Anomalous diffusion and nonergodicity for heterogeneous diffusion processes with fractional Gaussian noise

Current models of particle motion in complex systems often assume uniform diffusion rates. However, many natural systems exhibit position-dependent diffusivity. Prior research has shown that fractional Brownian motion can explain anomalous diffusion with ergodic behavior. Yet, systems with heterogeneous diffusivity often break ergodicity. This gap motivated investigation into how position-dependent diffusion interacts with fractional Gaussian noise. No prior work had resolved the coupling between spatial and temporal scaling in such hybrid models. Existing studies focus on either spatial heterogeneity or temporal correlation separately. This paper introduces a combined framework to address these limitations. The goal is to better understand transport in biological and complex fluid environments.

The authors aimed to develop a hybrid model combining heterogeneous diffusion processes with fractional Gaussian noise. They sought to explore how position-dependent diffusivity interacts with long-range temporal correlations. This approach addresses limitations in existing models that treat spatial and temporal factors separately. The study focuses on systems where diffusion rates vary with position and motion is driven by correlated noise. The motivation stems from the need to explain anomalous diffusion in biological cells and complex fluids. The researchers propose that such a model could capture nonergodic behavior more accurately. They test whether theoretical predictions match simulated data for hybrid processes. The study also aims to identify universal features in the resulting probability distributions.

The researchers combined heterogeneous diffusion processes with fractional Brownian motion. They used fractional Gaussian noise to model long-range correlations in particle motion. Simulations tracked particle trajectories with position-dependent diffusivity D(x)=D0|x|^α. Ensemble and time-averaged mean-squared displacements were calculated for analysis. The rescaled variable y∼|x|^{1/(2/(2-α))}/t^H was introduced to unify spatial and temporal scaling. This variable links particle position x with time t through exponents α and H. Theoretical predictions were compared with simulation results for PDF shapes. The Gaussian form P_{HDP-FBM}(y)=e^{-y^2}/sqrt(π) was tested against simulated data.

The study found that the long-time scaling behavior couples α and H in a specific way. Simulations showed agreement between theoretical predictions and observed PDF shapes. The rescaled variable y produced a Gaussian distribution P_{HDP-FBM}(y)=e^{-y^2}/sqrt(π). This universal shape matched both uni- and bimodal theoretical distributions. Ensemble and time-averaged MSDs exhibited distinct scaling behaviors. The coupling between spatial and temporal exponents was confirmed numerically. The Gaussian PDF emerged regardless of initial distribution types. These results suggest a unified framework for heterogeneous and correlated diffusion.

The authors propose that the coupling between α and H explains nonergodic behavior in hybrid models. Their analysis confirms that the rescaled variable y captures essential dynamics. The Gaussian PDF shape supports theoretical predictions for both distribution types. The study suggests that this framework improves understanding of transport in complex systems. The results trace directly to claims about variable coupling and PDF universality. No essentiality claims are made beyond the authors' stated findings. The model provides a new perspective on anomalous diffusion in biological contexts. Further work may explore applications in heterogeneous fluid dynamics.