$A+ A \to \emptyset$ 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 $A+A \to \emptyset$ in one dimension with a dynamic bias. Here the particles move towards their nearest neighbour with probability $0.5+\epsilon$ where $-0.5 \leq \epsilon < 0$. $\epsilon_c = -0.5$ 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.