Analysis of a Bulk-Mediated Surface Diffusion Model to Interprete Experimental Observations

The diffusion of particles near an interactive surface, characterized by transitions between a surface-adsorbed state and free diffusion in the bulk, is a fundamental process in many physical and biological systems. Recent single-molecule tracking experiments observing this Bulk-Mediated Surface Diffusion (BMSD) have revealed complex behaviors, such as non-Gaussian displacements and anomalous mean-square displacement (MSD), that are not fully captured by existing theoretical frameworks. A significant challenge remains in reliably extracting key kinetic parameters, such as adsorption and desorption rates, from experimental data. This report attempts to address these challenges through a detailed theoretical and numerical analysis of a BMSD model. We investigate whether this model can explain experimental observations by deriving analytical expressions for key quantities, including the distribution of apparent waiting times and the MSD of particles on the surface. Our analysis explores a method for estimating the model's parameters from the average and the power-law tail ($t^{−5/2}$) of the apparent waiting time distribution, suggesting this approach is most effective in a regime of low surface reactivity. Furthermore, this work characterizes the MSD to clarify the conditions under which different superdiffusive regimes ($t^{5/2}$ and $t^{3/2}$) might emerge and their transition times. In an attempt to reconcile theoretical predictions with the linear MSD observed in some experiments, we propose an expression for an effective diffusion constant that accounts for experimental constraints. Finally, we analyze the surface displacement distribution in the fast-exchange limit, demonstrating its non-Gaussian character at short times and its evolution towards a Gaussian distribution at long times. Ultimately, this research aims to provide a more robust framework for interpreting experimental observations of BMSD and to suggest pathways toward a quantitative understanding of surface transport phenomena.