Standard fluctuation-dissipation process from a deterministic mapping

We illustrate a derivation of a standard fluctuation-dissipation process from a discrete deterministic dynamical model. This model is a three-dimensional mapping, driving the motion of three variables, w, {ξ}, and {π}. We show that for suitable values of the parameters of this mapping, the motion of the variable w is indistinguishable from that of a stochastic variable described by a Fokker-Planck equation with well-defined friction {γ} and diffusion D. This result can be explained as follows. The bidimensional system of the two variables {ξ} and {π} is a nonlinear, deterministic, and chaotic system, with the key property of resulting in a finite correlation time for the variable {ξ} and in a linear response of {ξ} to an external perturbation. Both properties are traced back to the fully chaotic nature of this system. When this subsystem is coupled to the variable w, via a very weak coupling guaranteeing a large-time-scale separation between the two systems, the variable w is proven to be driven by a standard fluctuation-dissipation process. We call the subsystem a booster whose chaotic nature triggers the standard fluctuation-dissipation process exhibited by the variable w. The diffusion process is a trivial consequence of the central-limit theorem, whose validity is assured by the finite time scale of the correlation function of {ξ}. The dissipation affecting the variable w is traced back to the linear response of the booster, which is evaluated adopting a geometrical procedure based on the properties of chaos rather than the conventional perturbation approach.