Tagged particle dynamics in one dimensional ${A+ A \to kA}$ models with the particles biased to diffuse towards their nearest neighbour

Abstract

Dynamical features of tagged particles are studied in a one dimensional A + A → kA system for k = 0 and 1, where the particles A have a bias ǫ (0 ≤ ǫ ≤ 0.5) to hop one step in the direction of their nearest neighboring particle. ǫ = 0 represents purely diffusive motion and ǫ = 0.5 represents purely deterministic motion of the particles. We show that for any ǫ, there is a time scale t * which demarcates the dynamics of the particles. Below t * , the dynamics are governed by the annihilation of the particles, and the particle motions are highly correlated, while for t ≫ t * , the particles move as independent biased walkers. t * diverges as (ǫc − ǫ) −γ , where γ = 1 and ǫc = 0.5. ǫc is a critical point of the dynamics. At ǫc, the probability S(t), that a walker changes direction of its path at time t, decays as S(t) ∼ t −1 and the distribution D(τ ) of the time interval τ between consecutive changes in the direction of a typical walker decays with a power law as D(τ ) ∼ τ −2 .

$A+ A \to \emptyset$ reaction for particles with a dynamic bias to move away from their nearest neighbour in one dimension

“…The annihilation process is not affected by the bias but this extension leads to drastic changes in bulk dynamical properties. In a previous work [5], tagged particle dynamics have been reported in the one dimensional A + A → ∅ system where the particle A diffuses towards its nearest neighbour with a probability 0.5 + ǫ (0 < ǫ ≤ 0.5) and otherwise in the opposite direction [5]. To generalize the problem, in the present paper, the results for a negative bias are reported, i.e., when ǫ < 0.…”

“…For purely diffusive system (ǫ = 0), S(t) is independent of time, S(t) = p 0 . p 0 turns out to be ∼ 0.27 numerically with the updating rule used here [5].…”

“…As asynchronous dynamics have been used, the net displacement of a particle can be zero or more than one after a MC step [5]. The studies were performed on lattices of maximum size L = 24000 and the maximum number of configurations taken is 2000.…”

“…As for ǫ < 0 at large time S(t) is a constant, D(τ ) is expected to show an exponential decay [5]. Therefore, the tail of the distribution D(τ ) is fitted according to…”

“…One can assume that the particles A occupy the sites of a lattice and at each time step they hop to a nearest neighbouring site. The A + A → ∅ system has been studied in the recent past where the particles A move with a bias towards their nearest neighbours [3][4][5] in one dimension. The model, in its deterministic limit, maps to a opinion dynamics model studied earlier [6].…”