A renormalization group study of a class of reaction-diffusion model, with particles input

Abstract

We study a class of reaction-diffusion model extrapolating continuously between the pure coagulation-diffusion case ($A+A\to A$) and the pure annihilation-diffusion one ($A+A\to \varnothing$) with particles input ($\varnothing \to A$) at a rate $J$. For dimension $d\le 2$, the dynamics strongly depends on the fluctuations while, for $d>2$, the behaviour is mean-field like. The models are mapped onto a field theory which properties are studied in a renormalization group approach. Simple relations are found between the time-dependent correlation functions of the different models of the class. For the pure coagulation-diffusion model the time-dependent density is found to be of the form $c(t,J,D)=(J/D{)}^{1/\delta }\mathcal{F}[(J/D{)}^{\Delta }Dt]$, where $D$ is the diffusion constant. The critical exponent $\delta$ and $\Delta$ are computed to all orders in $\epsilon =2-d$, where $d$ is the dimension of the system, while the scaling function $\mathcal{F}$ is computed to second order in $\epsilon$. For the one-dimensional case an exact analytical solution is provided which predictions are compared with the results of the renormalization group approach for $\epsilon =1$.

Ten pages, using Latex and IOP macro. Two latex figures. Submitted to Journal of Physics A. Also available at http://mykonos.unige.ch/~rey/publi.html