Decay Forms of the Time Correlation Functions for Turbulence and Chaos

Taking the Rubin model for the one-dimensional Brownian motion and the chaotic Kuramoto-Sivashinsky equation for the one-dimensional turbulence, we derive a generalized Langevin equation in terms of the projection operator formalism, and then investigate the decay forms of the time correlation function Uk(t) and its memory function Γk(t )f or a normal mode uk(t) of the system with a wavenumber k .L etτ (u) k and τ (γ) k be the decay times of Uk(t )a ndΓk(t), respectively, with τ (u) k ≥ τ (γ) k . Here, τ (u) k is a macroscopic time scale if k � 1, but a microscopic time scale if k & 1, whereas τ (γ) k is always a microscopic time scale. Changing the length scale k −1 and the time scales τ (u) k , τ (γ) k , we can obtain various aspects of