Simple physical systems as a reference for multivariate information dynamics

Understanding a complex system entails capturing the non-trivial collective phenomena that arise from interactions between its different parts. Information theory is a flexible and robust framework to study such behaviors, with several measures designed to quantify and characterize the interdependencies among the system's components. However, since these estimators rely on the statistical distributions of observed quantities, it is crucial to examine the relationships between information-theoretic measures and the system's underlying mechanistic structure. To this end, here, we present an information-theoretic analytical investigation of an elementary system of interactive random walkers subject to Gaussian noise. Focusing on partial information decomposition, causal emergence, and integrated information, our results help us develop some intuitions on their relationship with the physical parameters of the system considered. Specifically, our findings clarify how coarse-graining affects information measures, as well as the mechanistic origin of higher-order behaviors and statistical emergence. Overall, we observe that in this simple scenario, information measures align more reliably with the system's mechanistic properties when calculated at the level of microscopic components, rather than their coarse-grained counterparts, and over timescales comparable with the system's intrinsic dynamics. Moreover, we show that approaches that separate the contributions of the system's dynamics and a steady-state distribution (e.g., via causal perturbations) may help strengthen the interpretation of information-theoretic analyses.