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Waiting time distribution for continuous stochastic systems.

Robert Gernert1, Clive Emary2, Sabine H L Klapp1

  • 1Institut für Theoretische Physik, Sekr. EW 7-1, Technische Universität Berlin, Hardenbergstrasse 36, D-10623 Berlin, Germany.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|January 24, 2015
PubMed
Summary

We generalize the waiting time distribution (WTD) for continuous stochastic processes, offering a new method to analyze particle motion across potential barriers. This approach bridges discrete and continuous dynamics for enhanced physical system analysis.

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Area of Science:

  • Statistical Physics
  • Physical Chemistry
  • Computational Physics

Background:

  • The waiting time distribution (WTD) is crucial for analyzing discrete stochastic processes.
  • Continuous dynamics in systems like particle motion through potential barriers are often approximated as discrete.
  • A unified approach for both discrete and continuous dynamics is needed.

Purpose of the Study:

  • To generalize the waiting time distribution (WTD) for continuous stochastic processes.
  • To introduce a method for calculating WTD from the Fokker-Planck (Smoluchowski) equation for continuous barrier crossing.
  • To analyze the directionality of motion, a feature lacking in the first passage time distribution (FPTD).

Main Methods:

  • Generalizing the WTD for continuous barrier crossing dynamics.
  • Calculating WTD using the Fokker-Planck (Smoluchowski) equation.
  • Comparing WTD from the Smoluchowski equation with Langevin simulations and master equation models.

Main Results:

  • A consistent generalization of WTD for continuous processes is proposed.
  • The WTD derived from the Smoluchowski equation shows full consistency with Langevin simulations.
  • For large energy barriers, the WTD aligns with results from a two-state master equation model, interpolating between stochastic motion types.

Conclusions:

  • The proposed method consistently generalizes WTD for continuous stochastic processes.
  • This approach provides insights into particle dynamics across potential barriers, incorporating directional information.
  • The method bridges discrete and continuous stochastic models, applicable to systems with and without external forces.