Microscopic Origin of the Logarithmic Time Evolution of Aging Processes in Complex Systems

There exists compelling experimental evidence in numerous systems for logarithmically slow time evolution, yet its full theoretical understanding remains elusive. We here introduce and study a generic transition process in complex systems, based on nonrenewal, aging waiting times. Each state n of the system follows a local clock initiated at t=0. The random time τ between clock ticks follows the waiting time density ψ(τ). Transitions between states occur only at local clock ticks and are hence triggered by the local forward waiting time, rather than by ψ(τ). For power-law forms ψ(τ) τ ^-1 ^-α (0<α<1) we obtain a logarithmic time evolution of the state number âŸ̈n(t) ⟠logâ¡(t/t ₀), while for α>2 the process becomes normal in the sense that âŸ̈n(t)⟠t. In the intermediate range 1<α<2 we find the power-law growth âŸ̈n(t)⟠tα ^-1. Our model provides a universal description for transition dynamics between aging and nonaging states.