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Although gaseous molecules travel at tremendous speeds (hundreds of meters per second), they collide with other gaseous molecules and travel in many different directions before reaching the desired target. At room temperature, a gaseous molecule will experience billions of collisions per second. The mean free path is the average distance a molecule travels between collisions. The mean free path increases with decreasing pressure; in general, the mean free path for a gaseous molecule will be...
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Related Experiment Video

Updated: Dec 28, 2025

Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level
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Universal formula for extreme first passage statistics of diffusion.

Sean D Lawley1

  • 1Department of Mathematics, University of Utah, Salt Lake City, Utah 84112, USA.

Physical Review. E
|February 20, 2020
PubMed
Summary

This study introduces a universal formula for the fastest first passage time (FPT) for diffusion processes. It reveals that the time for the quickest searcher to find a target is much faster than for a single searcher.

Area of Science:

  • Statistical Physics
  • Stochastic Processes
  • Mathematical Biology

Background:

  • First passage times (FPTs) govern the timescales of diffusion-limited processes in nature.
  • Existing research predominantly focuses on the FPT for a single diffusing particle.
  • The time for the fastest particle in a group to reach a target is often more relevant but less understood.

Purpose of the Study:

  • To derive a general formula for the moments of the fastest first passage time (FFPT).
  • To establish the universality of this FFPT formula across diverse diffusion scenarios.

Main Methods:

  • Analytical derivation of a formula for FFPT moments.
  • Consideration of d-dimensional diffusion with space-dependent properties.
  • Inclusion of complex environments like Riemannian manifolds and obstacles.

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Related Experiment Videos

Last Updated: Dec 28, 2025

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Main Results:

  • A simple, explicit formula for every moment of the FFPT is proven.
  • The formula demonstrates remarkable universality across various diffusion models.
  • The FFPT is shown to be significantly faster than single-particle FPTs due to rare event statistics.

Conclusions:

  • The derived formula provides a rigorous and unified framework for understanding FFPT.
  • This work corrects and generalizes previous heuristics and conjectures.
  • The findings have broad implications for physical, chemical, and biological systems governed by diffusion.