Kinetic and Related Models (KRM)

Kinetic theory and numerical simulations of two-species coagulation

Pages: 253 - 290, Volume 7, Issue 2, June 2014      doi:10.3934/krm.2014.7.253

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Carlos Escudero - Departamento de Matemáticas & ICMAT (CSIC-UAM-UC3M-UCM), Universidad Autónoma de Madrid, Ciudad Universitaria de Cantoblanco, 28049 Madrid, Spain (email)
Fabricio Macià - Universidad Politécnica de Madrid, ETSI Navales, Avda. Arco de la Victoria s/n, 28040 Madrid, Spain (email)
Raúl Toral - IFISC (Instituto de Física Interdisciplinar y Sistemas Complejos), CSIC-UIB, Campus UIB, 07122 Palma de Mallorca, Spain (email)
Juan J. L. Velázquez - Hausdorff Center for Mathematics, Rheinischen Friedrich-Wilhelms-Universität Bonn, 53115 Bonn, Germany (email)

Abstract: 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.

Keywords:  Smoluchowsky equations, self-similar asymptotics, coagulation, generating functions, numerical experiments.
Mathematics Subject Classification:  Primary: 35C20, 35Q20; Secondary: 82C22, 82C80.

Received: March 2013;      Revised: November 2013;      Available Online: March 2014.