Fluctuation relation beyond linear response theory

The Fluctuation Relation (FR) is an asymptotic result on the distribution of certain observables averaged over time intervals tau as t --> infinity and it is a generalization of the fluctuation - dissipation theorem to far from equilibrium systems in a steady state, which reduces to the usual Green - Kubo (GK) relation in the limit of small external non-conservative forces. FR is a theorem for smooth uniformly hyperbolic systems, and it is assumed to be true in all dissipative 'chaotic enough' systems in a steady state. In this paper, we develop a theory of finite time corrections to FR, needed to compare the asymptotic prediction of FR with numerical observations, which necessarily involve fluctuations of observables averaged over finite time intervals tau. We perform a numerical test of FR in two cases in which non-Gaussian fluctuations are observable, while GK does not apply and we get a non-trivial veri. cation of FR that is independent of and different from linear response theory. Our results are compatible with the theory of finite time corrections to FR, while FR would be observably violated, well within the precision of our experiments, if such corrections were neglected.