### 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 1 2012 |

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### All Science Journal Classification (ASJC) codes

- Physics and Astronomy (miscellaneous)

### Cite this

*Progress of Theoretical Physics*,

*127*(4), 615-629. https://doi.org/10.1143/PTP.127.615

**Decay forms of the time correlation functions for turbulence and chaos.** / Mori, Hazime; Okamura, Makoto.

Research output: Contribution to journal › Article

*Progress of Theoretical Physics*, vol. 127, no. 4, pp. 615-629. https://doi.org/10.1143/PTP.127.615

}

TY - JOUR

T1 - Decay forms of the time correlation functions for turbulence and chaos

AU - Mori, Hazime

AU - Okamura, Makoto

PY - 2012/4/1

Y1 - 2012/4/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84860123908&partnerID=8YFLogxK

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U2 - 10.1143/PTP.127.615

DO - 10.1143/PTP.127.615

M3 - Article

AN - SCOPUS:84860123908

VL - 127

SP - 615

EP - 629

JO - Progress of Theoretical Physics

JF - Progress of Theoretical Physics

SN - 0033-068X

IS - 4

ER -