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.
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