A general model for collision-based abrasion processes

By relying on existing results about touching probabilities of convex objects, we derive a partial differential equation for general abrasion processes based on collision sequences. Our model shows that collisions with generic abrading objects can be modelled as a linear combination of collisions with three special abrading objects. One of these corresponds to bouncing on a plane, and the Gauss curvature flow, one to abrasion by sand blasting and the uniform normal flow, while the third one to abrasion by random impacts of sticks, and the mean curvature flow. Our model lends itself to a natural discretization scheme where the three special collisions correspond to three random events. We discuss some applications to natural processes.