Random walk to a nonergodic equilibrium concept.

Area of Science:

• Statistical Mechanics
• Non-equilibrium Physics
• Complex Systems

Background:

• Random walk models like trap, continuous time, and comb models exhibit weak ergodicity breaking when average waiting times are infinite.
• A key open question in statistical mechanics is identifying the appropriate theory for systems with infinite average waiting times, replacing the standard Boltzmann-Gibbs theory.

Purpose of the Study:

• To investigate a nonergodic equilibrium concept for a continuous time random walk model within a potential field.
• To analyze the behavior of occupation time distributions in both ergodic and nonergodic phases of the model.

Main Methods:

• Analysis of a continuous time random walk model in a potential field.
• Investigation of occupation time distributions in finite spatial regions.
• Application of detailed balance conditions to establish connections with canonical statistical mechanics.

Main Results:

• In the nonergodic phase, occupation time distributions approach U- or W-shaped forms, related to the arcsine law.
• Under detailed balance, these nonergodic distributions are shown to depend on the partition function, linking nonergodic dynamics to canonical statistical mechanics.
• In the ergodic phase, occupation time distributions converge to a delta function, consistent with Boltzmann-Gibbs statistics.

Conclusions:

• The study establishes a framework for understanding nonergodic equilibrium in continuous time random walks.
• A direct relationship is demonstrated between nonergodic dynamics and canonical statistical mechanics through the partition function.
• The findings have potential implications for interpreting single-molecule experiments involving complex dynamics.