  We address, on the example of a simple solvable model, the issue of whether
the stationary state of dissipative systems converges to an equilibrium state
in the low dissipation limit. We study a driven dissipative Zero Range Process
on a tree, in which particles are interpreted as finite amounts of energy
exchanged between degrees of freedom. The tree structure mimicks the hierarchy
of length scales; energy is injected at the top of the tree ('large scales'),
transferred through the tree and dissipated mostly in the deepest branches of
the tree ('small scales'). Varying a parameter characterizing the transfer
dynamics, a transition is observed, in the low dissipation limit, between a
quasi-equilibrated regime and a far-from-equilibrium one, where the dissipated
flux does not vanish.
