Effect of presence of rigid impurities in a system of annihilating domain walls with dynamic bias

The dynamics of interacting domain walls, regarded as a system of particles which are biased to move towards their nearest neighbours and annihilate when they meet, have been studied in the recent past. We study the effect of the presence of a fraction $r$ of quenched impurities (which act as rigid walkers) on the dynamics. Here, in case two domain walls or one impurity and one domain wall happen to be on the same site, both get simultaneously annihilated. It is found that for any non-zero value of $r$, the dynamical behaviour changes as the surviving fraction of particles $ρ(t)$ attains a constant value. $ρ(t)t^α$ shows a universal behaviour when plotted against $r^βt$ with $α, β$ values depending on whether the particles are rigid or nonrigid. Also, the values differ for the biased and unbiased cases. The time scale associated with the particle decay obtained in several ways shows that it varies with $r$ in a power law manner with a universal exponent.

💡 Research Summary

The paper investigates how a finite fraction r of immobile “rigid” impurities influences the dynamics of annihilating domain walls in a one‑dimensional Ising‑Glauber setting, where domain walls are mapped onto diffusing particles that annihilate upon encounter (A + A → ∅). In addition to the usual unbiased random walk, the particles may be subjected to a dynamic bias ε (0 ≤ ε ≤ 0.5) that makes them preferentially step toward their nearest neighbour; ε = 0 corresponds to pure diffusion, while ε = 0.5 gives deterministic motion toward the nearest neighbour.

The model is implemented on a lattice of size L = 2 × 10⁴ with periodic boundaries. Initially the lattice is half‑filled (ρ₀ = 0.5) with a mixture of mobile and rigid walkers; the rigid fraction is r, and the mobile walkers follow the biased hopping rule. Updates are asynchronous: a site is chosen at random, a mobile particle (if present) attempts a hop according to the bias, and if the destination site already contains any particle (rigid or mobile) both are removed. One Monte‑Carlo step consists of L such attempts. The authors monitor the total surviving particle density ρ(t), the densities of rigid (ρ_r) and non‑rigid (ρ_nr) particles, and the persistence probability P(t) (the fraction of sites never visited).

Key findings:

1. Crossover to Saturation – For any non‑zero r, the density ρ(t) initially follows the power‑law decay known for the pure system (ρ ∼ t^{-α}) but then crosses over to a constant plateau ρ_sat. The crossover occurs earlier and the plateau value is larger for larger r. This behavior is observed both for ε = 0 (unbiased) and ε > 0 (biased).

2. Universal Scaling Form – The authors demonstrate that the data collapse onto a single scaling function: \