# Local Time Statistics and Permeable Barrier Crossing: from Poisson to Birth-Death Diffusion Equations

arxiv（2024）

Abstract

Barrier crossing is a widespread phenomenon across natural and engineering
systems. While an abundant cross-disciplinary literature on the topic has
emerged over the years, the stochastic underpinnings of the process are yet to
be fully understood. We fill this knowledge gap by considering a diffusing
particle and presenting a stochastic definition of Brownian motion in the
presence of a permeable barrier. This definition relies on reflected Brownian
motion and on the crossing events being Poisson processes subordinated by the
local time of the underlying motion at the barrier. Within this paradigm we
derive the exact expression for the distribution of the number of crossings,
and find an experimentally measurable statistical definition of permeability.
We employ Feynman-Kac theory to derive and solve a set of governing birth-death
diffusion equations and extend them to when barrier permeability is asymmetric.
As an application we study a system of infinite, identical and periodically
placed asymmetric barriers for which we derive analytically effective transport
parameters. This periodic arrangement induces an effective drift at long times
whose magnitude depends on the difference in the permeability on either side of
the barrier as well as on their absolute values. As the asymmetric
permeabilities act akin to localised “ratchet” potentials that break spatial
symmetry and detailed balance, the proposed arrangement of asymmetric barriers
provides an example of a noise-induced drift without the need to time-modulate
any external force or create temporal correlations on the motion of a diffusing
particle.

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