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 U _k (t) and its memory function λ _k (t) for a normal mode u _k(t) of the system with a wavenumber k. Let τ _k ^(u) and τ _k ^(γ) be the decay times of U _k(t) and λ _k(t), respectively, with τ _k ^(u) ≥ τ _k ^(γ). Here, τ _k ^(u) is a macroscopic time scale if k << 1, but a microscopic time scale if k ≳ 1, whereas τ _k ^(γ) is always a microscopic time scale. Changing the length scale k ^-1 and the time scales τ _k ^(u), τ _k ^(γ), we can obtain various aspects of the systems as follows. If ττ _k ^(γ) >> τ _k ^(γ), then the time correlation function U _k (t) exhibits the decay of macroscopic fluctuations, leading to an exponential decay U _k(t) ∞ exp(-t/τ _k ^(u)). At the singular point where τ _k ^(u) = τ _k ^(γ), however, both U _k(t) and λk(t) exhibit anomalous microscopic fluctuations, leading to the power-law decay U _k(t) ∞ t ^-3/2 cos[(2t/τ _k ^(u)) - (3π/4)] for t → ∞. The above decay forms give us important information on the macroscopic and microscopic fluctuations in the systems and their dissipations.