Unusual response to a localized perturbation in a generalized elastic model

The generalized elastic model encompasses several physical systems such as polymers, membranes, single-file systems, fluctuating surfaces, and rough interfaces. We consider the case of an applied localized potential, namely, an external force acting only on a single (tagged) probe, leaving the rest of the system unaffected. We derive the fractional Langevin equation for the tagged probe, as well as for a generic (untagged) probe, where the force is not directly applied. Within the framework of the fluctuation-dissipation relations, we discuss the unexpected physical scenarios arising when the force is constant and time periodic, whether or not the hydrodynamic interactions are included in the model. For short times, in the case of the constant force, we show that the average drift is linear in time for long-range hydrodynamic interactions and behaves ballistically or exponentially for local hydrodynamic interactions. Moreover, it can be opposite to the direction of the external disturbance for some values of the model's parameters. When the force is time periodic, the effects are macroscopic: the system splits into two distinct spatial regions whose size is proportional to the value of the applied frequency. These two regions are characterized by different amplitudes and phase shifts in the response dynamics.