Construction of coarse-grained order-parameters in non-equilibrium systems

TL;DR

This paper introduces a generalized renormalization group method for non-equilibrium systems, incorporating system-specific features through a control-theoretic coarse graining approach, leading to new RG equations.

Contribution

It develops a systematic coarse graining procedure based on control and operator theory, extending the Wilsonian RG to non-equilibrium contexts with novel RG equations.

Findings

Derived a new form of the projection operator from nonlinear wave dynamics.

Renormalized initial condition distributions using Boltzmann weights.

Found breakdown of naive power counting in the generalized RG framework.

Abstract

We develop a renormalization group (RG) procedure that includes important system-specific features. The key ingredient is to systematize the coarse graining procedure that generates the RG flow. The coarse graining technology comes from control and operator theoretic model reduction. The resulting "generalized" RG is a consistent generalization of the Wilsonian RG. We derive the form of the projection operator from the dynamics of a nonlinear wave equation and renormalize the distribution of initial conditions. The probability density of the initial conditions is chosen to be the Boltzmann weight for a standard $\phi^4$-theory. In our calculation, we find that in contrast to conventional implementations of the RG, na\"ive power counting breaks down. The RG-equations that we derive are different from those derived from the conventional RG.