Abstract: In 1949 Fermi proposed a mechanism for generation of the background cosmic radiation. One of the natural mathematical models for this phenomena is a particle which elastically bounces from the boundary inside a domain. If the boundary is not stationary the particle may loose or gain energy depending on the direction of the boundary motion at the point of collision. In this talk we will discuss recent result on the long term behaviour of the particle, including the possibility of unbounded energy growth.