Fluctuation-dissipation relations under Lévy noises

For systems close to equilibrium, the relaxation properties of measurable physical quantities are described by the linear response theory and the fluctuation-dissipation theorem (FDT). Accordingly, the response or the generalized susceptibility, which is a function of the unperturbed equilibrium system, can be related to the correlation between spontaneous fluctuations of a given conjugate variable. There have been several attempts to extend the FDT far from equilibrium, introducing new terms or using effective temperatures. Here, we discuss applicability of the generalized FDT to out-of-equilibrium systems perturbed by time-dependent deterministic forces and acting under the influence of white Lévy noise. For the linear and Gaussian case, the equilibrium correlation function provides a full description of the dynamic properties of the system. This is, however, no longer true for non-Gaussian Lévy noises, for which the second and sometimes also the first moments are divergent, indicating absence of underlying physical scales. This self-similar behavior of Lévy noises results in violation of the classical dissipation theorem for the stability index α<2. We show that by properly identifying appropriate variables conjugated to external perturbations and analyzing time-dependent distributions, the generalized FDT can be restored also for systems subject to Lévy noises. As a working example, we test the use of the generalized FDT for a linear system subject to Cauchy white noise.