Construction of coarse-grained order parameters in nonequilibrium systems

Area of Science:

• Theoretical Physics
• Nonlinear Dynamics
• Statistical Mechanics

Background:

• The Wilsonian renormalization-group (RG) is a powerful tool for understanding physical systems across different scales.
• Conventional RG methods often struggle with systems exhibiting complex dynamics or specific initial conditions.
• Model reduction techniques from control and operator theory offer potential for refining RG procedures.

Purpose of the Study:

• To develop a generalized renormalization-group (RG) procedure that systematically incorporates system-specific features.
• To apply this new RG method to a deterministic nonlinear wave equation (NLWE) with probabilistic initial conditions.
• To investigate the breakdown of conventional RG assumptions in this context.

Main Methods:

• Systematized coarse-graining procedure derived from control and operator theoretic model reduction.
• Application to a nonlinear wave equation (NLWE) with probabilistic initial conditions.
• Derivation of a projection operator from NLWE dynamics to generate RG flow for initial condition distributions.

Main Results:

• The developed "generalized" RG is a consistent extension of the Wilsonian RG.
• Naïve power counting breaks down in this specific application, contrary to conventional RG expectations.
• The derived RG equations differ from those obtained through conventional RG methods.

Conclusions:

• The generalized RG procedure provides a more consistent framework for analyzing complex systems.
• System-specific coarse-graining is crucial for accurate RG flow generation in nonlinear dynamics.
• This approach offers new insights into the behavior of nonlinear wave equations with probabilistic initial conditions.