Gaspard and Gilbert (2008) suggested a two-step strategy to derive the "macroscopic" heat equation from the "microscopic" kinetic equation. Their model consisted of a chain of localized and rarely interacting hard disks or balls. We consider a mathematically more tractable model by replacing every other hard ball with a piston. In order to complete the first, truly dynamical part of the GG-strategy, our aim is to prove that the evolution of the energies is governed by a Markov jump process in the "mesoscopic" limit. Based on joint work with P. Bálint, T. Gilbert, I.P. Toth and D. Szász.