Microscopic origin of the logarithmic time evolution of aging processes in complex systems
Michael A. Lomholt, Ludvig Lizana, Ralf Metzler, and Tobias Ambjörnsson
Accepted
There exists compelling experimental evidence in numerous systems for logarithmically slow time evolution, yet its theoretical understanding remains elusive. We here introduce and study a generic transition process in complex systems, based on non-renewal, aging waiting times. Each state $n$ of the system follows a local clock initiated at $t=0$. The random time $\tau$ between clock ticks follows the waiting time density $\psi(\tau)$. Transitions between states occur only at local clock ticks and are hence triggered by the local forward waiting time, rather than by $\psi(\tau)$. For power-law forms $\psi(\tau)\simeq \tau^{-1-\alpha}$ ($02$ the process becomes normal in the sense that $\langle n(t)\rangle\simeq t$. In the intermediate range $1