Brownian motion in time-dependent logarithmic potential: Exact results for dynamics and first-passage properties

TL;DR

This paper develops an asymptotic theory for Brownian motion in a time-dependent logarithmic potential, analyzing survival probabilities and particle dynamics, with applications to biological and chemical systems involving entropic barriers.

Contribution

It provides the first comprehensive asymptotic analysis of survival and dynamic properties for Brownian particles in time-dependent logarithmic potentials.

Findings

Survival probability exhibits three distinct decay regimes: power-law, marginal, and exponential.

The asymptotic theory aligns well with numerical simulations.

Different potential strength functions g(t) lead to varied long-time behaviors.

Abstract

The paper addresses Brownian motion in the logarithmic potential with time-dependent strength, $U(x,t) = g(t) \log(x)$, subject to the absorbing boundary at the origin of coordinates. Such model can represent kinetics of diffusion-controlled reactions of charged molecules or escape of Brownian particles over a time-dependent entropic barrier at the end of a biological pore. We present a simple asymptotic theory which yields the long-time behavior of both the survival probability (first-passage properties) and the moments of the particle position (dynamics). The asymptotic survival probability, i.e., the probability that the particle will not hit the origin before a given time, is a functional of the potential strength. As such it exhibits a rather varied behavior for different functions $g(t)$. The latter can be grouped into three classes according to the regime of the asymptotic decay of the survival probability. We distinguish 1. the regular (power-law decay), 2. the marginal (power law times a slow function of time), and 3. the regime of enhanced absorption (decay faster than the power law, e.g., exponential). Results of the asymptotic theory show good agreement with numerical simulations.