Estimation of energy-dependent parameter in the coupled continuous time random walk model

In recent years, anomalous diffusion dynamics has become a popular research area. Continuous Time Random Walk (CTRW) is a suitable model for describing anomalous diffusion. The statistical inference of anomalous diffusion processes is critical for exposing transport mechanisms in complex media, with precise estimation of essential parameters, such as the diffusion coefficient, posing a significant problem. This paper conducts research based on the proposed coupled CTRW model, which introduces a quadratic dependence relationship between the waiting time and the previous jump length to characterize anomalous diffusion behavior under the energy interaction mechanism. We focus on the Monte Carlo sampling of the trajectories over time, as well as the estimation of energy dependent parameter in this model, which are closely related to the diffusion coefficient. Two situations were examined: one with a flow field and the other without a flow field. Assuming a Gaussian distribution for jump length, we find the analytical solution to the relevant generalized diffusion equation which help us perform parameter estimation. Based on the time-varying migration paths of a large number of particles obtained through random simulation, we have constructed a dual-track research framework that includes model validation and parameter estimation. We employed maximum likelihood estimation (MLE) and two-step generalized method of moments (GMM) for statistical inference of the energy-dependent parameters. We analyzed the impact of several parameter alterations on the estimation results and compared the performance of the two techniques.