Microscopic origin of the logarithmic time evolution of aging processes in complex systems

Area of Science:

• Complex Systems
• Statistical Physics
• Theoretical Physics

Background:

• Logarithmically slow time evolution is observed in many systems, but lacks complete theoretical explanation.
• Existing models often fail to capture the nuances of aging processes in complex systems.

Purpose of the Study:

• To develop a generic theoretical model for transition processes in complex systems.
• To explain the observed logarithmic time evolution using aging waiting times.
• To provide a universal description for dynamics between aging and non-aging states.

Main Methods:

• Introduction of a generic transition process based on non-renewal and aging waiting times.
• Analysis of system state evolution triggered by local clock ticks and forward waiting times.
• Mathematical derivation of time evolution for different power-law forms of waiting time densities.

Main Results:

• Logarithmic time evolution ⟨n(t)⟩ ≃ log(t/t(0)) is derived for power-law waiting times with 0 < α < 1.
• Normal time evolution ⟨n(t)⟩ ≃ t is observed for α > 2.
• Intermediate power-law growth ⟨n(t)⟩ ≃ t(α-1) is found for 1 < α < 2.

Conclusions:

• The proposed model successfully explains logarithmic time evolution in complex systems.
• The model offers a universal framework applicable to systems exhibiting aging and non-aging dynamics.
• The findings bridge the gap between experimental observations and theoretical understanding of slow dynamics.