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

TL;DR

This paper employs renormalization group techniques to analyze reaction-diffusion models interpolating between coagulation and annihilation processes with particle input, revealing dimension-dependent behaviors and deriving critical exponents.

Contribution

It introduces a unified field theory framework for reaction-diffusion models with input, calculating critical exponents to all orders in epsilon expansion and providing exact solutions in one dimension.

Findings

Dimension d ≤ 2 shows fluctuation-dominated dynamics.

For d > 2, behavior is mean-field like.

Exact solution for 1D case matches RG predictions.

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 ($A+A\to\emptyset$) 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 $c(t,J,D) = (J/D)^{1/\delta}{\cal F}[(J/D)^{\Delta} Dt]$, 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 $\cal F$ 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 $\epsilon=1$.