## Abstract

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.

Original language | English |
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Pages (from-to) | 615-629 |

Number of pages | 15 |

Journal | Progress of Theoretical Physics |

Volume | 127 |

Issue number | 4 |

DOIs | |

Publication status | Published - Apr 2012 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Physics and Astronomy (miscellaneous)