A Plus A -> Empty Set System In One Dimension With Particle Motion Determined By Nearest Neighbour Distances: Results For Parallel Updates

A one dimensional A +A -> empty set 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 + epsilon towards their nearest neighbours with -0.5 <= epsilon <= 0.5. This implies a stochastic motion towards the nearest neighbour or away from it for positive and negative values of epsilon respectively, with epsilon = +/- 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 epsilon = 0. The decay of particle density shows departure from the usual power law behaviour as found in epsilon = 0, on both sides of epsilon = 0. The persistence probability P(t) is also calculated that shows a power law decay, P(t) proportional to t(-theta), for epsilon = 0, where theta approximate to 0.75, twice of what is obtained in asynchronous updating. For epsilon < 0, P(t) decays in a stretched exponential manner and switches over to a behaviour compatible with P(t) proportional to t(-theta) In t for epsilon > 0. The epsilon = 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 epsilon = 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 epsilon > 0. For the tagged particle, the probability distribution Pi(x, t) that the particle has undergone a displacement x at time t shows the existence of a scaling variable x/t(v) where v = 0.55 +/- 0.05 for epsilon > 0 and varies with epsilon for epsilon < 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 epsilon > 0 while the results are qualitatively similar for epsilon < 0. (C) 2021 Elsevier B.V. All rights reserved.