Survival Probability in a Random Velocity Field

TL;DR

This paper investigates the survival probability of particles in a two-dimensional system with a random velocity field, deriving its asymptotic behavior and scaling properties through qualitative and heuristic arguments supported by numerical simulations.

Contribution

It provides a novel analysis of survival probability decay and scaling forms for particles driven by a random velocity field, supported by numerical evidence.

Findings

Survival probability scales as t^{-1/4}.

The longitudinal distribution follows a specific scaling form with a t^{-5/4} dependence.

The proposed continuum equation reproduces the asymptotic distribution.

Abstract

The time dependence of the survival probability, S(t), is determined for diffusing particles in two dimensions which are also driven by a random unidirectional zero-mean velocity field, v_x(y). For a semi-infinite system with unbounded y and x>0, and with particle absorption at x=0, a qualitative argument is presented which indicates that S(t)~t^{-1/4}. This prediction is supported by numerical simulations. A heuristic argument is also given which suggests that the longitudinal probability distribution of the surviving particles has the scaling form P(x,t)~ t^{-1}u^{1/3}g(u). Here the scaling variable u is proportional to x/t^{3/4}, so that the overall time dependence of P(x,t) is proportional to t^{-5/4}, and the scaling function g(u) has the limiting dependences g(u) approaching a constant as u--->0 and g(u)~exp(-u^{4/3}) as u--->infinity. This argument also suggests an effective continuum equation of motion for the infinite system which reproduces the correct asymptotic longitudinal probability distribution.