  We study a scalar harmonic network with pair interactions and a binary
collision rule, exchanging the momenta of a randomly-chosen couple of sites. We
consider the case of the isolated network where the total energy is conserved.
In the first part, we recast the dynamics as a stochastic map in normal modes
(or action-angle) coordinates and provide a geometric interpretation of it. We
formulate the problem for generic networks but, for completeness, also
reconsider the translation-invariant lattices. In the second part, we examine
the kinetic limit and its range of validity. A general form of the linear
collision operator in terms of eigenstates of the network is given. This
defines an \textit{action network}, whose connectivity gives information on the
out-of-equilibrium dynamics. We present a few examples (ordered and disordered
chains and elastic networks) where the topology of connections in action spaces
can be determined in a neat way. As an application, we consider the classic
problem of relaxation to equipartition from the point of view of the dynamics
of linear actions. We compare the results based on the spectrum of the
collision operator with numerical simulation, performed with a novel scheme
based on direct solution of the equation of motion in normal modes coordinates.

Statistical Mechanics;Chaotic Dynamics