抄録
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.
| 元の言語 | 英語 |
|---|---|
| ページ（範囲） | 615-629 |
| ページ数 | 15 |
| ジャーナル | Progress of Theoretical Physics |
| 巻 | 127 |
| 発行部数 | 4 |
| DOI | |
| 出版物ステータス | 出版済み - 4 1 2012 |
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All Science Journal Classification (ASJC) codes
- Physics and Astronomy (miscellaneous)
これを引用
Decay forms of the time correlation functions for turbulence and chaos. / Mori, Hazime; Okamura, Makoto.
：: Progress of Theoretical Physics, 巻 127, 番号 4, 01.04.2012, p. 615-629.研究成果: ジャーナルへの寄稿 › 記事
}
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
UR - http://www.scopus.com/inward/citedby.url?scp=84860123908&partnerID=8YFLogxK
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 -