Irreducible memory function and slow dynamics in disordered systems

Area of Science:

• Statistical Mechanics
• Condensed Matter Physics

Background:

• Understanding the slow dynamics and relaxation processes in complex systems like glass-forming materials is crucial.
• Existing models often struggle to capture the transition from ergodic to nonergodic behavior and the emergence of dynamical heterogeneities.

Purpose of the Study:

• To develop a straightforward method for calculating the irreducible memory function.
• To represent the irreducible memory function using parallel decay channels.
• To connect the irreducible memory function to stochastic processes and explain universal behaviors in glass-forming systems.

Main Methods:

• Derivation of the irreducible memory function.
• Expression of the memory function in terms of two parallel decay channels.
• Analysis of the connection between the irreducible memory function and regenerative stochastic processes.

Main Results:

• A straightforward method to obtain the irreducible memory function is demonstrated.
• The irreducible memory function is shown to consist of two parallel decay channels.
• Under mild conditions, the irreducible memory function defines a regenerative stochastic process or a two-level stochastic system.
• This framework explains dynamical heterogeneities and the asymptotic statistical properties governed by limit processes.

Conclusions:

• The proposed representation of the irreducible memory function offers a valuable tool for studying systems with slow time dependence and internal degrees of freedom.
• The connection to stochastic processes provides a theoretical basis for understanding dynamical heterogeneities in glass-forming systems.
• The findings can explain the universal behaviors observed across various glass-forming materials.