  Recent numerical simulations of a disordered system (Preprint
arXiv:condmat/0307554) have shown the existence of two different relaxational
processes (called stimulated and spontaneous) characterizing the relaxation
observed in structural glasses. The existence of these two processes has been
claimed to be at the roots of the intermittency phenomenon observed in recent
experiments. Here we consider a generic system put in contact with a bath at
temperature T and characterized by an adiabatic slow relaxation (i.e. by a
negligible net heat flow from the system to the bath) in the aging state. We
focus on a simplified scenario (termed as partial equilibration) characterized
by the fact that $=0 (where only the spontaneous process is observable) and
whose microscopic stochastic dynamics is ergodic when constrained to the
constant energy surface. Three different effective temperatures can be defined:
a) from the fluctuation-dissipation theorem (FDT), b) from a fluctuation
theorem describing the statistical distribution of heat exchange events between
system and bath and c) from a set of observable-dependent microcanonical
relations in the aging state. In a partial equilibration scenario we show how
all three temperatures coincide reinforcing the idea that a statistical (rather
than thermometric) definition of a non-equilibrium temperature is physically
meaningful in aging systems. These results are explicitly checked in a simple
model system
