Waiting time dependence of aging

Many non-equilibrium systems show a delayed response to a sudden perturbation and the response is represented by a relaxation function ψ(t). I first define a temporal decay constant of the relaxation by λ(t)=–∂lnψ(t)/∂t . Writing it as λ(t_w)=–Cψ(t_w)/t_w +δ(t_w) for the observation at t_w , I classify the aging into two types: Type I when δ(t_w)=0 and type II when δ(t_w)≠0. Exploiting the free-energy-landscape (FEL) theory [1], I show that (1) the relaxation of the FEL manifests itself as type II aging, (2) the temporal relaxation time can be either increasing or decreasing function of t_w and (3) the relaxation time of the FEL can be deduced from t_w dependence of λ(t_w) .
As an example, I investigate aging phenomena of dielectric relaxation by a two-level model and show that relaxation functions of the polarization and of the linear susceptibility depend on t_w when the FEL has a delayed response. I also study the intermediate scattering function for a random walk model and show that the delayed response of the FEL gives rise to type II aging.
[1] T. Odagaki, J. Phys. Soc. Jpn. 86, 082001(2017).