Stationarity and energy transfer in out-of-equilibrium systems

Abstract

We define a characteristic energy density based on the measurement of the two first moments of the extrinsic injected power smoothed over time. Using the stationarity, we show that this definition characterizes an energy per degrees freedom of the intrinsic dissipation. Our framework can be applied to systems in contact with thermostats put out of equilibrium by an external driving but it holds also for intrinsically dissipative macroscopic systems that go at rest when the forcing is stopped. Moreover, we are not concerned about the fluctuations around zero of the smoothed injected power that can be extremely rare and difficult to catch experimentally. Then we show that the characteristic energy density we defined, reduces to the kinetic energy of a Brownian-like particle described by a set of Langevin equations with a viscous damping term. The particle can be either in contact with a thermostat or intrinsically dissipative and driven by a random force. In the first case, we recover the result obtained in the framework of the fluctuation relation but extended to a correlated thermal noise. Our characteristic energy density is measured in an experimental system of nonlinear waves generated by a large shaker in a thin elastic plate. A smaller shaker attached to the moving plate is used as a probe to measure the energy exchanged with the plate excited by the large shaker. For both, the proportionality of our characteristic energy density with the kinetic energy is demonstrated. It is a consequence of the viscous damping driving the dissipation in this system. Another system with nonlinear frictional dissipation is investigated numerically model. In this case, our definition of energy density deduced from fluctuations of injected power still characterizes the dissipation but is no more proportional to the kinetic energy because the dissipative process is not a viscous damping.