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

## TL;DR

This study investigates the dynamics of biased particles in one-dimensional annihilation models, revealing a critical bias where particle behavior transitions from correlated to independent motion, with diverging time scales and power-law distributions at the critical point.

## Contribution

It introduces a critical bias point in one-dimensional $A+A
ightarrow kA$ models, characterizing the transition from correlated to independent particle dynamics and analyzing associated scaling laws.

## Key findings

- Existence of a critical bias $oldsymbol{oldsymbol{ extepsilon_c=0.5}}$ separating dynamic regimes.
- Divergence of time scale $oldsymbol{t^*}$ as $oldsymbol{( extepsilon_c- extepsilon)^{-1}}$ near the critical point.
- Power-law decay of direction change probability and interval distribution at $oldsymbol{ extepsilon_c}$.

## Abstract

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 $\epsilon$ $(0 \leq \epsilon \leq 0.5)$ 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 $t^*$ 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 $(\epsilon_c-\epsilon)^{-\gamma}$, 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 $D(\tau) \sim \tau^{-2}$.