A + A → 0̸ system in one dimension with particle motion determined by nearest neighbour distances: Results for parallel updates

Abstract A one dimensional A + A → 0 system where the direction of motion of the particles is determined by the position of the nearest neighbours is studied. The particles move with a probability 0 . 5 + ϵ towards their nearest neighbours with − 0 . 5 ≤ ϵ ≤ 0 . 5 . This implies a stochastic motion towards the nearest neighbour or away from it for positive and negative values of ϵ respectively, with ϵ = ± 0 . 5 the two deterministic limits. The position of the particles are updated in parallel. The macroscopic as well as tagged particle dynamics are studied which show drastic changes from the diffusive case ϵ = 0 . The decay of particle density shows departure from the usual power law behaviour as found in ϵ = 0 , on both sides of ϵ = 0 . The persistence probability P ( t ) is also calculated that shows a power law decay, P ( t ) ∝ t − θ , for ϵ = 0 , where θ ≈ 0 . 75 , twice of what is obtained in asynchronous updating. For ϵ 0 , P ( t ) decays in a stretched exponential manner and switches over to a behaviour compatible with P ( t ) ∝ t − θ ln t for ϵ > 0 . The ϵ = 0 . 5 point is characterised by the presence of permanent dimers, which are isolated pairs of particles on adjacent sites. Under the parallel dynamics and for attractive interaction these particles may go on swapping their positions for a long time, in particular, for ϵ = 0 . 5 these may survive permanently. Interestingly, for a chosen special initial condition that inhibits the formation of dimers, one recovers the asynchronous behaviour, manifesting the role of the dimers in altering the scaling behaviour for ϵ > 0 . For the tagged particle, the probability distribution Π ( x , t ) that the particle has undergone a displacement x at time t shows the existence of a scaling variable x ∕ t ν where ν = 0 . 55 ± 0 . 05 for ϵ > 0 and varies with ϵ for ϵ 0 . Finally, a comparative analysis for the behaviour of all the relevant quantities for the system using parallel and asynchronous dynamics (studied recently) shows that there are significant differences for ϵ > 0 while the results are qualitatively similar for ϵ 0 .