Construction of coarse-grained order parameters in nonequilibrium systems

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 the control and operator theoretic model reduction. The resulting "generalized" RG is a consistent generalization of the Wilsonian RG. We apply the procedure to a deterministic nonlinear wave equation (NLWE) with probabilistic initial conditions. We derive the form of the projection operator from the dynamics of the NLWE and then use it to generate the RG flow for the distribution of initial conditions. The probability density of the initial conditions is chosen to be a Boltzmann weight that is quartic in the field variables. In our calculation, we find that in contrast to conventional implementations of the RG, naive power counting breaks down. We also show that the resulting RG equations are different from those derived from the conventional RG.