$A+A\to \varnothing$ reaction for particles with a dynamic bias to move away from their nearest neighbour in one dimension

Abstract

We consider the dynamics of particles undergoing the reaction $A+A\to \varnothing$ in one dimension with a dynamic bias. Here the particles move towards their nearest neighbour with probability $0.5+\epsilon$ where $-0.5\le \epsilon <0$. ${\epsilon }_c=-0.5$ is the deterministic limit where the nearest neighbour interaction is strictly repulsive. We show that the negative bias changes drastically the behaviour of the fraction of surviving particles $\rho (t)$ and persistence probability $P(t)$ with time $t$. $\rho (t)$ decays as $a/(\log t{)}^b$ where $b$ increases with $\epsilon -{\epsilon }_c$. $P(t)$ shows a stretched exponential decay with non-universal decay parameters. The probability $\Pi (x,t)$ that a tagged particle is at position $x$ from its origin is found to be Gaussian for all $\epsilon <0$; the associated scaling variable is $x/t^{\alpha }$ where $\alpha$ approaches the known limiting value $1/4$ as $\epsilon \to {\epsilon }_c$, in a power law manner. Some additional features of the dynamics by tagging the particles are also studied. The results are compared to the case of positive bias, a well studied problem.

12 pages, 9 figures. Version accepted for publication in J. Phys. A