Probing dissipative processes in stationary out-of-equilibrium systems through the fluctuations of injected power

We consider out-of-equilibrium systems in a stationary state. They can be systems in contact with a thermostat pushed out-of-equilibrium or athermal intrinsically dissipative systems. For both types of systems we define an energy, E-c, that characterizes the intrinsic dissipative processes although it is estimated from the fluctuations of the extrinsic injected power. Our definition involves only the two first moments of the injected power smoothed over a large time tau. Hence it is easy to measure it experimentally. Our reasoning lies only on properties of stationary processes. Thus it applies to any kind of dissipative systems. We test our definition on various systems. We show that for Langevin dynamics, we recover some results that are usually given by the fluctuation relation. We also probe this energy experimentally on bending waves in a thin elastic plate. In all these cases, we establish the fact that it coincides with the kinetic energy per degrees of freedom. It is however not always the case as shown on numerical simulations on frictional systems.