  We analyze the convergence to equilibrium in a family of Kac-like kinetic
equations in multiple space dimensions. These equations describe the change of
the velocity distribution in a spatially homogeneous gas due to binary
collisions between the particles. We consider a general linear mechanism for
the exchange of the particles' momenta, with interaction coefficients that are
random matrices with a distribution that is {independent} of the velocities of
the colliding particles. Applying a synthesis of probabilistic methods and
Fourier analysis, we are able to identify sufficient conditions for the
existence and uniqueness of a stationary state, we characterize this stationary
state as a mixture of Gaussian distributions, and we prove equilibration of
transient solutions under minimal hypotheses on the initial conditions. In
particular, we are able to classify the high-energy tails of the stationary
distribution, which might be of Pareto type. We also discuss several examples
to which our theory applies, among them models with a non-symmetric stationary
state.
