Kubo Fluctuation Relations in the Generalized Elastic Model

Figure 1

(Color online) Time periodic force . (a) 3D rendering of a membrane described by (1), under the effect of an applied time periodic force in (black arrow). represents the height of the fluctuating membrane on a 2-dimensional substrate (). Regions I and II correspond, respectively, to the inner and outer region in which the membrane separates when the force is applied. The color code, red for region I and green for region II, has been drawn for the reader's convenience: increasing the frequency entails the shrinkage of the red region (I). (b) Schematic representation of the untagged response amplitude as a function of the distance : since it is a schematic drawing no scale is needed on the -axis. In region I the universal behaviour holds for any kind of hydrodynamic interactions. In region II the decay of the response's amplitude is for long range hydrodynamic systems (grey (upper) solid line), for local hydrodynamic systems with with (blue (middle) solid line), and exponentially fast for local hydrodynamic systems with (orange (bottom) line). (c) Schematic representation of the untagged response phase as a function of the distance . No scale is needed on the -axis. Region I: the system displays a universal phase delay for long range and local hydrodynamic interactions; that is, . Region II: for long range hydrodynamic systems the phase is absent; that is, (grey (middle) solid line); for local hydrodynamics the phase is approximately if with (upper blue line), while it is approximately if or (bottom blue line); if the hydrodynamic interactions are local and the phase shows a linear dependence on the distance ; that is, (orange (bright linear) solid line).

(a)

(b)

(c)