Mean First Passage Time for a Random Walker and Its Application to Chemical Kinetics

Abstract

The mean first passage time for a random walk with reflecting and absorbing barriers is computed by assuming Onsager's reciprocal relation for the transition probabilities. The result, which is valid for an arbitrary dimensional random walk, appears as a quotient of determinants whose elements are the transition probabilities and the initial distribution. For the one‐dimensional case, this result is written as a series which converges rapidly when nearest neighbor transitions predominate; in fact, the series reduces to a closed form for the case when only nearest neighbor transitions are allowed, and always converges for the case when only nearest and next‐nearest transitions are allowed. Montroll and Shuler's computation of the mean first passage time for the case of the truncated harmonic oscillator model of a diatomic molecule is simplified and extended to an arbitrary initial distribution. In addition, the mean first passage time for the case of an‐harmonic oscillator model is computed by using the transition probability obtained recently by Shuler, Bazley, and Montroll. In terms of the mean first passage time, the range of validity of the equilibrium theory in chemical kinetics is discussed.