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First-passage and first-arrival problems in continuous-time random walks: Beyond the diffusion approximation.

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Exact solutions for continuous-time random-walks reveal diffusion approximations can be inaccurate. Mean first-passage times vary by ensemble, and point-like targets have zero first-arrival probability with continuous jump lengths.

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Area of Science:

  • Statistical Physics
  • Stochastic Processes
  • Mathematical Modeling

Background:

  • The continuous-time random-walk (CTRW) model is fundamental in describing anomalous diffusion and transport phenomena.
  • First-passage and first-arrival problems are critical for understanding the dynamics of particles and systems.
  • Existing approximations, like diffusion approximation, are widely used but their accuracy needs rigorous validation.

Purpose of the Study:

  • To derive exact solutions for first-passage and first-arrival problems within the CTRW framework.
  • To rigorously assess the accuracy of diffusion approximations under various conditions.
  • To investigate the influence of ensemble properties and jump-length distributions on first-arrival statistics.

Main Methods:

  • Derivation of exact analytical solutions for first-passage and first-arrival times in CTRW.
  • Comparison of exact solutions with results from diffusion approximation for specific jump-length distributions.
  • Analysis of first-arrival probabilities for point-like targets using continuous jump-length distributions.

Main Results:

  • Diffusion approximations can significantly deviate from exact solutions for CTRWs with finite variance jump-length distributions.
  • Mean first-passage and first-arrival times are sensitive to the ensemble (e.g., stationary vs. non-aged), with stationary ensembles exhibiting longer mean first-passage times.
  • The probability of first arrival at a point-like target is zero for any continuous jump-length distribution, challenging prior assumptions.

Conclusions:

  • Exact solutions are crucial for accurately characterizing CTRW dynamics, highlighting limitations of diffusion approximations.
  • Ensemble choice critically impacts first-passage time statistics in CTRWs.
  • The zero probability of arrival at a point target for continuous jump lengths necessitates a re-evaluation of certain theoretical frameworks.