Survival of a Diffusing Particle in a Transverse Shear Flow: A First-Passage Problem with Continuously Varying Persistence Exponent

TL;DR

This paper analyzes the survival probability of a diffusing particle in a shear flow with a boundary, revealing a universal decay exponent when flow velocities are equal and a variable exponent otherwise.

Contribution

It introduces a model for particle survival in shear flow with a boundary, deriving the persistence exponent and its dependence on flow parameters.

Findings

Survival probability decays as t^{-1/4} when flow velocities are equal.

Persistence exponent varies with flow asymmetry and flow profile exponent.

Explicit dependence of the persistence exponent on flow parameters is determined.

Abstract

We consider a particle diffusing in the y-direction, dy/dt=\eta(t), subject to a transverse shear flow in the x-direction, dx/dt=f(y), where x \ge 0 and x=0 is an absorbing boundary. We treat the class of models defined by f(y) = \pm v_{\pm}(\pm y)^\alpha where the upper (lower) sign refers to y>0 (y<0). We show that the particle survives with probability Q(t) \sim t^{-\theta} with \theta = 1/4, independent of \alpha, if v_{+}=v_{-}. If v_{+} \ne v_{-}, however, we show that \theta depends on both \alpha and the ratio v_{+}/v_{-}, and we determine this dependence.