A kinetic reaction model: decay to equilibrium and macroscopic limit

TL;DR

This paper introduces a kinetic relaxation model for a generation-recombination reaction, demonstrating exponential decay to equilibrium and deriving a nonlinear diffusion limit through hypocoercivity and entropy methods.

Contribution

It presents a new kinetic model for reactions, proves decay to equilibrium using hypocoercivity, and rigorously derives the macroscopic limit as a nonlinear diffusion equation.

Findings

Exponential decay of small perturbations established.

Rigorous derivation of macroscopic nonlinear diffusion limit.

Application of hypocoercivity methods to reaction models.

Abstract

We propose a kinetic relaxation-model to describe a generation-recombination reaction of two species. The decay to equilibrium is studied by two recent methods for proving hypocoercivity of the linearized equations. Exponential decay of small perturbations can be shown for the full nonlinear problem. The macroscopic/fast-reaction limit is derived rigorously employing entropy decay, resulting in a nonlinear diffusion equation for the difference of the position densities.