Waiting time distribution for continuous stochastic systems

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.

Feedback Control of Colloidal Transport

“…These oscillations can be traced back to the periodic reconstruction of the effective energy landscape, V DF (z, t) + u(z), which consists of a periodic increase and decrease of the energy barriers [43]. The period T of oscillations roughly coincides with the inverse Kramers rate [21,39], which is the relevant time scale for the slow barrier-crossing. Also, the regime of pronounced oscillations partly coincides with the regime where a "speed up" of the motion occurs.…”

Force-linearization closure for non-Markovian Langevin systems with time delay

“…However, in the limit of small noise intensities, we can find an approximation for τ K by using Kramers theory for (Markovian) systems characterized by a static potential landscape U . When the potential barriers ∆U are large compared to the thermal energy, i.e., when γD 0 ∆U , the Kramers theory provides an Arrhenius formula [35,42,44] for the escape rate, r K = U (x min )|U (x max )|/(2πγ) exp (−∆U /γD 0 ), (with U (x ex ) = ∂ xx U (x)| x=xex , with x ex ∈ {x min , x max }). This estimate for r K relies on an equilibrium approximation for ρ 1 and does not involve ρ 2 .…”

“…The control "target" is the particle position which, due to the typical (micron-scale) size of a colloid, is readily accessible in real space experiments and simulations [14][15][16]. Feedback control can be implemented, e.g., as a co-moving "optical tweezer" [24,25,[33][34][35][36], which can be well approximated by a co-moving quadratic potential [25,33,37]. Depending on the experimental setup, a time delay naturally arises during the position measurement and the adjustment of the tweezer, or it can be intentionally implemented as a feature.…”

“…by optical landscapes [39]. The impact of time delay on diffusion in washboard potentials has already been studied on the basis of a heuristic delayed FPE [24,35,36] and by simulations [16]. Our second example is a colloidal particle in a double-well potential.…”

Time- and ensemble-averages in evolving systems: the case of Brownian particles in random potentials

“…[23][24][25] The dynamics of individual Brownian particles exposed to a stationary random potential energy landscape (PEL), as illustrated in Fig. [32][33][34][35][36][37] In particular, almost any external potential can be imposed on colloidal particles using laser light fields. Colloidal model systems can be used to systematically and quantitatively study particle dynamics in experiments [26][27][28][29][30][31] and simulations.…”