The out of equilibrium response function in sub-diffusive systems

We study the Einstein relation between spontaneous fluctuations and the response to an external perturbation for the comb model and the single file, which are examples of systems with sub-diffusive transport properties. The relevance of nonequilibrium conditions is investigated: when a stationary current (in the form of a drift or an energy flux) is present, the Einstein relation breaks down. In the case of the comb model, a general relation-appearing in the recent literature-between the response function and an unperturbed suitable correlation function allows us to explain the obtained results. This suggests that the relevant ingredient in breaking the Einstein formula, for stationary regimes, is not anomalous diffusion but the presence of currents driving the system out of equilibrium.