Ergodic Measure and Potential Control of Anomalous Diffusion.

Area of Science:

• Statistical Mechanics
• Complex Systems

Background:

• The ergodic hypothesis, equating long-time and ensemble averages, is crucial for understanding anomalous diffusion, irreversibility, and entropy.
• Finite measurement times pose challenges to establishing ergodicity, as time averages fluctuate and their convergence rate requires investigation.

Purpose of the Study:

• To investigate the time-dependent fluctuation width of time averages for velocity and kinetic energy in a force-free particle system.
• To determine the time scale for a system transitioning from a stationary state to an effective ergodic state.
• To analyze the influence of a logarithmic spatial potential on diffusion processes.

Main Methods:

• Utilizing the generalized Langevin equation to model particle dynamics.
• Analyzing the stationary velocity autocorrelation function.
• Introducing a logarithmic spatial potential to modulate diffusion.

Main Results:

• The study quantifies the time-dependent fluctuation width for velocity and kinetic energy time averages.
• An estimation of the shortest time scale for ergodic establishment is provided.
• Logarithmic potential was shown to modulate free ballistic diffusion and control diffusion processes, realizing power-law regimes.

Conclusions:

• Non-ergodicity in diffusion processes can effectively mimic the sparseness of a medium.
• Logarithmic potentials play a unique role in modulating diffusion behavior and achieving specific power-law regimes.