First Passage Times in Compact Domains Exhibit Biscaling

The study of first passage time for diffusing particles reaching target states is foundational in various practical applications, including diffusion-controlled reactions. In large systems, first passage times statistics exhibit a biscaling behavior, challenging the use of a single timescale. In this work, we present a biscaling theory for the probability density function of first passage times in confined compact processes, applicable to both Euclidean and fractal domains and for diverse geometries. Our theory employs two distinct scaling functions: one for short times, capturing initial dynamics in unbounded systems, and the other for long times, which is sensitive to finite size effects. The combined framework is argued to provide a complete expression for first passage time statistics across all timescales. As our detailed calculations show, the theory describes various scenarios with and without external force fields, for active and thermal settings, and in the presence of resetting when a nonequilibrium steady state emerges.