  Micro-reversibility, that is, the time reversal symmetry exhibited by
microscopic dynamics, plays a central role in thermodynamics and statistical
mechanics. It is used to prove fundamental results such as Onsager reciprocal
relations or fluctuation theorems. From micro-reversibility one can also prove
that isolated systems and systems in contact with a thermal bath relax to
micro-canonical and canonical ensembles, respectively. However, a number of
problems arise when trying to reproduce this proof for classical and quantum
collisional baths, consisting of particles from an equilibrium reservoir
interacting with a localized system via collisions. In particular, it is not
completely clear which distribution for the velocities of the incident
particles warrants thermalization. Here, we clarify these issues by showing
that Liouville's theorem is a necessary condition for micro-reversibility in
classical and semi-classical scenarios. As a consequence, one must take into
account all the canonical coordinates and momenta, including the position of
the incident particles. Taking into account the position modifies the effective
probability distribution of the velocity of the particles that interact with
the system, which is no longer Maxwellian. We finally show an example of
seemingly plausible collision rules that nonetheless violate the Liouville
theorem and allow one to design machines that beat the second law of
thermodynamics.
