The first‐passage time approach

Daniel T. Gillespie and Effrosyni Seitaridou

in Simple Brownian Diffusion

Published in print October 2012 | ISBN: 9780199664504
Published online January 2013 | e-ISBN: 9780191748516 | DOI:
The first‐passage time approach

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Previous chapters have focused mainly on the problem of computing where a diffusing solute molecule will be at a specified future time. In this concluding chapter the chapter takes a look at the subtly different problem of computing when a diffusing solute molecule will first reach a specified location. This is known as the first-passage time problem. Within the framework of the Einstein model of diffusion, the first-passage time problem has been solved analytically in only a few relatively simple cases. In the context of the Langevin model the problem is even more challenging, and apparently no exact results are known at present. The chapter considers the first-passage time problem only within the framework of the Einstein model, and the chapter examines only its two simplest forms. The aim is to illuminate the concepts and issues involved, and to demonstrate some solution strategies. In the course of doing that, the chapter will uncover some results that will deepen the understanding of the cell-jumping hypothesis that underlies the discrete- stochastic model of diffusion.

Keywords: Einstein diffusion; first-passage time; backward Fokker-Planck equation; simulating diffusion; discrete-stochastic model

Chapter.  11722 words.  Illustrated.

Subjects: Physics

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