On the relaxation rate of short chains of rotors interacting with Langevin thermostats

In this short note, we consider a system of two rotors, one of which interacts with a Langevin heat bath. We show that the system relaxes to its invariant measure (steady state) no faster than a stretched exponential $\exp (-c{t}^{1\mathrm{/}2})$. This indicates that the exponent $1\mathrm{/}2$ obtained earlier by the present authors and J.-P. Eckmann for short chains of rotors is optimal.