Renormalization group theory for global asymptotic analysis.

We show with several examples that renormalization group (RG) theory can be used to understand singular and reductive perturbation methods in a unified fashion. Amplitude equations describing slow motion dynamics in nonequilibrium phenomena are RG equations. The renormalized perturbation approach may be simpler to use than other approaches, because it does not require the use of asymptotic matching, and yields practically superior approximations. Pacs numbers: 47.20.Ky, 02.30.Mv, 64.60.Ak The essence of the renormalization group (RG) is to extract those structurally stable features of a system which are insensitive to details. 1,2 Thus, RG methods may be regarded as a means of asymptotic analysis. The usefulness of this point of view has been amply demonstrated 3 by the relation between the RG and intermediate asymptotics, 4 which showed that the anomalous exponents appearing in (e.g.) the long-time behavior of certain hydrodynamic systems were calculable using RG. Many different techniques for asymptotic analysis have been developed including the multiple scaling (MS) method (which actually subsumes all the others), the boundary layer (BL) method, the WKB approximation, and others. 5 Reductive perturbation methods 6 have been used to extract the dynamics describing the global space-time behavior of complicated systems near bifurcation points. 7 At a purely technical level, the starting point for both perturbative RG methods and conventional asymptotic methods is the removal of divergences from a perturbation series. Given the above similarities, a natural question arises: what is the relation, if any, between conventional asymptotic methods and the RG? In this Letter, we demonstrate that many singular perturbation methods may be understood as renormalized perturbation theory, and that amplitude equations obtainable by the reductive perturbation methods are renormalization group equations. 8 One of the advantages of the RG approach is that the starting point is simply a straightforward naive perturbation expansion, for which very little a priori knowledge is required; we will see that the RG approach seems to be more efficient and accurate, in practice, than standard methods in extracting global information from the perturbation expansion.