Continuously varying exponents in $A+B \to 0$ reaction with long-ranged attractive interaction

TL;DR

This paper studies how long-range attractive interactions affect the reaction kinetics in a one-dimensional $A+B o 0$ system, revealing continuously varying exponents and crossover behaviors depending on the interaction range.

Contribution

It analytically and numerically demonstrates that dynamical exponents vary continuously with interaction range parameter $\sigma$, identifying two critical crossover points and recovering diffusive behavior beyond them.

Findings

Dynamical exponents for particle density vary with $\sigma$ when $\sigma < 1/2$.

Exponents for domain size vary with $\sigma$ when $\sigma < 7/6$.

Beyond critical $\sigma$, diffusive behavior dominates, matching standard diffusion results.

Abstract

We investigate the kinetics of the $A+B \to 0$ reaction with long-range attractive interaction $V(r) \sim -r^{-2\sigma}$ between $A$ and $B$ or with the drift velocity $v \sim r^{-\sigma}$ in one dimension, where $r$ is the closest distance between $A$ and $B$. It is analytically show that the dynamical exponents for density of particles ($\rho$) and the size of domains ($\ell$) continuously vary with $\sigma$ when $\sigma < \sigma_c =/1/2$, while that for the distance between adjacent opposite species ($\ell_{AB}$) varies when $\sigma < \sigma_c^{AB}= 7/6$. Beyond $\sigma_c^{AB}$, diffusive motions dominate the kinetics, so that the dynamical behavior for diffusive systems is completely recovered. These anomalous behaviors with the two crossover values of $\sigma$ are supported by numerical simulations and the argument of effective repulsion between the opposite species domains.