Linear response functions of chaotic systems and equilibrium moments

For systems in thermal equilibrium or systems with a Gaussian equilibrium distribution the response to small perturbations can be computed via a fluctuation-dissipation theorem from the correlations. For general open systems, especially systems with a low-dimensional chaotic attractor, this approach fails. With regard to such systems in the present study two alternative methods to compute response functions from properties of the unperturbed system are discussed. Here response functions are related to equilibrium moments instead of equilibrium correlations. The first method is based on the moment expansion of a generalized Kubo theory. The second is a new approach starting from a shadowing assumption. Both methods are independent of the particular equilibrium statistics. In concrete applications only a finite number of equilibrium moments is known. These allow to compute the exact high-frequency or short-time response, but not the low-frequency or static response. To obtain this low-frequency response from a finite number of moments, approximations are introduced. The methods are exemplified with the Lorenz system.