%0 Book Section %1 statphys23_0584 %A Kobayashi, M.U. %A Fujisaka, H.F. %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 chaos coefficients correlation diffusion dynamical functions orbits periodic statphys23 topic-5 unstable %T Time correlation functions and diffusion coefficients in chaotic sytems in terms of unstable periodic orbits %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=584 %X 0.6cm Time correlation functions and diffusion coefficients are the most fundamental quantities for studying fluctuation statistics in chaotic systems. Although deriving time correlation functions and diffusion coefficients from fundamental equations of motion describing chaotic dynamics is an important and fundamental problem in statistical mechanics, it is generally quite difficult to do this, because of the nonlinearity of chaotic systems and the singular structure of the invariant measure. These can be obtained, in principle, by solving the eigenvalue problems of the time evolution operator. The eigenvalue problem is, however, generally meaningless, because the eigenfunctions of the time evolution operators in chaotic systems are singular almost everywhere in the state space, due to the fractal structure of the manifolds. In order to avoid this problem, one of the present authors proposed a new method of determining the time correlation functions that does not require solving the eigenvalue problem. This new method, called the Markov method, extends Mori's original projection operator approach in a more tractable way and can be used to associate the dynamical correlation functions with static quantities. Meanwhile static quantities are determined in terms of unstable periodic orbits. Thus the dynamical correlation functions can be obtained in terms of unstable periodic orbits. 0.6cm Furthermore diffusion coefficients can be obtained by integrating time correlation functions. So we can determine diffusion coefficients in terms of unstable periodic orbits by using the above method. 0.6cm In this presentation, applying this method to various chaotic dynamical systems, we prove the usefulness of the present approach.

@incollection{statphys23_0584, abstract = {\hskip 0.6cm Time correlation functions and diffusion coefficients are the most fundamental quantities for studying fluctuation statistics in chaotic systems. Although deriving time correlation functions and diffusion coefficients from fundamental equations of motion describing chaotic dynamics is an important and fundamental problem in statistical mechanics, it is generally quite difficult to do this, because of the nonlinearity of chaotic systems and the singular structure of the invariant measure. These can be obtained, in principle, by solving the eigenvalue problems of the time evolution operator. The eigenvalue problem is, however, generally meaningless, because the eigenfunctions of the time evolution operators in chaotic systems are singular almost everywhere in the state space, due to the fractal structure of the manifolds. In order to avoid this problem, one of the present authors proposed a new method of determining the time correlation functions that does not require solving the eigenvalue problem. This new method, called the {\it Markov method}, extends Mori's original projection operator approach in a more tractable way and can be used to associate the {\it dynamical} correlation functions with {\it static} quantities. Meanwhile {\it static} quantities are determined in terms of unstable periodic orbits. Thus the {\it dynamical} correlation functions can be obtained in terms of unstable periodic orbits. \hskip 0.6cm Furthermore diffusion coefficients can be obtained by integrating time correlation functions. So we can determine diffusion coefficients in terms of unstable periodic orbits by using the above method. \hskip 0.6cm In this presentation, applying this method to various chaotic dynamical systems, we prove the usefulness of the present approach.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Kobayashi, M.U. and Fujisaka, H.F.}, biburl = {https://www.bibsonomy.org/bibtex/206da3eabf1faabf272dc900e102880d8/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {8ddc349408dd68340623bcea30e776e8}, intrahash = {06da3eabf1faabf272dc900e102880d8}, keywords = {chaos coefficients correlation diffusion dynamical functions orbits periodic statphys23 topic-5 unstable}, month = {9-13 July}, timestamp = {2007-06-20T10:16:24.000+0200}, title = {Time correlation functions and diffusion coefficients in chaotic sytems in terms of unstable periodic orbits}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=584}, year = 2007 }