A Toy-Model Study of the Grazing Collisions in the Kinetic Theory

Abstract

As a toy model study for the grazing collision effect in the Boltzmann equation, a spatially homogeneous problem for the two-dimensional Lorentz gas from the initial data with a jump discontinuity is investigated. The collision kernel of our model has the same angular singularity as the Boltzmann collision kernel. Thanks to the simplicity of the model, the jump propagation and its regularization are clearly seen, depending on whether the collision cross-section is finite or infinite. Three types of behavior are observed: a jump propagation, a jump propagation accompanied with a divergent derivative, and no jump propagation. For the model with a critical parameter, the switching from the second type to the last is observed.