First Passage Times in Compact Domains Exhibit Biscaling

Area of Science:

• Physics
• Physical Chemistry
• Statistical Mechanics

Background:

• First passage time (FPT) is crucial for understanding diffusion-controlled reactions and particle dynamics.
• Large systems often display biscaling behavior in FPT, defying single-timescale analysis.
• Existing models struggle to capture the full spectrum of FPT statistics across diverse systems.

Purpose of the Study:

• To develop a comprehensive biscaling theory for the probability density function of first passage times.
• To unify the description of FPT statistics across short and long timescales.
• To provide a framework applicable to confined compact processes in various domains and geometries.

Main Methods:

• Development of a novel biscaling theory for FPT probability density functions.
• Inclusion of two distinct scaling functions: one for initial dynamics and one for finite-size effects.
• Application of the theory to analyze diverse scenarios, including external forces and resetting mechanisms.

Main Results:

• The proposed theory successfully describes FPT statistics across all timescales.
• It captures initial dynamics in unbounded systems and finite-size effects in confined systems.
• The framework is validated for active, thermal, and non-equilibrium steady-state scenarios.

Conclusions:

• The biscaling theory offers a complete framework for understanding FPT statistics in confined systems.
• It reconciles short-time and long-time dynamics, providing a unified perspective.
• This work advances the study of diffusion processes in complex and realistic settings.