Logarithmic speed-up of relaxation in A-B annihilation with exclusion

TL;DR

This paper demonstrates that in a one-dimensional A-B annihilation process with exclusion, periodic initial conditions induce logarithmic corrections to the decay rate of active particles, revealing new slow relaxation dynamics.

Contribution

It introduces a detailed analysis of how periodic initial conditions affect decay laws, showing a logarithmic correction for even-length blocks using Monte Carlo simulations and a coarsening model.

Findings

Decay follows t^{-1/2}(log t)^{-1} for even-length blocks

Decay follows t^{-1/2} for odd-length blocks

Crossover behavior observed in mixed initial conditions

Abstract

We show that the decay of the density of active particles in the reaction $A+B \rightarrow 0$ in one dimension, with exclusion interaction, results in logarithmic corrections to the expected power law decay, when the starting initial condition (i.c.) is periodic. It is well-known that the late-time density of surviving particles goes as $t^{-1/4}$ with random initial conditions, and as $t^{-1/2}$ with alternating initial conditions ($ABABAB$...). We show that the decay for periodic i.c.s made of longer blocks ($A^{n}B^{n}A^{n}B^{n}$...) do not show a pure power-law decay when $n$ is even. By means of first-passage Monte Carlo simulations, and a mapping to a q-state coarsening model which can be solved in the Independent Interval Approximation (IIA), we show that the late-time decay of the density of surviving particles goes as $t^{-1/2}(\log{(t)})^{-1}$ for $n$ even, but as $t^{-1/2}$ when $n$ is odd. We relate this kinetic symmetry breaking in the Glauber Ising model. We also see a very slow crossover from a $t^{-1/2}(\log{(t)})^{-1}$ regime to eventual $t^{-1/2}$ behaviour for i.c.s made of mixtures of odd- and even-length blocks.