First passage times in compact domains exhibit bi-scaling

Abstract: The study of first passage times for diffusing particles reaching target states is foundational in various practical applications, including diffusion-controlled reactions. In this work, we present a bi-scaling theory for the probability density function of first passage times in confined compact processes, applicable to both Euclidean and Fractal domains, diverse geometries, and scenarios with or without external force fields, accommodating Markovian and semi-Markovian random walks. In large systems, first passage time statistics exhibit a bi-scaling behavior, challenging the use of a single time scale. Our theory employs two distinct scaling functions: one for short times, capturing initial dynamics in unbounded systems, and the other for long times is sensitive to finite size effects. The combined framework provides a complete expression for first passage time statistics across all time scales.