  The Fluctuation Relation (FR) is an asymptotic result on the distribution of
certain observables averaged over time intervals T as T goes to 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 T. 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 verification 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.
