  The aging regime of the trap model, observed for a temperature T below the
glass transition temperature T_g, is a prototypical example of non-stationary
out-of-equilibrium state. We characterize this state by evaluating its
"distance to equilibrium", defined as the Shannon entropy difference \Delta S
(in absolute value) between the non-equilibrium state and the equilibrium state
with the same energy. We consider the time evolution of \Delta S and show that,
rather unexpectedly, \Delta S(t) continuously increases in the aging regime, if
the number of traps is infinite, meaning that the "distance to equilibrium"
increases instead of decreasing in the relaxation process. For a finite number
N of traps, \Delta S(t) exhibits a maximum value before eventually converging
to zero when equilibrium is reached. The time t* at which the maximum is
reached however scales in a non-standard way as t* ~ N^(T_g/2T), while the
equilibration time scales as N^(T_g/T). In addition, the curves \Delta S(t) for
different N are found to rescale as ln t/ln t*, instead of the more familiar
scaling t/t*.
