An Irreversible Discrete Causal Framework Driven by Local Newtonian Information for Propagating Dissipative Open Systems

This study introduces a mathematical formulation, termed the causal recording (CR) framework, to elucidate propagating dissipative open systems such as surface waves. In this framework, spatiotemporally independent local states deterministically transition via an operator and irreversibly register force information. By sequentially recording these states from previously registered causal steps, the model inherently incorporates time-irreversibility and physical stability regardless of spatial domain boundaries. The coherence of the CR formulation with natural patterns is confirmed through direct comparison with real-time water ripple experiments. Numerical reconstructions reveal that the emergent macroscopic wave evolution is autonomously driven by the sequential accumulation of localized Newtonian force events. Furthermore, dynamic analysis reveals that dissipative Lagrangian trajectories emerge from the collective history of these local force events. Based on the difference between localized internal force depletion and external propagation, a dual dynamic structure for propagating open systems is proposed, suggesting a total-energy formulation: Uprop = Ui + Ue. Finally, by correlating these emergent dynamics during propagation with entropic force, the entropy at the initial force source is offset by the continuous forward energy flux. When the continuous external energy flows are included, the total entropy of the propagating dissipative system continuously increases.