Outflow probability for drift--diffusion dynamics

TL;DR

This paper analytically derives the outflow probability for a one-dimensional drift-diffusion process with specific boundary conditions, applicable to fields like traffic flow, using eigenfunction expansion methods.

Contribution

It provides a comprehensive analytical solution for the first passage time distribution in drift-diffusion models with mixed boundary conditions, covering all drift scenarios.

Findings

Explicit formulas for outflow probability for all drift values

Analytical solution using eigenfunction expansion and Sturm--Liouville analysis

Application to traffic flow breakdown probability

Abstract

The presented explanations are provided for the one--dimensional diffusion process with constant drift by using forward Fokker--Planck technique. We are interested in the outflow probability in a finite interval, i.e. first passage time probability density distribution taking into account reflecting boundary on left hand side and absorbing border on right hand side. This quantity is calculated from balance equation which follows from conservation of probability. At first, the initial--boundary--value problem is solved analytically in terms of eigenfunction expansion which relates to Sturm--Liouville analysis. The results are obtained for all possible values of drift (positive, zero, negative). As application we get the cumulative breakdown probability which is used in theory of traffic flow.