Renormalization Group Method Applied to Kinetic Equations: roles of initial values and time

TL;DR

This paper applies the renormalization group method to derive kinetic and transport equations from microscopic models, emphasizing the importance of initial conditions and showing how irreversibility emerges from time-reversible equations.

Contribution

It clarifies the role of initial values in the RG derivation of kinetic equations and unifies the reduction processes for various equations.

Findings

RG method naturally determines initial distribution functions

Averaging leads to time-irreversible equations from reversible ones

Unified derivation of Boltzmann, Fokker-Planck, and fluid equations

Abstract

The so-called renormalization group (RG) method is applied to derive kinetic and transport equations from the respective microscopic equations. The derived equations include Boltzmann equation in classical mechanics, Fokker-Planck equation, a rate equation in a quantum field theoretical model. Utilizing the formulation of the RG method which elucidates the important role played by the choice of the initial conditions, the general structure and the underlying assumptions in the derivation of kinetic equations in the RG method is clarified. It is shown that the present formulation naturally leads to the choice for the initial value of the microscopic distribution function at arbitrary time $t_0$ to be on the averaged distribution function to be determined. The averaged distribution function may be thought as an integral constant of the solution of microscopic evolution equation; the RG equation gives the slow dynamics of the would-be initial constant, which is actually the kinetic equation governing the averaged distribution function. It is further shown that the averaging as given above gives rise to a coarse-graining of the time-derivative which is expressed with the initial time $t_0$, thereby leads to time-irreversible equations even from a time-reversible equation. It is shown that a further reduction of Boltzmann equation to fluid dynamical equations and the adiabatic elimination of fast variables in Fokker-Planck equation are also performed in a unified way in the present method.