Dynamic structures of the time correlation functions of chaotic nonequilibrium fluctuations.

Using the projection operator formalism we explore the decay form of the time correlation function U-n(t)equivalent to <(u) over cap (n)(t)u(n)*(0)> of the state variable (u) over cap (n)(t) in the chaotic Kuramoto-Sivashinsky equation. The decay form turns out to be the algebraic decay 1/[1+(gamma(na)t)(2)] in the initial regime t < 1/gamma(ne) and the exponential decay exp(-gamma(ne)t) in the final regime t>1/gamma(ne). The memory function Gamma(n)(t) that represents the chaos-induced transport is found to obey the Gaussian decay exp[-(beta(ng)t)(2)] in the case of large wave numbers, but the 3/2 power decay exp[-(beta(n3)t)(3/2)] in the case of small wave numbers. The power spectrum of (u) over cap (n)(t) is given by the real part U-n(')(omega) of the Fourier-Laplace transform of U-n(t) and has a dominant peak at omega=0. This peak within the linewidth (gamma) over bar (ne)(approximate to gamma(ne)) is given by the Lorentzian spectrum (gamma) over bar (2)(ne)/(omega(2)+(gamma) over bar (2)(ne)). However, the wings of the peak outside the width (gamma) over bar (ne) turn out to take the exponential spectrum exp(-omega/gamma(na)). Thus it is found that the exponential decay exp(-gamma(ne)t) appears to lead to the universal Lorentzian peak, while the algebraic decay 1/[1+(gamma(na)t)(2)] arises to bring about the exponential wing.