First passage time distribution in heterogeneity controlled kinetics: going beyond the mean first passage time.

Aljaž Godec,Ralf Metzler

doi:10.1038/srep20349

Abstract

The first passage is a generic concept for quantifying when a random quantity such as the position of a diffusing molecule or the value of a stock crosses a preset threshold (target) for the first time. The last decade saw an enlightening series of new results focusing mostly on the so-called mean and global first passage time (MFPT and GFPT, respectively) of such processes. Here we push the understanding of first passage processes one step further. For a simple heterogeneous system we derive rigorously the complete distribution of first passage times (FPTs). Our results demonstrate that the typical FPT significantly differs from the MFPT, which corresponds to the long time behaviour of the FPT distribution. Conversely, the short time behaviour is shown to correspond to trajectories connecting directly from the initial value to the target. Remarkably, we reveal a previously overlooked third characteristic time scale of the first passage dynamics mirroring brief excursion away from the target.

Highlights

The first passage is a generic concept for quantifying when a random quantity such as the position of a diffusing molecule or the value of a stock crosses a preset threshold for the first time
How fast does the amplitude or position of a random process reach a given threshold value for the first time? This so-called first-passage time (FPT)[1,2] is central to the description of the kinetics in a large variety of systems across many disciplines, including diffusion controlled chemical reactions[3], signalling cascades in biological cells[4,5], transport in disordered media[6] including the breakthrough dynamics in hydrological aquifers[7], the location of food by foraging bacteria and animals[8,9] up to the global spreading of diseases[10,11] or stock market dynamics[12]
In order to allow for a meaningful comparison of the FPT kinetics at various degrees φ of heterogeneity we introduce a constraint on the conservation of the spatially averaged diffusion coefficient, D

Summary

Results

The particle experiences a space dependent friction Γ(r) originating from spatial variations in the long range hydrodynamic coupling to the motion of the medium[33]. We focus on the high friction limit corresponding to overdamped motion and assume that the particle diffuses with the isotropic position dependent diffusion coefficient D(r) = 2kBT × Γ(r). The diffusing particle experiences the fluctuation induced thermal drift F (r) ~ kBT ∇Γ(r) ensuring thermodynamic consistency in the sense that D(r) has a purely stochastic origin and does not reflect any heterogeneity in the entropic potential of mean force[34]. From this result the Laplace transform of the FPT density is obtained from the corresponding probability flux into the target,

We introduce dimensionless variables x

Despite being

Discussion

Additional Information