Violation of local detailed balance upon lumping despite a clear timescale separation

Abstract

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 timescale separation local detailed balance is believed to inherently 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 lumping leads to a violation of local detailed balance despite a clear timescale 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 as far as we know 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.