STANDARD FLUCTUATION-DISSIPATION PROCESS FROM A DETERMINISTIC MAPPING

We illustrate a derivation of a standard fluctuation-dissipation proce ss from a discrete deterministic dynamical model. This model is a thre e-dimensional mapping, driving the motion of three variables, omega, x i, and pi. We show that for suitable values of the parameters of this mapping, the motion of the variable omega is indistinguishable from th at of a stochastic variable described by a Fokker-Planck equation with well-defined friction gamma and diffusion D. This result can be expla ined as follows. The bidimensional system of the two variables xi and pi is a nonlinear, deterministic, and chaotic system, with the key pro perty of resulting in a finite correlation time for the variable xi an d in a linear response of xi to an external perturbation. Both propert ies are traced back to the fully chaotic nature of this system. When t his subsystem is coupled to the variable omega, via a very weak coupli ng guaranteeing a large-time-scale separation between the two systems, the variable omega is proven to be driven by a standard fluctuation-d issipation process. We call the subsystem a booster whose chaotic natu re triggers the standard fluctuation-dissipation process exhibited by the variable omega. The diffusion process is a trivial consequence of the central-limit theorem, whose validity is assured by the finite tim e scale of the correlation function of xi. The dissipation affecting t he variable omega is traced back to the linear response of the booster , which is evaluated adopting a geometrical procedure based on the pro perties of chaos rather than the conventional perturbation approach.