Aging in the trap model as a relaxation further away from equilibrium

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 Δ S (in absolute value) between the non-equilibrium state and the equilibrium state with the same energy. We consider the time evolution of Δ S and show that, rather unexpectedly, Δ 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, Δ 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 Δ S(t) for different N are found to rescale as ln t/ln t*, instead of the more familiar scaling t/t*.