Application of the renormalization-group method to the reduction of transport equations

Abstract

We first give a comprehensive review of the renormalization group method for\nglobal and asymptotic analysis, putting an emphasis on the relevance to the\nclassical theory of envelopes and on the importance of the existence of\ninvariant manifolds of the dynamics under consideration. We clarify that an\nessential point of the method is to convert the problem from solving\ndifferential equations to obtaining suitable initial (or boundary)\nconditions:The RG equation determines the slow motion of the would-be integral\nconstants in the unperturbative solution on the invariant manifold. The RG\nmethod is applied to derive the Navier-Stokes equation from the Boltzmann\nequation, as an example of the reduction of dynamics. We work out to obtain the\ntransport coefficients in terms of the one-body distribution function.\n