Alternating Kinetics of Annihilating Random Walks Near a Free Interface

TL;DR

This paper studies the decay dynamics of annihilating random walks near a free boundary, revealing alternating decay exponents for particles at the interface and providing bounds and estimates for these exponents.

Contribution

It introduces the concept of alternating decay exponents for particles near a free interface and derives bounds and heuristic estimates for the primary decay exponent.

Findings

Survival probability decays as t^{-alpha_n} with alternating alpha_n values.

Alpha_1 is bounded above by 1/4 and heuristically estimated at approximately 0.2067.

The first particle's average position advances as approximately 1.7 times t^{1/2}.

Abstract

The kinetics of annihilating random walks in one dimension, with the half-line x>0 initially filled, is investigated. The survival probability of the nth particle from the interface exhibits power-law decay, S_n(t)~t^{-alpha_n}, with alpha_n approximately equal to 0.225 for n=1 and all odd values of n; for all n even, a faster decay with alpha_n approximately equal to 0.865 is observed. From consideration of the eventual survival probability in a finite cluster of particles, the rigorous bound alpha_1<1/4 is derived, while a heuristic argument gives alpha_1 approximately equal to 3 sqrt{3}/8 = 0.2067.... Numerically, this latter value appears to be a stringent lower bound for alpha_1. The average position of the first particle moves to the right approximately as 1.7 t^{1/2}, with a relatively sharp and asymmetric probability distribution.