  We analyse the process of energy exchanges generated by the elastic
collisions between a point-particle, confined to a two-dimensional cell with
convex boundaries, and a `piston', i.e. a line-segment, which moves back and
forth along a one-dimensional interval partially intersecting the cell. This
model can be considered as the elementary building block of a spatially
extended high-dimensional billiard modeling heat transport in a class of hybrid
materials exhibiting the kinetics of gases and spatial structure of solids.
Using heuristic arguments and numerical analysis, we argue that, in a regime of
rare interactions, the billiard process converges to a Markov jump process for
the energy exchanges and obtain the expression of its generator.
