Randomization and Memory Functions of Chaos and Turbulence

The chaotic orbits of dynamical systems become random and stochastic on long timescales due to the orbital instability of chaos. This randomization of chaotic orbits gives rise to the dissipation of the macroscopic kinetic energy supplied by an external force into the random kinetic energy of chaos or turbulence and leads to various transport processes. The randomization of chaotic orbits is formulated in terms of a memory function which describes the loss of memory of the initial states due to the orbital instability. Then the nonlinear term of the evolution equation that causes chaos or turbulence is transformed into the sum of a fluctuating force and a memory-function term. Thus the deterministic evolution equation is found to become a Markovian stochastic equation on long timescales. Then, considering the chaos-induced friction of a forced pendulum and the molecular and turbulent viscosity of incompressible fluids, we explore the memory functions and the fluctuation-dissipation formula for transport coefficients in order to establish a statistical-mechanical approach to chaos and turbulence. There are two kinds of time-correlation functions defined by the long time-average over a chaotic orbit and the long time-average over its intersection points on a Poincare section. The two are shown to have different statistical properties.