Self-organization in a diversity induced thermodynamics

Area of Science:

• Physics
• Complex Systems
• Statistical Mechanics

Background:

• Classical thermodynamics describes macroscopic systems, but understanding self-organization in disordered microscopic systems remains a challenge.
• The Kuramoto model is a standard framework for studying synchronization in coupled oscillators, but often lacks spatial dynamics and particle properties.

Purpose of the Study:

• To investigate the emergence of global self-organized patterns from disordered point oscillators via deterministic cooperation.
• To explore the thermodynamic characteristics and pattern evolution within a modified Kuramoto model incorporating spatial degrees of freedom and particle polarity.
• To analyze the role of disorder (diversity) as a temperature-like parameter influencing pattern formation and system dynamics.

Main Methods:

• Introduction of a modified Kuramoto model with Euclidean degrees of freedom and particle polarity.
• Utilizing the standard deviation of frequency distribution as a disorder parameter (diversity), analogous to temperature.
• Analyzing system dynamics across a range of diversity values, from zero to high levels, to observe pattern formation and transitions.

Main Results:

• At low diversity, static phase-synchronized patterns (crystals) emerge, behaving as a dissipative many-body system.
• Increasing diversity leads to crystal vibrations, disintegration into smaller, internally synchronized dynamic patterns, and a wide variety of self-organized shapes.
• High diversity results in erratic dynamics, short-lived patterns, and eventual disappearance, indicating a phase transition and critical behavior at a specific diversity value.

Conclusions:

• Global self-organized patterns can arise from local interactions in disordered systems through deterministic cooperative processes.
• The modified Kuramoto model successfully captures thermodynamic-like characteristics and demonstrates pattern evolution across different scales.
• System behavior, including pattern formation and stability, is robust and not critically dependent on specific interaction functions or frequency distributions.