Diffusion-Annihilation in the Presence of a Driving Field

TL;DR

This paper investigates how an external driving field influences a one-dimensional stochastic reaction-diffusion system, revealing that density fluctuations are unaffected by the drive and that its effects can be transformed away in the scaling regime.

Contribution

It demonstrates that the external driving field does not alter density fluctuations and that its effects can be absorbed via a Galilei transformation in the scaling regime.

Findings

Density fluctuations are independent of the driving field.

Local fluctuations do not depend on the driving field under certain averages.

The effect of the driving can be transformed away in the scaling regime.

Abstract

We study the effect of an external driving force on a simple stochastic reaction-diffusion system in one dimension. In our model each lattice site may be occupied by at most one particle. These particles hop with rates $(1\pm\eta)/2$ to the right and left nearest neighbouring site resp. if this site is vacant and annihilate with rate 1 if it is occupied. We show that density fluctuations (i.e. the $m^{th}$ moments $\langle N^m \rangle$ of the density distribution at time $t$) do not depend on the spatial anisotropy $\eta$ induced by the driving field, irrespective of the initial condition. Furthermore we show that if one takes certain translationally invariant averages over initial states (e.g. random initial conditions) even local fluctuations do not depend on $\eta$. In the scaling regime $t \sim L^2$ the effect of the driving can be completely absorbed in a Galilei transformation (for any initial condition). We compute the probability of finding a system of $L$ sites in its stationary state at time $t$ if it was fully occupied at time $t_0 = 0$.