# Hypocoercivity and Fast Reaction Limit for Linear Reaction Networks with Kinetic Transport

## TL;DR

This paper studies the long-term behavior of a kinetic transport model for chemical reaction networks, proving convergence to equilibrium using hypocoercivity methods, with results depending on spatial domain and particle mass separation.

## Contribution

It introduces hypocoercivity techniques to analyze reaction networks with kinetic transport, including cases with separated particle masses and different spatial domains.

## Key findings

- Exponential convergence in flat torus
- Algebraic decay in whole space
- Macroscopic behavior governed by diffusion equation

## Abstract

The long time behavior of a model for a first order, weakly reversible chemical reaction network is considered, where the movement of the reacting species is described by kinetic transport. The reactions are triggered by collisions with a nonmoving background with constant temperature, determining the post-reactional equilibrium velocity distributions. Species with different particle masses are considered, with a strong separation between two groups of light and heavy particles. As an approximation, the heavy species are modeled as nonmoving. Under the assumption of at least one moving species, long time convergence is proven by hypocoercivity methods for the cases of positions in a flat torus and in whole space. In the former case the result is exponential convergence to a spatially constant equilibrium, and in the latter it is algebraic decay to zero, at the same rate as solutions of parabolic equations. This is no surprise since it is also shown that the macroscopic (or reaction dominated) behavior is governed by the diffusion equation.