  In this paper we investigate the possibility of a dynamical definition of an
effective temperature for compacting granular media in the framework of the
Fluctuation-Dissipation (FD) relations. We have studied two paradigmatic models
for the compaction of granular media, which consider particles diffusing on a
lattice, with either geometrical (Tetris model) or dynamical (Kob-Andersen
model) constraints. Idealized compaction without gravity has been implemented
for the Tetris model, and compaction with a preferential direction imposed by
gravity has been studied for both models. In the ideal case of an homogeneous
compaction, the obtained FD ratio is clearly shown to be in agreement with the
prediction of Edwards' measure at various densities. Similar results are
obtained with gravity only when the homogeneity of the bulk is imposed. In this
case the FD ratio obtained dynamically for horizontal displacements and
mobility and from Edwards' measure coincide. Finally, we propose experimental
tests for the validity of the Edwards' construction through the comparison of
various types of dynamical measurements.
