Order parameter dynamics in complex systems: From models to data.

Area of Science:

• Complex Systems Science
• Statistical Physics
• Nonlinear Dynamics

Background:

• Collective ordering behaviors are common in complex systems, arising from self-organization and couplings.
• Order parameters quantify transitions to collective states, emerging from numerous degrees of freedom.
• Synergetics provides a framework for understanding self-organization and collective dynamics.

Purpose of the Study:

• To review collective dynamics of complex systems through the lens of order-parameter dynamics.
• To present methods for constructing order-parameter dynamics in both model-based and data-based scenarios.
• To introduce the eigen-microstate approach (EMP) for analyzing complex systems with challenging modeling.

Main Methods:

• Synergetic theory and the slaving principle to define order parameters from slow modes.
• Analytical reduction procedures (e.g., Ott-Antonsen, Lorentz ansatz) for model-based systems.
• The eigen-microstate approach (EMP) to reconstruct order-parameter dynamics from big data via eigenmode decomposition.

Main Results:

• Order-parameter dynamics successfully describe synchronization, chimera states, and neuron network dynamics.
• The EMP effectively captures macroscopic collective behavior, including Bose-Einstein condensation-like transitions and dominant eigenmode emergence.
• EMP applications demonstrated success in phase transitions (Ising model), climate dynamics, stock market fluctuations, and living systems' collective motion.

Conclusions:

• Order-parameter dynamics offer a powerful framework for understanding collective behaviors in complex systems.
• The eigen-microstate approach (EMP) provides a novel data-driven method for analyzing complex systems.
• This approach unifies the study of collective phenomena across diverse scientific domains.