Logarithmic speed-up of relaxation in $A\text{−}B$ annihilation with exclusion

We show that the decay of the density of active particles in the reaction $A+B\to 0$ 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 ($ABABAB\cdots$). We show that the decay for periodic i.c.'s made of longer blocks ($A^n{B}^n{A}^n{B}^n\cdots$) 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 $t^{-1/2}{\left[\ln \left(t\right)\right]}^{-1}$ 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 $t^{-1/2}{\left[\ln \left(t\right)\right]}^{-1}$ regime to eventual $t^{-1/2}$ behavior for i.c.'s made of mixtures of odd- and even-length blocks.

Figure 1

A plot of $1/\left[N\left(t\right)\sqrt{t}\right]$ (arbitrary units) against $\ln \left(t\right)$, which asymptotes to either a straight line with a positive slope (even $n$) or to a constant (odd $n$). A pure power law decay is plotted for comparison; note the upward curvature of the line. For a lattice size $L=2\times {10}^7$, averaged over 500 realizations.

Figure 2

A plot of $1/\left[N\left(t\right)\sqrt{t}\right]$ (arbitrary units) against $\ln \left(t\right)$, for initial conditions made of the units ${\left(AB\right)}^m{\left(AABB\right)}^n$, denoted by $m:n$. The results exhibit a crossover between intermediate-time $t^{-1/2}/\ln \left(t\right)$ behavior and eventual $t^{-1/2}$ behavior, with crossover time depending on the ratio $n/m$. For a lattice size $L={10}^7$, averaged over 100 realizations.