Application of the Renormalization-group Method to the Reduction of Transport Equations

TL;DR

This paper reviews the renormalization group method for analyzing differential equations, emphasizing invariant manifolds, and demonstrates its application in deriving the Navier-Stokes equation from the Boltzmann equation, including transport coefficients.

Contribution

It provides a comprehensive review of the RG method and applies it to reduce the Boltzmann equation to the Navier-Stokes equation, highlighting the derivation of transport coefficients.

Findings

RG method clarifies the reduction of complex dynamics to slow motions.

Derived transport coefficients from the Boltzmann equation.

Demonstrated the RG approach's effectiveness in fluid dynamics.

Abstract

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