We discuss, in the context of energy flow in high-dimensional systems the behavior of a chain of rotators (rotors) which is purely Hamiltonian, apart from dissipation at just one end. We derive bounds on the dissipation rate which become arbitrarily small in certain physical regimes, and we present numerical evidence that these bounds are sharp. We also demonstrate in a special model that the very slow dissipation is due to the slow drift along a family of approximate breather solutions.