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

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

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