Fluctuation-response relation in systems with chaotic behaviour

The statistics of systems with good chaotic properties obey a formal fluctuation-response relation which gives the average linear response of a dynamical system to an external perturbation in terms of two-time correlation functions. Unfortunately, except for particularly simple cases, the appropriate form of correlation function is unknown because an analytic expression for the invariant density is lacking. The simplest situation is that in which the probability distribution is Gaussian. In that case, the fluctuation-response relation is a linear relation between the response matrix and the two-time two-point correlation matrix. Some numerical computations have been carried out in low-dimensional models of hydrodynamic systems. The results show that fluctuation-response relation for Gaussian distributions is not a useful approximation. Nevertheless, these calculations show that, even for non-Gaussian statistics, the response function and the two-time correlations can have similar qualitative features, which may be attributed to the existence of the more general fluctuation-response relation.