Kinetic theory and numerical simulations of two-species coagulation

Abstract

In this work we study the stochastic process of two-speciescoagulation. This process consists in the aggregation dynamicstaking place in a ring. Particles and clusters of particles are setin this ring and they can move either clockwise or counterclockwise.They have a probability to aggregate forming larger clusters whenthey collide with another particle or cluster. We study thestochastic process both analytically and numerically. Analytically,we derive a kinetic theory which approximately describes the processdynamics. One of our strongest assumptions in this respectis the so called well--stirred limit, that allows neglecting theappearance of spatial coordinates in the theory, so this becomeseffectively reduced to a zeroth dimensional model.We determine the long time behavior of such a model, making emphasisin one special case in which it displays self-similar solutions.In particular these calculationsanswer the question of how the system gets ordered, with allparticles and clusters moving in the same direction, in the longtime. We compare our analytical results with direct numericalsimulations of the stochastic process and both corroborate itspredictions and check its limitations. In particular, we numericallyconfirm the ordering dynamics predicted by the kinetic theory andexplore properties of the realizations of the stochastic processwhich are not accessible to our theoretical approach.