First passage time problems in time-dependent fields

John E. Fletcher, Shlomo Havlin, George H. Weiss

Research output: Contribution to journalArticlepeer-review

50 Scopus citations


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 L2 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 languageEnglish
Pages (from-to)215-232
Number of pages18
JournalJournal of Statistical Physics
Issue number1-2
StatePublished - Apr 1988


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


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