  We introduce a global thermostat on Kac's 1D model for the velocities of
particles in a space-homogeneous gas subjected to binary collisions, also
interacting with a (local) Maxwellian thermostat. The global thermostat
rescales the velocities of all the particles, thus restoring the total energy
of the system, which leads to an additional drift term in the corresponding
nonlinear kinetic equation. We prove ergodicity for this equation, and show
that its equilibrium distribution has a density that, depending on the
parameters of the model, can exhibit heavy tails, and whose behaviour at the
origin can range from being analytic, to being $C^k$, and even to blowing-up.
Finally, we prove propagation of chaos for the associated $N$-particle system,
with a uniform-in-time rate of order $N^{-\eta}$ in the squared $2$-Wasserstein
metric, for an explicit $\eta \in (0, 1/3]$.
