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First passage on disordered intervals.

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We found surprising first-passage time behaviors in disordered hopping systems. Our new method using backward equations simplifies calculations for these complex random walk properties.

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Area of Science:

  • Physics
  • Statistical Mechanics
  • Mathematical Physics

Background:

  • Nearest-neighbor hopping on finite intervals is a fundamental model in statistical mechanics.
  • Disordered hopping rates introduce significant complexity to system dynamics.
  • Understanding first-passage times is crucial for analyzing transport phenomena.

Purpose of the Study:

  • To derive and characterize unexpected first-passage properties in disordered hopping systems.
  • To develop a simpler analytical approach for calculating first-passage time moments and distributions.
  • To investigate the impact of disorder on the statistical behavior of first-passage times.

Main Methods:

  • Utilizing the backward equation in conjunction with probability generating functions.
  • Deriving all moments and the distribution of first-passage times.
  • Comparing the simplicity of the backward equation approach to traditional forward equation methods.

Main Results:

  • Revealed highly variable spatial dependence of first-passage times.
  • Observed huge disparities in first-passage times across different disorder realizations.
  • Identified significant discrepancies between the first moment and the square root of the second moment.
  • Discovered bimodal first-passage time distributions.

Conclusions:

  • The backward equation approach offers a more straightforward method for analyzing first-passage properties compared to the forward equation.
  • Disordered hopping rates lead to complex and counterintuitive first-passage behaviors.
  • The derived properties provide new insights into random walk dynamics in disordered media.