Detecting phase transitions in collective behavior using manifold's curvature

Area of Science:

• Complex Systems
• Dynamical Systems Theory
• Data Analysis

Background:

• Multi-agent systems exhibit complex behaviors often constrained to invariant manifolds.
• Phase transitions in these systems are characterized by abrupt changes in physical properties like speed and structure.
• Identifying these transitions is crucial for understanding and controlling system dynamics.

Purpose of the Study:

• To propose and validate a new method for detecting phase transitions in multi-agent systems.
• To characterize phase transitions as the splitting of an invariant manifold into distinct sub-manifolds.
• To develop a technique for separating phase transition regions from phase transition-free sub-manifolds.

Main Methods:

• Utilizing the relationship between curvature and singular value ratios of sampled points on a curve.
• Extending the analysis to higher dimensions using the shape operator.
• Approximating phase transitions via locally computed singular value ratios on the manifold.

Main Results:

• The proposed Phase Transition Detection (PTD) method successfully identifies phase transitions.
• The method effectively splits the underlying manifold into distinct phase transition-free sub-manifolds.
• Validation was performed using a particle simulation and three real-world examples.

Conclusions:

• The PTD method provides a robust approach for detecting and analyzing phase transitions in complex systems.
• This technique allows for the isolation of system dynamics unaffected by phase transitions.
• The findings have implications for understanding emergent behaviors and control strategies in multi-agent systems.