Dynamical independence: Discovering emergent macroscopic processes in complex dynamical systems

We introduce a notion of emergence for macroscopic variables associated with highly multivariate microscopic dynamical processes. Dynamical independence instantiates the intuition an emergent process as one possessing characteristics system ``in its own right,'' laws distinct from those underlying dynamics. quantify (departure from) by transformation-invariant Shannon information-based measure dependence. emphasize data-driven discovery dynamically independent variables, and idea multiscale ``emergence portrait'' complex systems. show how dependence may be computed explicitly linear systems in both time frequency domains, facilitating phenomena across spatiotemporal scales, outline application operationalization to inference portraits neural neurophysiological time-series data. discuss discrete- continuous-time deterministic dynamics, potential Hamiltonian mechanics classical such flocking cellular automata.