### Abstract

Using the projection operator formalism we explore the decay form of the time correlation function Un (t) un (t) un (0) of the state variable u n (t) in the chaotic Kuramoto-Sivashinsky equation. The decay form turns out to be the algebraic decay 1 [1+ (na t) 2] in the initial regime t ne and the exponential decay exp (a ne t) in the final regime t>1â• I ne. The memory function Î"n (t) that represents the chaos-induced transport is found to obey the Gaussian decay exp [a (ng t) 2] in the case of large wave numbers, but the 3/2 power decay exp [a (I n3 t) 3a 2] in the case of small wave numbers. The power spectrum of u n (t) is given by the real part Una (Ï‰) of the Fourier-Laplace transform of Un (t) and has a dominant peak at Ï‰=0. This peak within the linewidth ne (ane) is given by the Lorentzian spectrum ne 2 a (Ï‰2 + ne 2). However, the wings of the peak outside the width ne turn out to take the exponential spectrum exp (na). Thus it is found that the exponential decay exp (ne t) appears to lead to the universal Lorentzian peak, while the algebraic decay 1 [1+ (na t) 2] arises to bring about the exponential wing.

Original language | English |
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Article number | 061104 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 76 |

Issue number | 6 |

DOIs | |

Publication status | Published - Dec 5 2007 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Statistics and Probability
- Condensed Matter Physics

### Cite this

**Dynamic structures of the time correlation functions of chaotic nonequilibrium fluctuations.** / Mori, Hazime; Okamura, Makoto.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Dynamic structures of the time correlation functions of chaotic nonequilibrium fluctuations

AU - Mori, Hazime

AU - Okamura, Makoto

PY - 2007/12/5

Y1 - 2007/12/5

N2 - Using the projection operator formalism we explore the decay form of the time correlation function Un (t) un (t) un (0) of the state variable u n (t) in the chaotic Kuramoto-Sivashinsky equation. The decay form turns out to be the algebraic decay 1 [1+ (na t) 2] in the initial regime t ne and the exponential decay exp (a ne t) in the final regime t>1â• I ne. The memory function Î"n (t) that represents the chaos-induced transport is found to obey the Gaussian decay exp [a (ng t) 2] in the case of large wave numbers, but the 3/2 power decay exp [a (I n3 t) 3a 2] in the case of small wave numbers. The power spectrum of u n (t) is given by the real part Una (Ï‰) of the Fourier-Laplace transform of Un (t) and has a dominant peak at Ï‰=0. This peak within the linewidth ne (ane) is given by the Lorentzian spectrum ne 2 a (Ï‰2 + ne 2). However, the wings of the peak outside the width ne turn out to take the exponential spectrum exp (na). Thus it is found that the exponential decay exp (ne t) appears to lead to the universal Lorentzian peak, while the algebraic decay 1 [1+ (na t) 2] arises to bring about the exponential wing.

AB - Using the projection operator formalism we explore the decay form of the time correlation function Un (t) un (t) un (0) of the state variable u n (t) in the chaotic Kuramoto-Sivashinsky equation. The decay form turns out to be the algebraic decay 1 [1+ (na t) 2] in the initial regime t ne and the exponential decay exp (a ne t) in the final regime t>1â• I ne. The memory function Î"n (t) that represents the chaos-induced transport is found to obey the Gaussian decay exp [a (ng t) 2] in the case of large wave numbers, but the 3/2 power decay exp [a (I n3 t) 3a 2] in the case of small wave numbers. The power spectrum of u n (t) is given by the real part Una (Ï‰) of the Fourier-Laplace transform of Un (t) and has a dominant peak at Ï‰=0. This peak within the linewidth ne (ane) is given by the Lorentzian spectrum ne 2 a (Ï‰2 + ne 2). However, the wings of the peak outside the width ne turn out to take the exponential spectrum exp (na). Thus it is found that the exponential decay exp (ne t) appears to lead to the universal Lorentzian peak, while the algebraic decay 1 [1+ (na t) 2] arises to bring about the exponential wing.

UR - http://www.scopus.com/inward/record.url?scp=36849014293&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=36849014293&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.76.061104

DO - 10.1103/PhysRevE.76.061104

M3 - Article

AN - SCOPUS:36849014293

VL - 76

JO - Physical Review E

JF - Physical Review E

SN - 2470-0045

IS - 6

M1 - 061104

ER -