Mind the memory: Consistent time reversal removes artefactual scaling of energy dissipation rate and provides more accurate and reliable thermodynamic inference

Abstract: It has been proposed that an observed inverse power-law dependence of the Markovian estimate for the steady-state dissipation rate on the coarse-graining scale in self-similar networks reflects a scale-dependent energy dissipation. By explicit examples, it is demonstrated here that there are in general no relations between such an apparent power-law dependence and the actual dissipation on different length scales. We construct fractal networks with a single dissipative scale and networks with a true inverse energy-dissipation cascade, and show that they display the same scaling behavior. Moreover, we show that a self-similar network structure does not imply an inverse power-law scaling but may be mistaken for one in practice. When no dissipative cycles become hidden by the coarse graining, any scale dependence of the dissipation estimate vanishes if the memory is correctly accounted for in the time-reversal operation. A $k$-th order estimator is derived and necessary and sufficient conditions are proved for a guaranteed lower bound on dissipation. These higher-order estimators saturated in the order are proved to provide sharper lower bounds on dissipation and their scale dependence signifies hidden dissipative cycles. It is shown that estimators not saturated in the order may erroneously overestimate the microscopic dissipation. Our results underscore the still underappreciated importance of correctly accounting for memory in analyzing coarse observations.