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Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level
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Crossover between Lévy and Gaussian regimes in first-passage processes.

Jun-ichi Inoue1, Naoya Sazuka

  • 1Complex Systems Engineering, Graduate School of Information Science and Technology, Hokkaido University, N14-W9, Kita-ku, Sapporo 060-0814, Japan. j_inoue@complex.eng.hokudai.ac.jp

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 13, 2007
PubMed
Summary

We introduce a new method for first-passage time analysis applicable to various stochastic processes. Our study reveals a crossover in scaling laws for truncated Lévy flights, transitioning from non-Gaussian to Gaussian regimes.

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Area of Science:

  • Statistical Physics
  • Stochastic Processes
  • Probability Theory

Background:

  • First-passage time problems are fundamental in analyzing stochastic processes.
  • Lévy flights exhibit heavy-tailed distributions, posing challenges for traditional analysis.
  • Truncated Lévy flights (KoBoL processes) modify Lévy distributions by cutting off extreme values.

Purpose of the Study:

  • To propose a novel approach for solving first-passage time problems.
  • To investigate the behavior of first-passage time distributions in truncated Lévy flights.
  • To analyze the crossover phenomena between non-Gaussian and Gaussian regimes.

Main Methods:

  • Development of a general method applicable to Wiener processes, Lévy flights, and stable distributions.
  • Focus on truncated Lévy flights (KoBoL processes) to demonstrate the method's utility.
  • Analysis of asymptotic scaling laws for first-passage time distributions.

Main Results:

  • Identified a crossover in the asymptotic scaling law of the first-passage time distribution.
  • Observed a transition from a t(-(alpha+1)/alpha)-law (non-Gaussian Lévy regime) to a t(-3/2)-law (Gaussian regime).
  • Demonstrated ultraslow convergence from the non-Gaussian to the Gaussian regime in truncated Lévy flights.

Conclusions:

  • The proposed method effectively analyzes first-passage time problems in complex stochastic systems.
  • Truncated Lévy flights exhibit a distinct crossover in scaling laws, bridging non-Gaussian and Gaussian behaviors.
  • The findings have implications for understanding anomalous diffusion and related phenomena.