## 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{variant}* 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 length L the mean residence time goes like L^{2} for large L when π{variant}≠π{variant}*, but when π{variant}=π{variant}* the dependence is proportional to L. 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.

Original language | English |
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Pages (from-to) | 215-232 |

Number of pages | 18 |

Journal | Journal of Statistical Physics |

Volume | 51 |

Issue number | 1-2 |

DOIs | |

State | Published - Apr 1988 |

## Keywords

- Random walks
- diffusion processes
- diffusive coherence
- first passage times
- residence times