  We consider a chain of anharmonic oscillators immersed in a heat bath with a
temperature gradient and a time varying tension applied to one end of the chain
while the other side is fixed to a point. We prove that under diffusive
space-time rescaling the volume strain distribution of the chain evolves
following a non-linear diffusive equation. The stationary states of the
dynamics are of non-equilibrium and have a positive entropy production, so the
classical relative entropy methods cannot be used. We develop new estimates
based on entropic hypocoercivity, that allows to control the distribution of
the positions configurations of the chain. The macroscopic limit can be used to
model isothermal thermodynamic transformations between non-equilibrium
stationary states.
