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

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 ($A+A\to\emptyset$) with particles input ($\emptyset\to A$) at a rate $J$. For dimension $d\leq 2$, 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 $c(t,J,D) = (J/D)^{1/δ}{\cal F}[(J/D)^Δ Dt]$, where $D$ is the diffusion constant. The critical exponent $δ$ and $Δ$ are computed to all orders in $ε=2-d$, where $d$ is the dimension of the system, while the scaling function $\cal F$ is computed to second order in $ε$. For the one-dimensional case an exact analytical solution is provided which predictions are compared with the results of the renormalization group approach for $ε=1$.