  We present a method for bounding, and in some cases computing, the spectral
gap for systems of many particles evolving under the influence of a random
collision mechanism. In particular, the method yields the exact spectral gap in
a model due to Mark Kac of energy conserving collisions with one dimensional
velocities. It is also sufficiently robust to provide qualitatively sharp
bounds also in the case of more physically realistic momentum and energy
conserving collisions in three dimensions, as well as a range of related
models.
