Inference of time irreversibility from incomplete information: Linear systems and its pitfalls

Data from experiments and theoretical arguments are the two pillars sustaining the job of modeling physicalsystems through inference. In order to solve the inference problem, the data should satisfy certain conditionsthat depend also upon the particular questions addressed in a research. Here we focus on the characterization ofsystems in terms of a distance from equilibrium, typically the entropy production (time-reversal asymmetry) orthe violation of the Kubo fluctuation-dissipation relation. We show how general, counterintuitive and negativefor inference, is the problem of the impossibility to estimate the distance from equilibrium using a series ofscalar data which have a Gaussian statistics. This impossibility occurs also when the data are correlated intime, and that is the most interesting case because it usually stems from a multi-dimensional linear Markoviansystem where there are many timescales associated to different variables and, possibly, thermal baths. Observinga single variable (or a linear combination of variables) results in a one-dimensional process which is alwaysindistinguishable from an equilibrium one (unless a perturbation-response experiment is available). In a settingwhere only data analysis (and not new experiments) is allowed, we propose as a way out the combined useof different series of data acquired with different parameters. This strategy works when there is a sufficientknowledge of the connection between experimental parameters and model parameters. We also briefly discusshow such results emerge, similarly, in the context of Markov chains within certain coarse-graining schemes. Ourconclusion is that the distance from equilibrium is related to quite a fine knowledge of the full phase space, andtherefore typically hard to approximate in real experiments.