Reaction Diffusion and Ballistic Annihilation Near an Impenetrable Boundary

TL;DR

This paper investigates how an impenetrable boundary affects the density profiles of reacting particles undergoing diffusion or ballistic motion, revealing contrasting effects and universal behaviors in different dimensions.

Contribution

It introduces a field-theoretic renormalization group analysis for reaction-diffusion systems near boundaries and provides exact solutions for ballistic annihilation in one dimension.

Findings

Diffusion causes a density excess near the boundary.

Ballistic annihilation results in a density deficit at the boundary.

Exact solutions are obtained for one-dimensional cases in both diffusion and ballistic scenarios.

Abstract

The behavior of the single-species reaction process $A+A\to O$ is examined near an impenetrable boundary, representing the flask containing the reactants. Two types of dynamics are considered for the reactants: diffusive and ballistic propagation. It is shown that the effect of the boundary is quite different in both cases: diffusion-reaction leads to a density excess, whereas ballistic annihilation exhibits a density deficit, and in both cases the effect is not localized at the boundary but penetrates into the system. The field-theoretic renormalization group is used to obtain the universal properties of the density excess in two dimensions and below for the reaction-diffusion system. In one dimension the excess decays with the same exponent as the bulk and is found by an exact solution. In two dimensions the excess is marginally less relevant than the bulk decay and the density profile is again found exactly for late times from the RG-improved field theory. The results obtained for the diffusive case are relevant for Mg$^{2+}$ or Cd$^{2+}$ doping in the TMMC crystal's exciton coalescence process and also imply a surprising result for the dynamic magnetization in the critical one-dimensional Ising model with a fixed spin. For the case of ballistic reactants, a model is introduced and solved exactly in one dimension. The density-deficit profile is obtained, as is the density of left and right moving reactants near the impenetrable boundary.