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Prog. Theor. Phys. Vol. 127 No. 4 (2012) pp. 615-629

Decay Forms of the Time Correlation Functions for Turbulence and Chaos

Hazime Mori and Makoto Okamura^*

Research Institute for Applied Mechanics, Kyushu University, Kasuga 816-8580, Japan

(Received January 26, 2012; Revised March 5, 2012)

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 Γ_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 Γ_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 \gtrsim1, 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 τ_k^(u) ≫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 Γ_k(t) exhibit anomalous microscopic fluctuations, leading to the power-law decay U_k(t) ∝t^-3/2cos [(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.

Subject Index : 051, 056
URL : http://ptp.ipap.jp/link?PTP/127/615/
DOI : 10.1143/PTP.127.615

^*E-mail: okamura@riam.kyushu-u.ac.jp

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