Fluctuation-response relations for multitime correlations

We show that time-correlation functions of arbitrary order for any random variable in a statistical dynamical system can be calculated as higher-order response functions of the mean history of the variable. The response is to a "control term" added as a modification to the master equation for statistical distributions. The proof of the relations is based upon a variational characterization of the generating functional of the time correlations. The same fluctuation-response relations are preserved within moment closures for the statistical dynamical system, when these are constructed via the variational Rayleigh-Ritz procedure. For the two-time correlations of the moment variables themselves, the fluctuation-response relation is equivalent to an "Onsager regression hypothesis" for the small fluctuations. For correlations of higher order, there is a further effect in addition to such linear propagation of fluctuations present instantaneously: the dynamical generation of correlations by nonlinear interaction of fluctuations. In general, we discuss some physical and mathematical aspects of the Ansatze required for an accurate calculation of the time correlations. We also comment briefly upon the computational use of these relations, which is well suited for automatic differentiation tools. An example will be given of a simple closure for turbulent energy decay, which illustrates the numerical application of the relations.