  We consider a chain of three rotors (rotators) whose ends are coupled to
stochastic heat baths. The temperatures of the two baths can be different, and
we allow some constant torque to be applied at each end of the chain. Under
some non-degeneracy condition on the interaction potentials, we show that the
process admits a unique invariant probability measure, and that it is ergodic
with a stretched exponential rate. The interesting issue is to estimate the
rate at which the energy of the middle rotor decreases. As it is not directly
connected to the heat baths, its energy can only be dissipated through the two
outer rotors. But when the middle rotor spins very rapidly, it fails to
interact effectively with its neighbors due to the rapid oscillations of the
forces. By averaging techniques, we obtain an effective dynamics for the middle
rotor, which then enables us to find a Lyapunov function. This and an
irreducibility argument give the desired result. We finally illustrate
numerically some properties of the non-equilibrium steady state.
