Quantifying scale-dependent irreversibility using persistent homology

Irreversibility is a measure of whether an observer can distinguish a process from its time reversed version. For physical systems, irreversibility is a fundamental property related to dissipation, breaking of detailed balance and non-equilibrium phenomena. But in any real non-equilibrium system, such as in vivo studies of oocytes or in vitro reconstituted actomyosin, irreversibility is associated with the specific timescales of the system’s non-equilibrium dynamics: an observer can be fooled into believing a process is reversible if they watch on the wrong timescales. Here, we generalize persistence homology, a scale-dependent topological characterization method, to quantify irreversibility on different scales. While persistence homology is usually used to detect undirected loops, we define a similarity score inspired by statistical physics that captures information about directed circular fluxes. The resulting persistence barcode quantifies irreversibility on different timescales without any prior knowledge of what the relevant variables are.