  We consider a heavy piston in an infinite cylinder surrounded by ideal gases
on both sides. The piston moves under elastic collisions with gas atoms. We
assume here that the gases always exert equal pressures on the piston, hence
the piston remains at the so called mechanical equilibrium. However, the
temperatures and densities of the gases may differ across the piston. In that
case some earlier studies by Gruber, Piasecki and others reveal a very slow
motion (drift) of the piston in the direction of the hotter gas. At the same
time the hotter gas slowly transfers its energy (heat) across the piston to the
cooler gas. While the previous studies of this interesting phenomenon were only
heuristic or experimental, we provide first rigorous proofs assuming that the
velocity distribution of the ideal gas satisfies a certain ``cutoff''
condition.
