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 Gamma(k)(t) for a normal mode u(k)(t) of the system with a wavenumber k. Let tau((u))(k) and tau((gamma))(k) be the decay times of U-k(t) and Gamma(k)(t), respectively, with tau((u))(k) >= tau((gamma))(k). Here, tau((u))(k) is a macroscopic time scale if k << 1, but a microscopic time scale if k greater than or similar to 1, whereas tau((gamma))(k) is always a microscopic time scale. Changing the length scale k(-1) and the time scales tau((u))(k), tau((gamma))(k), we can obtain various aspects of the systems as follows. If tau((u))(k) >> tau((gamma))(k), then the time correlation function U-k(t) exhibits the decay of macroscopic fluctuations, leading to an exponential decay U-k(t) proportional to exp(-t/tau((u))(k)). At the singular point where tau((u))(k) = tau((gamma))(k), however, both U-k(t) and Gamma(k)(t) exhibit anomalous microscopic fluctuations, leading to the power-law decay U-k(t) proportional to t(-3/2) cos[(2t/tau((u))(k)) - (3 pi/4)] for t -> infinity. The above decay forms give us important information on the macroscopic and microscopic fluctuations in the systems and their dissipations.