Kinetic Theory of Simple Reacting Spheres: An Application to Coloring Processes

Abstract

We consider a simplified version of the kinetic model of simple reacting spheres (SRS) for a quaternary reactive mixture of hard-spheres in the dilute-gas limit. The model mimics a coloring process occurring with probability $\alpha _{R}$, described by the reversible chemical law $A_1+A_2\rightleftharpoons A_3+A_4$. We provide the linearized collisional operators of our model and investigate some of their mathematical properties. In particular we obtain an explicit and symmetric representation of the elastic and reactive kernels and use this to prove the compactness of the linearized collisional operator in $(L^2(\mathbb {R}^3))^4$.