Summary: 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 \(\Gamma _{k}(t)\) for a normal mode \(u_{k}(t)\) of the system with a wavenumber \(k\). Let \(\tau_k^{(<u>)}\) and\(\tau_k^{(\gamma )}\) be the decay times of \(<U>_k(<t>)\) and \(\Gamma_k(<t>)\), respectively, with \(\tau_{k}^{(u)} \geq \tau_k^{(\gamma )}\). Here, \(\tau_k^{(<u>)}\) is a macroscopic time scale if \(<k> \ll 1\), but a microscopic time scale if \(<k> \gtrsim1\), whereas \(\tau_k^{(\gamma )}\) is always a microscopic time scale.
Changing the length scale \(<k>^{-1}\) and the time scales \(\tau_{k}^{(u)}\), \(\tau_k^{(\gamma )}\), we can obtain various aspects of the systems as follows. If \(\tau_k^{(<u>)} \gg \tau_k^{(\gamma )}\), then the time correlation function \(<U>_k(<t>)\) exhibits the decay of macroscopic fluctuations, leading to an exponential decay \(<U>_k(<t>) \propto \exp (-<t>/\tau_k^{(u)})\).
At the singular point where \(\tau_k^{(<u>)} = \tau_k^{(\gamma )}\), however, both \(<U>_k(<t>)\) and \(\Gamma_k(<t>)\) exhibit anomalous microscopic fluctuations, leading to the power-law decay \(<U>_k(<t>) \propto <t>^{-3/2}\cos [(2<t>/\tau_k^{(u)})-(3\pi /4)] \) for \(t \rightarrow \infty \).
The above decay forms give us important information on the macroscopic and microscopic fluctuations in the systems and their dissipations.