Chaotic Diffusion in Non-Dissipative Mappings

Abstract

We discuss in this Chapter three different procedures to investigate the chaotic diffusion for a family of discrete mappings. The first of them involves a phenomenological investigation obtained from scaling hypotheses leading to a scaling law relating three critical exponents among them. The second one transforms the equation of differences into an ordinary differential equation which integration for short time leads to a good description of the time evolution obtained analytically and the numerical findings. For long enough time the stationary state is obtained via the localization of the lowest action invariant spanning curve allowing the determination of the critical exponents. Finally the third one considers the analytical solution of the diffusion equation, furnishing then the probability to observe a particle with a certain action at a given instant of time. From the knowledge of the probability all the momenta of the distribution are obtained including the three critical exponents describing the scaling properties of the dynamics.