On effective temperature in network models of collective behavior

Area of Science:

• Statistical Physics
• Complex Systems
• Non-equilibrium Physics

Background:

• Collective behavior in self-propelled units is a fundamental phenomenon in nature.
• The Vicsek model is a widely studied theoretical framework for such systems.
• Understanding phase transitions (order-disorder) in these systems remains a challenge.

Purpose of the Study:

• To develop a dynamical systems framework for analyzing the Vectorial Network Model (VNM) with additive noise.
• To identify a key parameter controlling macroscopic phase transitions in collective behavior.
• To establish a connection between noise properties and collective system dynamics, defining an effective temperature.

Main Methods:

• Analytical study of the Vectorial Network Model (VNM) using a mean-field approximation.
• Application of a dynamical systems framework to analyze stochastic dynamics under general additive noise.
• Derivation of a fluctuation-dissipation relation to define an effective temperature.

Main Results:

• A single parameter, linearly dependent on the circular mean of noise, dictates the system's ordered or disordered macroscopic phase.
• This parameter is identified as an effective temperature of collective behavior.
• The exact critical temperature is analytically derived for systems with low connectivity.

Conclusions:

• The concept of effective temperature provides a unified framework for understanding order-disorder phase transitions in collective behavior.
• This finding facilitates the development of coarse-graining techniques for complex systems.
• It offers insights into the physical mechanisms underlying the emergence of collective phenomena.