbitrarily far from equilibrium. Here we
present a transparent example of a Markov model of a molecular motor where
local detailed balance can be violated despite a clear time-scale separation
and hence Markovian dynamics. Driving the system far from equilibrium can lead
to a violation of local detailed balance against the driving force. We further
show that local detailed balance can be restored, even in the presence of
memory, if the coarse-graining is carried out as Milestoning. Our work
establishes Milestoning not only as a kinetically but for the first time also
as a thermodynamically consistent coarse-graining method. Our results are
relevant as soon as individual transition paths are appreciable or can be
resolved.
