Hierarchical Reconstruction of Time-arrow from Multi-time Correlations

Abstract

The entropy production rate (EPR), a key measure of thermodynamic irreversibility in thermodynamics of small systems, is difficult to determine directly in experiments, motivating lower-bound-based estimation from observations. However, a systematic framework for organizing increasing amounts of the irreversibility information in experimental state observables into progressively tighter bounds remains lacking. Here, we show that multi-time correlations of a class of state observations naturally encode this information to provide a hierarchy. By defining a reconstruction operation as a combination of correlations, we obtain a sequence of lower bounds on the EPR. Correlations of higher order capture the thermodynamic information at greater temporal depth, thereby capturing more irreversibility and yielding tighter bounds. Under ideal conditions, this hierarchy converges to the full EPR in the limit of infinitely dense observations over a finite time window.