Dynamical theory of complex systems with two-way micro–macro causation

In many complex systems encountered in the natural and social sciences, mechanisms governing system dynamics at a microscale depend upon the values of state variables characterizing the system at coarse-grained, macroscale (Goldenfeld and Woese, 2011, Noble et al., 2019, and Chater and Loewenstein, 2023). State variables, in turn, are averages over relevant probability distributions of the microscale variables. Neither inferential Top–Down nor mechanistic Bottom–Up modeling alone can predict responses of such scale-entwined systems to perturbations. We describe and explore the properties of a dynamic theory that combines Top–Down information-theoretic inference with Bottom–Up , state-variable-dependent mechanisms. The theory predicts the functional form of nonstationary probability distributions over microvariables and relates the trajectories of time-evolving macrovariables to the form of those distributions. Analytic expressions for the time evolution of Lagrange multipliers from Maxent solutions allow for rapid calculation of the time trajectories of state variables even in high dimensional systems. Examples of possible applications to scale-entwined systems in nonequilibrium chemical thermodynamics, epidemiology, economics, and ecology exemplify the potential multidisciplinary scope of the theory. A worked-out low-dimension example illustrates the structure of the theory and demonstrates how scale entwinement can result in slowed recovery from perturbations, reddened time series spectra in response to white-noise input, and hysteresis upon parameter displacement and subsequent restoration.