Heterogeneous diffusion processes and nonergodicity with Gaussian colored noise in layered diffusivity landscapes

Diffusion processes in heterogeneous environments are widely studied in physics and biology. Traditional models assume uniform diffusion coefficients, but many real systems exhibit spatially varying diffusion rates. Prior research has shown that heterogeneous diffusion processes (HDPs) can exhibit nonergodic behavior when driven by uncorrelated noise. However, the effects of spatially layered diffusion coefficients with random scaling exponents remain underexplored. This gap motivated investigations into how layered structures influence diffusion dynamics. While ergodicity breaking in HDPs is known, the role of colored noise in such systems is less understood. Existing studies often assume uniform scaling exponents across the entire system. This paper introduces a novel framework where diffusion coefficients vary both in space and in scaling exponent. The lack of detailed analysis on layered systems with Gaussian colored noise represents a key limitation in current literature.

The study aims to explore how heterogeneous diffusion processes behave in systems with spatially layered diffusion coefficients and Gaussian colored noise. The specific problem addressed is the lack of understanding of how random scaling exponents and layered structures influence diffusion dynamics. The motivation stems from the need to better model diffusion in biological cells and geophysical systems. The research focuses on quantifying nonergodic and non-Gaussian behaviors in such systems. By varying the scaling exponents and layer thicknesses, the authors seek to identify conditions under which different diffusion regimes emerge. The study also examines the impact of noise correlation times on system behavior. This approach allows for a more realistic representation of diffusion in complex, disordered environments. The ultimate goal is to provide insights into weak ergodicity breaking in HDPs driven by colored noise.

The researchers employed numerical simulations to analyze heterogeneous diffusion processes in layered systems. The system was modeled with periodic intervals, each containing a randomly assigned scaling exponent from a Gaussian distribution. The diffusion coefficient was centered at the midpoint of each interval, introducing spatial randomness. Gaussian colored noise was used to drive the diffusion process, allowing for the study of time-correlated fluctuations. The authors varied the scaling exponents, layer thicknesses, and noise correlation times to explore their effects. Statistical analysis was performed to identify diffusion regimes such as superdiffusion, subdiffusion, and normal diffusion. Nonergodicity was assessed by comparing time-averaged and ensemble-averaged properties. The study also quantified the emergence of non-Gaussian behaviors in the system.

The study found that layered systems with randomly assigned scaling exponents can exhibit superdiffusion, subdiffusion, and normal diffusion depending on the parameters. Nonergodic behavior was observed in regions where the diffusion coefficient varied significantly across layers. The thickness of the layers had a direct impact on the diffusion regime, with thinner layers promoting more erratic diffusion. The correlation time of Gaussian colored noise influenced the degree of nonergodicity, with longer correlation times increasing the deviation from ergodicity. Non-Gaussian behaviors emerged when the scaling exponents showed high variability across layers. The researchers quantified the boundaries of these behaviors using statistical measures. The results showed that weak ergodicity breaking is more pronounced in systems with high layer variability and long noise correlation times. These findings provide a detailed map of diffusion regimes in layered HDPs driven by colored noise.

The authors concluded that heterogeneous diffusion processes in layered systems with Gaussian colored noise can exhibit a range of diffusion regimes. The presence of spatially varying scaling exponents and layer thicknesses significantly affects the ergodicity of the system. The study confirms that nonergodic behavior is more likely in systems with high variability in scaling exponents. The results suggest that the correlation time of the noise plays a critical role in determining the degree of nonergodicity. The findings provide a framework for understanding weak ergodicity breaking in disordered systems. The authors propose that these insights can be applied to model diffusion in biological cells and geophysical systems. The study highlights the importance of considering both spatial heterogeneity and noise correlation in HDP models. These conclusions align with the observed behaviors and numerical results presented in the paper.