Fluctuation Relation beyond Linear Response Theory

The Fluctuation Relation (FR) is an asymptotic result onthe distribution of certain observables averaged over timeintervals τ as τ → ∞ and it is a generalization of thefluctuation–dissipation theorem to far from equilibrium systemsin 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 isassumed to be true in all dissipative ‘chaotic enough’ systemsin a steady state. In this paper, we develop a theory of finitetime corrections to FR, needed to compare the asymptoticprediction of FR with numerical observations, which necessarilyinvolve fluctuations of observables averaged over finite timeintervals τ. We perform a numerical test of FR in two cases inwhich non-Gaussian fluctuations are observable, while GK does notapply and we get a non-trivial verification of FR that is independent of and different from linear response theory.Our results are compatible with the theory of finite timecorrections to FR, while FR would be observably violated,well within the precision of our experiments, if such correctionswere neglected.