Decay Forms of the Time Correlation Functions for Turbulence and Chaos(General and Mathematical Physics)

概要

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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 ⪎ 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 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^<-3/2> 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.
理論物理学刊行会の論文
2012-04-25