%0 Book Section %1 statphys23_0055 %A Okamura, M. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K chaotic coherent division eddy fluctuating method motion operator projection statphys23 topic-3 viscosity %T Validity of the essential assumption in a projection operator method %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=55 %X The projection operator method developed by Mori involves the essential assumption that chaotic motion is successfully divided into a coherent motion and a fluctuating motion. We investigate the validity of the assumption using the Kuramoto-Sivashinsky equation equation ut+uux +\partial^2 ux^2+\partial^4 ux^4=0 equation under the $L$-periodic boundary condition $(L=500)$. Using a projection operator $P$, we obtain the generalized Langevin equation equation d\hatu_n(t)dt = -ınt_0^t \Gamma_n(s)u_n(t-s)ds+r_n(t) equation from the Fourier transform of Eq.~(1), where \ \Gamma_n(t) =r_n(t) r_n^\ast\rangle u_nu_n^\ast\rangle,\qquad r_n(t) \equiv e^(1-P)Łambda t(1-P)N_n. \ Here $e^Łambda t$ denotes a time evolution operator, and $N_n$ is the Fourier transform of the nonlinear term in Eq.~(1). Equation~(2) seems to indicate that the chaotic motion \ f_n(t)du_n(t)/dt \ is divided into the coherent motion \ s_n(t)\equiv-ınt_0^t \Gamma_n(s)u_n(t-s)ds \ and the fluctuating motion $r_n(t)$. The most important question is whether $s_n(t)$ is coherent and $r_n(t)$ is fluctuating, because this is only an assumption. We now investigate the validity of the assumption in the case of $n=5$ as a representative of the long wave modes using two methods: a comparison of the time correlation functions \ F_n(t)f_n(t)f_n^\ast\rangle,\qquad R_n(t)r_n(t)r_n^\ast\rangle \ and another of time profiles $s_n(t)$ and $r_n(t)$. First, we compare the two time correlation functions $F_5(t)$ and $R_5(t)$. The numerical results indicate that the correlation time of $F_5(t)$ is larger than 40. This means that $f_5(t)$ includes a very slowly varying motion, as expected. However, the correlation time of $R_5(t)$ is about 15. This is much shorter than the correlation time of $F_5(t)$. As a result of this difference in the correlation time, the slowly varying motion is extracted by the projection operator. Therefore, we have found that the chaotic motion $f_5(t)$ is successfully divided into the coherent motion $s_5(t)$ and the fluctuating motion $r_5(t)$ using the projection operator for long wave modes. Second, we compare two types of motion: the motion $s_5(t)$ (broken curve) that is expected to be coherent; and the motion $r_5(t)$ (solid curve) that is expected to be fluctuating, as shown in Fig.~1. This comparison is more intuitive than the former comparison. We can strongly confirm from the figure that $s_5(t)$ is coherent and $r_5(t)$ is fluctuating; hence, the essential assumption is valid in the case of long wave modes. The assumption is also valid in the case of short wave modes, although their results are not shown. We have evaluated the eddy viscosity as $9.0$ by extracting the nonlinear term from the coherent part. This value is consistent with four former results with other methods: $7.3<\nu_T<9.9$ by Zaleski~(1989), $\nu_T10.5$ by Sneppen et al.~(1992), $\nu_T10$ by Sakaguchi~(2000), and $\nu_T8$ by Ueno et al.~(2005). We have also found that the eddy viscosity is expressed by \ \nu_T=łim_k_n01k_n^2ınt_0^\Gamma_n(s)ds \ in the projection operator formalism.

