  The existence of a generalized fluctuation-dissipation theorem observed in
simulations and experiments performed in various glassy materials is related to
the concepts of local equilibration and heterogeneity in space. Assuming the
existence of a dynamic coherence length scale up to which the system is locally
equilibrated, we extend previous generalizations of the FDT relating static to
dynamic quantities to the physically relevant domain where asymptotic limits of
large times and sizes are not reached. The formulation relies on a simple
scaling argument and has thus not the character of a theorem. Extensive
numerical simulations support this proposition. Our results quite generally
apply to systems with slow dynamics, independently of the space dimensionality,
the chosen dynamics, or the presence of disorder.
