Quantifying the most probable dynamics of a particle inside a sphere

Particle random walk phenomena manifest across diverse biophysical and physical systems. Motivated by receptor diffusion dynamics toward cellular membranes, this study investigates statistically dominant transition dynamics for particles undergoing confined diffusion within spherical geometries. We developed a computational framework combining stochastic process modeling with deep learning techniques to quantify two critical aspects of the diffusion process: most probable transition time for particles migrating from the spherical center to arbitrary boundary points, and most probable transition pathways connecting the central source to preferential terminal locations on the spherical surface. The methodology integrates numerical simulations of stochastic differential equations with neural network architectures trained on the Onsager-Machlup variational principle, enabling systematic identification of entropy-optimized diffusion trajectories. Through this integrated approach, we derive mechanistic insights into the spatiotemporal organization of membrane-bound receptors, proposing a predictive framework for receptor spatial distribution patterns influenced by constrained diffusion energetics.