Phase-Space Divergence as the Driver of Information Flow in Dynamical Systems

Understanding the mechanisms that generate information flow in dynamical systems is crucial for advancing causal inference and dependency characterization in natural and engineered systems. Information flow is defined as the exchange of predictive or uncertainty-reducing knowledge between variables in a coupled system, arising when fluctuations in one component influence the variability in another. This study establishes that information flow emerges as a direct result of trajectory divergence in phase-space, an effect encoded in the generalized dynamics of probability density functions. We show that when the divergence of flow fields in phase-space is non-zero, it induces temporal changes in the entropic structure of the system. This expands the traditional Liouville equation to non-conservative systems. This divergence creates, rather than merely propagates, informational dependencies among system components, highlighting the dynamic nature of mutual and multivariate information in such systems.

Our results reveal that in conservative systems, where phase-space volume is preserved, the system entropy remains invariant, and informational dependencies are determined solely by initial conditions. In contrast, dissipative systems—exemplified by the damped harmonic oscillator and the Lorenz system—exhibit significant entropic and informational evolution driven by non-zero divergence. The mathematical framework presented quantifies the role of divergence in shaping joint, marginal, and conditional entropy, as well as bivariate and higher-order mutual information. This approach provides a comprehensive understanding of how phase-space dynamics underpin the flow and transformation of information.

The findings have profound implications across multiple domains, including environmental science, climate dynamics, and engineered systems, where causal relationships often arise from interactions between variables in complex networks. By bridging physical principles with information theory, the work offers a new lens for exploring the dynamics of natural and artificial systems, with potential applications in predictive modeling, data assimilation, and the design of resilient systems under uncertainty.

This investigation not only addresses a longstanding question about the origin of information flow in coupled systems but also lays the groundwork for future studies incorporating time-lagged dependencies and higher-order interactions in both theoretical and applied contexts. The framework proposed herein enables a more refined analysis of information flow in complex systems, advancing our ability to interpret, predict, and engineer their behavior.