@incollection{statphys23_0055, abstract = {The projection operator method developed by Mori involves the essential assumption that chaotic motion is successfully divided into a coherent motion and a fluctuating motion. We investigate the validity of the assumption using the Kuramoto-Sivashinsky equation \begin{equation} \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x} +\frac{\partial^2 u}{\partial x^2}+\frac{\partial^4 u}{\partial x^4}=0 \end{equation} under the $L$-periodic boundary condition $(L=500)$. Using a projection operator $\mathcal{P}$, we obtain the generalized Langevin equation \begin{equation} \frac{d\hat{u}_{n}(t)}{dt} = -\int_0^t \Gamma_{n}(s)\hat{u}_{n}(t-s)ds+r_{n}(t) \end{equation} from the Fourier transform of Eq.~(1), where \[ \Gamma_{n}(t) =\frac{\langle r_n(t) r_n^\ast\rangle} {\langle \hat{u}_{n}\hat{u}_{n}^\ast\rangle},\qquad r_{n}(t) \equiv e^{(1-\mathcal{P})\Lambda t}(1-\mathcal{P})N_{n}. \] Here $e^{\Lambda t}$ denotes a time evolution operator, and $N_n$ is the Fourier transform of the nonlinear term in Eq.~(1). Equation~(2) seems to indicate that the chaotic motion \[ f_n(t)\equiv d\hat{u}_{n}(t)/dt \] is divided into the coherent motion \[ s_n(t)\equiv-\int_0^t \Gamma_{n}(s)\hat{u}_{n}(t-s)ds \] and the fluctuating motion $r_n(t)$. The most important question is whether $s_n(t)$ is coherent and $r_n(t)$ is fluctuating, because this is only an assumption. We now investigate the validity of the assumption in the case of $n=5$ as a representative of the long wave modes using two methods: a comparison of the time correlation functions \[ F_n(t)\equiv \langle f_{n}(t)f_{n}^\ast\rangle,\qquad R_n(t)\equiv \langle r_{n}(t)r_{n}^\ast\rangle \] and another of time profiles $s_n(t)$ and $r_n(t)$. First, we compare the two time correlation functions $F_5(t)$ and $R_5(t)$. The numerical results indicate that the correlation time of $F_5(t)$ is larger than 40. This means that $f_5(t)$ includes a very slowly varying motion, as expected. However, the correlation time of $R_5(t)$ is about 15. This is much shorter than the correlation time of $F_5(t)$. As a result of this difference in the correlation time, the slowly varying motion is extracted by the projection operator. Therefore, we have found that the chaotic motion $f_5(t)$ is successfully divided into the coherent motion $s_5(t)$ and the fluctuating motion $r_5(t)$ using the projection operator for long wave modes. Second, we compare two types of motion: the motion $s_5(t)$ (broken curve) that is expected to be coherent; and the motion $r_5(t)$ (solid curve) that is expected to be fluctuating, as shown in Fig.~1. This comparison is more intuitive than the former comparison. We can strongly confirm from the figure that $s_5(t)$ is coherent and $r_5(t)$ is fluctuating; hence, the essential assumption is valid in the case of long wave modes. The assumption is also valid in the case of short wave modes, although their results are not shown. We have evaluated the eddy viscosity as $9.0$ by extracting the nonlinear term from the coherent part. This value is consistent with four former results with other methods: $7.3<\nu_T<9.9$ by Zaleski~(1989), $\nu_T\approx 10.5$ by Sneppen et al.~(1992), $\nu_T\approx 10$ by Sakaguchi~(2000), and $\nu_T\approx 8$ by Ueno et al.~(2005). We have also found that the eddy viscosity is expressed by \[ \nu_T=\lim_{k_n\rightarrow 0}\frac{1}{k_n^2}\int_0^\infty \Gamma_{n}(s)ds \] in the projection operator formalism.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Okamura, M.}, biburl = {https://www.bibsonomy.org/bibtex/2f4445b271723de7dbdc5a6e5e3e01a56/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {ac0d1bd26daa31bdaea853999d80060d}, intrahash = {f4445b271723de7dbdc5a6e5e3e01a56}, keywords = {chaotic coherent division eddy fluctuating method motion operator projection statphys23 topic-3 viscosity}, month = {9-13 July}, timestamp = {2007-06-20T10:16:10.000+0200}, title = {Validity of the essential assumption in a projection operator method}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=55}, year = 2007 }