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Waiting time distribution for continuous stochastic systems

Gernert, Robert; Emary, Clive; Klapp, Sabine H.L.

Authors

Robert Gernert

Clive Emary

Sabine H.L. Klapp



Abstract

The waiting time distribution (WTD) is a common tool for analyzing discrete stochastic processes in classical and quantum systems. However, there are many physical examples where the dynamics is continuous and only approximately discrete, or where it is favourable to discuss the dynamics on a discretized and a continuous level in parallel. An example is the hindered motion of particles through potential landscapes with barriers. In the present paper we propose a consistent generalization of the WTD from the discrete case to situations where the particles perform continuous barrier crossing characterized by a finite duration. To this end, we introduce a recipe to calculate the WTD from the Fokker-Planck (Smoluchowski) equation. In contrast to the closely related first passage time distribution (FPTD), which is frequently used to describe continuous processes, the WTD contains information about the direction of motion. As an application, we consider the paradigmatic example of an overdamped particle diffusing through a washboard potential. To verify the approach and to elucidate its numerical implications, we compare the WTD defined via the Smoluchowski equation with data from direct simulation of the underlying Langevin equation and find full consistency provided that the jumps in the Langevin approach are defined properly. Moreover, for sufficiently large energy barriers, the WTD defined via the Smoluchowski equation becomes consistent with that resulting from the analytical solution of a (two-state) master equation model for the short-time dynamics developed previously by us [Phys. Rev. E 86, 061135 (2012)]. Thus, our approach “interpolates” between these two types of stochastic motion. We illustrate our approach for both symmetric systems and systems under constant force.

Citation

Gernert, R., Emary, C., & Klapp, S. H. (2014). Waiting time distribution for continuous stochastic systems. Physical review. E, Statistical, nonlinear, and soft matter physics, 90(6), Article 062115. https://doi.org/10.1103/PhysRevE.90.062115

Journal Article Type Article
Acceptance Date Jul 9, 2014
Online Publication Date Dec 8, 2014
Publication Date Dec 8, 2014
Deposit Date Jun 28, 2018
Publicly Available Date Mar 29, 2024
Journal Physical Review E
Print ISSN 1539-3755
Electronic ISSN 1550-2376
Publisher American Physical Society
Peer Reviewed Peer Reviewed
Volume 90
Issue 6
Article Number 062115
DOI https://doi.org/10.1103/PhysRevE.90.062115
Public URL https://hull-repository.worktribe.com/output/898455
Publisher URL https://journals.aps.org/pre/abstract/10.1103/PhysRevE.90.062115

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