  We give a new and elementary computation of the spectral gap of the Kac walk
on the N-sphere. The result is obtained as a by-product of a more general
observation which allows to reduce the analysis of the spectral gap of an
N-component system to that of the same system for N=3. The method applies to a
number of random 'binary collision' processes with complete-graph structure,
including non-homogeneous examples such as exclusion and colored exclusion
processes with site disorder.
