  We consider the stationary states of a chain of $n$ anharmonic coupled
oscillators, whose deterministic hamiltonian dynamics is perturbed by random
independent sign change of the velocities (a random mechanism that conserve
energy). The extremities are coupled to thermostats at different temperature
$T_\ell$ and $T_r$ and subject to constant forces $\tau_\ell$ and $\tau_r$. If
the forces differ $\tau_\ell \neq \tau_r$ the center of mass of the system will
move of a speed $V_s$ inducing a tension gradient inside the system. Our aim is
to see the influence of the tension gradient on the thermal conductivity. We
investigate the entropy production properties of the stationary states, and we
prove the existence of the Onsager matrix defined by Green-kubo formulas
(linear response). We also prove some explicit bounds on the thermal
conductivity, depending on the temperature.
