Tagged particle dynamics in one dimensional $A+A\to kA$ models with the particles biased to diffuse towards their nearest neighbour

Abstract

Dynamical features of tagged particles are studied in a one dimensional $A+A\to kA$ system for $k=0$ and 1, where the particles $A$ have a bias $\epsilon$ $(0\le \epsilon \le 0.5)$ to hop one step in the direction of their nearest neighboring particle. $\epsilon =0$ represents purely diffusive motion and $\epsilon =0.5$ represents purely deterministic motion of the particles. We show that for any $\epsilon$, there is a time scale $t^{\ast }$ which demarcates the dynamics of the particles. Below $t^{\ast }$, the dynamics are governed by the annihilation of the particles, and the particle motions are highly correlated, while for $t\gg t^{\ast }$, the particles move as independent biased walkers. $t^{\ast }$ diverges as $({\epsilon }_c-\epsilon {)}^{-\gamma }$, where $\gamma =1$ and ${\epsilon }_c=0.5$. ${\epsilon }_c$ is a critical point of the dynamics. At ${\epsilon }_c$, the probability $S(t)$, that a walker changes direction of its path at time $t$, decays as $S(t)\sim t^{-1}$ and the distribution $D(\tau )$ of the time interval $\tau$ between consecutive changes in the direction of a typical walker decays with a power law as $D(\tau )\sim {\tau }^{-2}$.

8 pages, 8 figures. Version accepted for publication in J. Phys. A