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) identical with < û_(n)(t)û*_(n)(0)> of the state variable û_(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 û_(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_(ne) (approximately equal to gamma_(ne)) is given by the Lorentzian spectrum (-)gamma2_(ne)/(omega2+(-)gamma2_(ne)) . However, the wings of the peak outside the width (-) gamma_(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.