Ergodic properties of heterogeneous diffusion processes in a potential well.

Ergodicity is a central concept in statistical mechanics, linking time averages to ensemble averages. In many physical systems, especially those with spatially varying properties, the assumption of ergodicity may break down. Prior research has shown that standard diffusion models assume uniform diffusivity and do not account for spatial heterogeneity. This gap motivated the investigation of how space-dependent diffusivity affects ergodic behavior. In particular, the role of potential wells in modifying diffusion dynamics remains underexplored. Existing models often simplify the system to uniform conditions, which may not reflect real-world scenarios. The behavior of systems with power-law diffusivity and potential functions is not well understood. This paper addresses the need for a more comprehensive framework to analyze ergodicity in heterogeneous environments. The study aims to clarify how the interplay between diffusivity and potential influences the ergodic properties of a system.

The study aims to analyze the ergodic behavior of heterogeneous diffusion processes in a potential well. It focuses on the observable-occupation time to determine whether time averages match ensemble averages. The researchers investigate how the system's ergodicity depends on the relative scaling of diffusivity and potential. The study considers three distinct cases based on the power-law exponents of diffusivity and potential. The motivation stems from the need to understand nonergodic behavior in complex physical systems. The researchers propose to use a simplified model with power-law forms for diffusivity and potential. The study's primary goal is to classify the ergodic properties under different conditions. The findings may provide insights into the behavior of systems with spatially varying properties.

The researchers use an overdamped Langevin equation to model the diffusion process. The diffusivity and potential are assumed to follow power-law forms for simplicity. The study considers three cases based on the relative values of the power-law exponents. Monte Carlo simulations are used to evaluate the probability density distribution of the time-averaged occupation time. The researchers analyze the competition between diffusivity and potential in determining ergodicity. The system is classified as ergodic, nonergodic, or conditionally ergodic based on the exponents. The time average and ensemble average are compared to assess ergodic behavior. The simulations help validate the theoretical predictions and provide a visual representation of the results.

The system is ergodic when the potential exponent is greater than the diffusivity exponent. In this case, the time average matches the ensemble average for long times. The steady-state solution determines both averages in the ergodic regime. The system becomes nonergodic when the diffusivity exponent is greater than the potential exponent. Infinite-ergodic theory explains the relationship between time and ensemble averages in this case. The middle case, where the exponents are equal, shows a more complex dependence on the prefactors. The probability density distribution of the time-averaged occupation time is evaluated for all three cases. The Monte Carlo simulations confirm the theoretical predictions and highlight the differences in ergodic behavior.

The ergodic properties of the system depend on the relative scaling of diffusivity and potential. The system is ergodic when the potential exponent is greater than the diffusivity exponent. The time and ensemble averages coincide in this case, determined by the steady-state solution. The system is nonergodic when the diffusivity exponent is greater than the potential exponent. Infinite-ergodic theory explains the nonergodic behavior in this regime. The middle case shows a more delicate dependence on the prefactors of diffusivity and potential. The probability density distribution of the time-averaged occupation time varies across the three cases. The Monte Carlo simulations support the theoretical analysis and provide a visual confirmation of the results. The study clarifies the conditions under which a system exhibits ergodic or nonergodic behavior.