On the waiting time distribution for continuous stochastic systems

TL;DR

This paper extends the waiting time distribution (WTD) concept from discrete to continuous stochastic systems, providing a method to calculate it from the Fokker-Planck equation and validating it through simulations and analytical models.

Contribution

It introduces a consistent way to define and compute the WTD for continuous systems, bridging the gap between discrete and continuous stochastic process analysis.

Findings

WTD can be derived from the Fokker-Planck equation for continuous systems.

The approach is validated against Langevin simulations and analytical models.

The method interpolates between discrete jump processes and continuous diffusion dynamics.

Abstract

The waiting time distribution (WTD) is a common tool for analysing 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 generalisation of the WTD from the discrete case to situations where the particles perform continuous barrier-crossing characterised 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 previous by us [PRE 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.