Beyond the linear Fluctuation-Dissipation Theorem: the Role of Causality
In this paper we re-examine the traditional problem of connecting the internal fluctuations of a system to its response to external forcings and extend the classical theory in order to be able to encompass also nonlinear processes. With this goal, we try to join on the results by Kubo on statistical mechanical systems close to equilibrium, i.e. whose unperturbed state can be described by a canonical ensemble, the theory of dispersion relations, and the response theory recently developed by Ruelle for non-equilibrium systems equipped with an invariant SRB measure. Our derivations highlight the strong link between causality and the possibility of connecting unambiguously fluctuation and response, both at linear and nonlinear level. We first show in a rather general setting how the formalism of the Ruelle response theory can be used to derive in a novel way Kramers-Kronig relations connecting the real and imaginary part of the linear and nonlinear response to external perturbations. We then provide a formal extension at each order of nonlinearity of the fluctuation-dissipation theorem (FDT) for general systems possessing a smooth invariant measure. Finally, we focus on the physically relevant case of systems close to equilibrium, for which we present explicit fluctuation-dissipation relations linking the susceptibility describing the $n^th$ order response of the system with the expectation value of suitably defined correlations of $n+1$ observables taken in the equilibrium ensemble. While the FDT has an especially compact structure in the linear case, in the nonlinear case joining the statistical properties of the fluctuations of the system to its response to external perturbations requires linear changes of variables, simple algebraic sums and multiplications, and a multiple convolution integral. These operations, albeit cumbersome, can be easily implemented numerically.