Abstract
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 and determine its asymptotic behavior. In particular we answer the question of how the system gets ordered, with all particles and clusters moving in the same direction, in the long time.
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The theoretical study of coagulation and its kinetic description is of broad interest because of its vast applicability in diverse topics such as aerosols [20], polymerization [22, 27], Ostwald ripening [5, 15], galaxies and stars clustering [21], and population biology [17] among many others. We propose a generalization of the stochastic process of coagulation. We consider the coagulation process among two different species: the aggregation takes place only when one element of one of the species interacts with an element of the other species. In particular, we place particles and clusters of particles in a ring, where they move with constant speed, either clockwise or counterclockwise. When two clusters (or two particles or one particle and one cluster) meet they have the chance to aggregate and form a cluster containing all particles involved in the collision. The direction of motion of the newborn cluster is chosen following certain probabilistic rules. We are interested in the properties of the realizations of such a stochastic process, and in particular in their long time behavior. Our main theoretical technique is the use of kinetic equations, an approach we have outlined in [10]. This is of course just one possible extension of the theory of coagulation. We have designed it getting inspiration from self-organizing systems and in particular from collective organism behavior. Let us note that this is a field that has been studied using a broad range of different theoretical techniques [2, 6, 7, 23, 25]. Another field which has inspired ourselves is the study of the dynamics of opinion formation and spreading [3, 8, 24], which is represented for instance by the classical voter model [4, 13, 16]. As a final influence, we mention that clustering has been previously studied in population dynamics models [12] including swarming systems [14], and coagulation equations have been used in both swarming [18] and opinion formation models [19]. Despite of its simplicity, the two-species coagulation model could be related to some of these systems.
A particular system that has influenced the current developments is the collective motion of locusts. The experiment performed in [1] revealed that locusts marching on a (quasi one dimensional) ring presented a coherent collective motion for high densities; low densities were characterized by a random behavior of the individuals and intermediate densities showed coherent displacements alternating with sudden changes of direction. The models that have been introduced to describe this experiment assume that the organisms behave like interacting particles [1, 11, 26]. Related interacting particle models have been used to describe the collective behavior of many different organisms and analyzing the mathematical properties of such models has been a very active research area [2, 6, 7]. The two-species coagulation model could be thought of as a particular limit of some of these models or as a simplified version of them which still retains some desirable features.
The goal of our current work is not to describe the detailed behavior of any specific system. Instead, we explore the mathematical properties of a stochastic process which has been designed by borrowing inspiration from different self-organizing systems. Therefore the focus is on mathematical tractability. We determine under which conditions consensus is reached and what form it adopts. We introduce the kinetic theory that approximately describes the stochastic process and concentrate on its mathematical analysis. We also study the system by means of direct numerical simulations of the stochastic process. We use them to check the predictions of our kinetic theory and explore the stochastic process beyond the kinetic level. Kinetic approximations neglect many sources of fluctuations and thus numerical simulations are required in order to describe many properties of the individual realizations of the stochastic process which are not reflected at the kinetic level.
Our analysis is limited to the one-dimensional spatial situation with periodic boundary conditions (the dynamics is taking place in a circumference). Our approach is based on coagulation equations. This kinetic description is in general invalid for one-dimensional systems because it is known that spatial correlations do propagate in this dimensionality. So we assume a collision takes place when two clusters meet with a very small probability. The probability should be so small that all the particles travel the whole system several times before one collision happens on average. This way the system becomes well-stirred, and so we can neglect spatial correlations and treat the system as if it were zero dimensional, what allows mathematical tractability. The content of this summary is based on the developments reported in a paper that is currently being considered for its possible publication [9].
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Escudero, C. (2013). Kinetic Theory of Two-Species Coagulation. In: Gilbert, T., Kirkilionis, M., Nicolis, G. (eds) Proceedings of the European Conference on Complex Systems 2012. Springer Proceedings in Complexity. Springer, Cham. https://doi.org/10.1007/978-3-319-00395-5_129
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DOI: https://doi.org/10.1007/978-3-319-00395-5_129
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