Kinetic theory and numerical simulations of two-species coagulation

In this work we study the stochastic process of two-species
coagulation. This process consists in the aggregation dynamics
taking place in a ring. Particles and clusters of particles are set
in this ring and they can move either clockwise or counterclockwise.
They have a probability to aggregate forming larger clusters when
they collide with another particle or cluster. We study the
stochastic process both analytically and numerically. Analytically,
we derive a kinetic theory which approximately describes the process
dynamics. One of our strongest assumptions in this respect
is the so called well--stirred limit, that allows neglecting the
appearance of spatial coordinates in the theory, so this becomes
effectively reduced to a zeroth dimensional model.
We determine the long time behavior of such a model, making emphasis
in one special case in which it displays self-similar solutions.
In particular these calculations
answer the question of how the system gets ordered, with all
particles and clusters moving in the same direction, in the long
time. We compare our analytical results with direct numerical
simulations of the stochastic process and both corroborate its
predictions and check its limitations. In particular, we numerically
confirm the ordering dynamics predicted by the kinetic theory and
explore properties of the realizations of the stochastic process
which are not accessible to our theoretical approach.