  We investigate the dynamics of a small long-range interacting system, in
contact with a large long-range thermal bath. Our analysis reveals the
existence of striking anomalies in the energy flux between the bath and the
system. In particular, we find that the evolution of the system is not
influenced by the kinetic temperature of the bath, as opposed to what happens
for short-range collisional systems. As a consequence, the system may get
hotter also when its initial temperature is larger than the bath temperature.
This observation is explained quantitatively in the framework of the
collisionless Vlasov description of toy models with long-range interactions and
shown to be valid whenever the Vlasov picture applies, from cosmology to plasma
physics.
