Wait-and-switch relaxation model: relationship between nonexponential relaxation patterns and random local properties...

Area of Science:

• Statistical Physics
• Complex Systems Dynamics
• Probability Theory

Background:

• Complex systems often exhibit relaxation phenomena characterized by fractional-power laws over short and long timescales.
• Understanding the underlying stochastic mechanisms governing these relaxation responses is crucial for characterizing system behavior.

Purpose of the Study:

• To present a novel wait-and-switch stochastic model for relaxation processes.
• To explain the universality of fractional-power laws in complex systems using probability theory.
• To elucidate the roles of different stochastic mechanisms in nonequilibrium state evolution.

Main Methods:

• Utilizing the "random-variable" formalism from limit theorems of probability theory.
• Analyzing the time evolution of macroscopic systems in a nonequilibrium state.
• Deriving established relaxation functions within the proposed theoretical framework.

Main Results:

• The study explains the universality of short- and long-time fractional-power laws in relaxation responses.
• It identifies two key stochastic mechanisms influencing relaxation: local statistical properties and the nature of relaxation contributions (random or deterministic).
• The Havriliak-Negami and Kohlrausch-Williams-Watts functions are derived, and the origins of the stretched-exponential integral kernel are shown.

Conclusions:

• The wait-and-switch stochastic model provides a unified framework for understanding relaxation in complex systems.
• The model successfully explains the emergence of fractional-power laws and specific relaxation functions.
• The characteristics of random walks of migrating defects are shown to influence relaxation scenarios and the integral kernel.