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, omega, xi, and pi. We show that for suitable values of the parameters of this mapping, the motion of the variable omega is indistinguishable from that of a stochastic variable described by a Fokker-Planck equation with well-defined friction gamma and diffusion D. This result can be explained as follows. The bidimensional system of the two variables xi and pi is a nonlinear, deterministic, and chaotic system, with the key property of resulting in a finite correlation time for the variable xi and in a linear response of xi 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 omega, via a very weak coupling guaranteeing a large-time-scale separation between the two systems, the variable omega 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 omega. 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 xi. The dissipation affecting the variable omega 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.

STANDARD FLUCTUATION-DISSIPATION PROCESS FROM A DETERMINISTIC MAPPING

MANNELLA, RICCARDO;
1993

Abstract

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, omega, xi, and pi. We show that for suitable values of the parameters of this mapping, the motion of the variable omega is indistinguishable from that of a stochastic variable described by a Fokker-Planck equation with well-defined friction gamma and diffusion D. This result can be explained as follows. The bidimensional system of the two variables xi and pi is a nonlinear, deterministic, and chaotic system, with the key property of resulting in a finite correlation time for the variable xi and in a linear response of xi 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 omega, via a very weak coupling guaranteeing a large-time-scale separation between the two systems, the variable omega 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 omega. 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 xi. The dissipation affecting the variable omega 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.
Bianucci, M; Mannella, Riccardo; Fan, Xm; Grigolini, P; West, Bj
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11568/25410
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