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

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.

The detailed derivations are added to section 5 (fluiddynamical limit of Boltzmann equation) and to Appendix B (Adiabatic elimination of fast variables in Fokker-Planck equation) which is moved to the text as a section. Other minor corrections are made all over the text including typos

Cited by in corpus (16)

• Constructive Methods of Invariant Manifolds for Kinetic Problems
• Sigma and hydrodynamic modes along the critical line
• Dynamical renormalization group approach to relaxation in quantum field theory
• Functional renormalisation group approach to far-from-equilibrium quantum field dynamics
• Far-from-equilibrium quantum many-body dynamics
• Stable First-order Particle-frame Relativistic Hydrodynamics for Dissipative Systems
• Second-order Relativistic Hydrodynamic Equations for Viscous Systems; how does the dissipation affect the internal energy?
• Uniqueness of Landau-Lifshitz Energy Frame in Relativistic Dissipative Hydrodynamics
• Derivation of relativistic hydrodynamic equations consistent with relativistic Boltzmann equation by renormalization-group method
• Relativistic Causal Hydrodynamics Derived from Boltzmann Equation: a novel reduction theoretical approach
• Application of the Renormalization-group Method to the Reduction of Transport Equations
• Maxwell-Juttner distribution for rigidly-rotating flows in spherically symmetric spacetimes using the tetrad formalism
• Derivation of second-order relativistic hydrodynamics for reactive multi-component systems
• Exact renormalization group approach to a nonlinear diffusion equation
• Overdamping Phenomena near the Critical Point in O($N$) Model
• Dynamic nature at the QCD Critical End Point