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