In this chapter, we will discuss three procedures for describing chaotic diffusionChaotic diffusion in a family of discrete maps. The first involves a phenomenological description derived from scaling hypotheses, leading to a generalized homogeneous function that results in a scaling law relating three critical exponents. The second transforms the difference equation of the discrete map into an ordinary differential equation (ODE). The integration of this ODE provides a temporal description that agrees excellently with numerical results in the short-time regime. For long times, the location of the first invariant spanning curve is determined, allowing for the calculation of the critical exponents. Finally, the third approach considers the analytical solution of the diffusion equation, which gives the probability of observing a particle at a given position in phase space at a given time. From the knowledge of this probability, one can compute the average observables that lead to the appropriate critical exponents.

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Chaotic Diffusion in Non-dissipative Maps

  • Edson Denis Leonel

摘要

In this chapter, we will discuss three procedures for describing chaotic diffusionChaotic diffusion in a family of discrete maps. The first involves a phenomenological description derived from scaling hypotheses, leading to a generalized homogeneous function that results in a scaling law relating three critical exponents. The second transforms the difference equation of the discrete map into an ordinary differential equation (ODE). The integration of this ODE provides a temporal description that agrees excellently with numerical results in the short-time regime. For long times, the location of the first invariant spanning curve is determined, allowing for the calculation of the critical exponents. Finally, the third approach considers the analytical solution of the diffusion equation, which gives the probability of observing a particle at a given position in phase space at a given time. From the knowledge of this probability, one can compute the average observables that lead to the appropriate critical exponents.