# Reaction Diffusion Models in One Dimension with Disorder

### Pierre Le Doussal 1, Cecile Monthus 2, 3

#### Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 60 (1999) 1212-1238

We study a large class of 1D reaction diffusion models with quenched disorder using a real space renormalization group method (RSRG) which yields exact results at large time. Particles (e.g. of several species) undergo diffusion with random local bias (Sinai model) and react upon meeting. We obtain the large time decay of the density of each specie, their associated universal amplitudes, and the spatial distribution of particles. We also derive the spectrum of exponents which characterize the convergence towards the asymptotic states. For reactions with several asymptotic states, we analyze the dynamical phase diagram and obtain the critical exponents at the transitions. We also study persistence properties for single particles and for patterns. We compute the decay exponents for the probability of no crossing of a given point by, respectively, the single particle trajectories ($\\theta$) or the thermally averaged packets ($\\bar{\\theta}$). The generalized persistence exponents associated to n crossings are also obtained. Specifying to the process $A+A \\to \\emptyset$ or A with probabilities $(r,1-r)$, we compute exactly the exponents $\\delta(r)$ and $\\psi(r)$ characterizing the survival up to time t of a domain without any merging or with mergings respectively, and $\\delta_A(r)$ and $\\psi_A(r)$ characterizing the survival up to time t of a particle A without any coalescence or with coalescences respectively. $\\bar{\\theta}, \\psi, \\delta$ obey hypergeometric equations and are numerically surprisingly close to pure system exponents (though associated to a completely different diffusion length). Additional disorder in the reaction rates, as well as some open questions, are also discussed.

• 1. Laboratoire de Physique Théorique de l'ENS (LPTENS),
CNRS : UMR8549 – Université Paris VI - Pierre et Marie Curie – Ecole Normale Supérieure de Paris - ENS Paris
• 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
CNRS : UMR8626 – Université Paris XI - Paris Sud
• 3. Division de Physique Théorique, IPN,
Université Paris XI - Paris Sud