First passage time problems in time-dependent fields

Abstract

This paper discusses the simplest first passage time problems for random walks and diffusion processes on a line segment. When a diffusing particle moves in a time-varying field, use of the adjoint equation does not lead to any simplification in the calculation of moments of the first passage time as is the case for diffusion in a time-invariant field. We show that for a discrete random walk in the presence of a sinusoidally varying field there is a resonant frequency ϖ* for which the mean residence time on the line segment is a minimum. It is shown that for a random walk on a line segment of lengthL the mean residence time goes likeL ² for largeL when ϖ≠ϖ*, but when ϖ=ϖ* the dependence is proportional toL. The results of our simulation are numerical, but can be regarded as exact. Qualitatively similar results are shown to hold for diffusion processes by a perturbation expansion in powers of a dimensionless velocity. These results are extended to higher values of this parameter by a numerical solution of the forward equation.