Dynamical Renormalization Group for Mode-Coupling Field Theories with Solenoidal Constraint.

Area of Science:

• Biophysics
• Theoretical Physics
• Complex Systems

Background:

• Empirical data on collective behavior in biological systems (e.g., bird flocks, insect swarms) necessitates advanced theoretical tools.
• Mode-coupling field theories are required to model the dynamics of these systems, where social forces influence particle velocity via spin.
• Existing theories struggle with the complexity of coupled fields (density, velocity, spin) and enforcing constraints like incompressibility (solenoidal constraint).

Purpose of the Study:

• To perform a dynamic renormalization group analysis of a mode-coupling field theory with a solenoidal constraint.
• To investigate the impact of the solenoidal constraint on the theoretical framework for collective biological behavior.
• To address the challenge of incorporating incompressibility into equilibrium mode-coupling theories.

Main Methods:

• Equilibrium dynamic renormalization group analysis.
• Mode-coupling field theory.
• Application of a solenoidal constraint to the velocity field.
• Classification using the Halperin- and Hohenberg scheme (solenoidal Model G).

Main Results:

• The solenoidal constraint introduces a novel vertex mixing static and dynamical coupling constants.
• This new vertex is essential for renormalization group closure and ensuring consistency between dynamics and statics.
• The constraint modifies the static universality class but, intriguingly, does not alter the dynamical universality class.

Conclusions:

• The developed framework provides a consistent theoretical approach for systems with solenoidal constraints.
• The findings suggest an exception to the typical relationship between static and dynamical universality classes.
• This work is a significant step towards developing off-equilibrium mode-coupling theories for biological collectives.