Emergent Continuity and Long-Time Exploration in Finite Nonlinear Networks

This work investigates the emergence of continuous macroscopic behavior and long-time dynamics in finite nonlinear systems. We model the system as a globally energy-conserving network of discrete oscillators and analyze its dynamics using a Kuramoto-based framework. We show that continuous macroscopic variables can emerge as effective coarse-grained representations through phase synchronization and relational entropy reduction. Despite the finite structural capacity of the system, conserved energy drives repeated cycles of structural dissolution and re-condensation. These cycles act as a dynamical mechanism enabling long-time ergodic exploration of macroscopic configurations. This framework provides a physically motivated perspective on how continuous descriptions and extended temporal dynamics can arise from fundamentally discrete, finite systems, with connections to coarse-graining and emergent collective behavior.