  The observed tightness of the mass discrepancy-acceleration relation (MDAR)
poses a fine-tuning challenge to current models of galaxy formation. We propose
that this relation could arise from collisional interactions between baryons
and dark matter (DM) particles, without the need for modification of gravity or
ad hoc feedback processes. We assume that these interactions satisfy the
following three conditions: (i) the relaxation time of DM particles is
comparable to the dynamical time in disk galaxies; (ii) DM exchanges energy
with baryons due to elastic collisions; (iii) the product between the baryon-DM
cross section and the typical energy exchanged in a collision is inversely
proportional to the DM number density. We present an example of a particle
physics model that gives a DM-baryon cross section with the desired density and
velocity dependence. Direct detection constraints require our DM particles to
be either very light ($m << m_b$) or very heavy ($m >> m_b$), corresponding
respectively to heating and cooling of DM by baryons. In both cases, our
mechanism applies and an equilibrium configuration can in principle be reached.
Here, we focus on the heavy DM/cooling case as it is technically simpler. Under
these assumptions, we find that rotationally-supported disk galaxies could
naturally settle to equilibrium configurations satisfying a MDAR at all radii
without invoking finely tuned feedback processes. We also discuss issues
related to the small scale clumpiness of baryons, as well as predictions for
pressure-supported systems. We argue in particular that galaxy clusters do not
follow the MDAR despite being DM-dominated because they have not reached their
equilibrium configuration. Finally, we revisit existing phenomenological,
astrophysical and cosmological constraints on baryon-DM interactions in light
of the unusual density dependence of the cross section.
