# American Institute of Mathematical Sciences

June  2014, 7(2): 253-290. doi: 10.3934/krm.2014.7.253

## Kinetic theory and numerical simulations of two-species coagulation

 1 Departamento de Matemáticas & ICMAT (CSIC-UAM-UC3M-UCM), Universidad Autónoma de Madrid, Ciudad Universitaria de Cantoblanco, 28049 Madrid, Spain 2 Universidad Politécnica de Madrid, ETSI Navales, Avda. Arco de la Victoria s/n, 28040 Madrid, Spain 3 IFISC (Instituto de Física Interdisciplinar y Sistemas Complejos), CSIC-UIB, Campus UIB, 07122 Palma de Mallorca, Spain 4 Hausdorff Center for Mathematics, Rheinischen Friedrich-Wilhelms-Universität Bonn, 53115 Bonn, Germany

Received  March 2013 Revised  November 2013 Published  March 2014

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.
Citation: Carlos Escudero, Fabricio Macià, Raúl Toral, Juan J. L. Velázquez. Kinetic theory and numerical simulations of two-species coagulation. Kinetic & Related Models, 2014, 7 (2) : 253-290. doi: 10.3934/krm.2014.7.253
##### References:
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Phys., 81 (2009), 591. doi: 10.1103/RevModPhys.81.591. Google Scholar [7] P. Clifford and A. Sudbury, A model for spatial conflict,, Biometrika, 60 (1973), 581. doi: 10.1093/biomet/60.3.581. Google Scholar [8] M. Conti, B. Meerson, A. Peleg and P. V. Sasorov, Phase ordering with a global conservation law: Ostwald ripening and coalescence,, Phys. Rev. E, 65 (2002). doi: 10.1103/PhysRevE.65.046117. Google Scholar [9] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the Lambert $W$ function,, Adv. Comput. Math., 5 (1996), 329. doi: 10.1007/BF02124750. Google Scholar [10] A. Czirók, A.-L. Barabási and T. Vicsek, Collective motion of self-propelled particles: Kinetic phase transition in one dimension,, Phys. Rev. Lett., 82 (1999), 209. Google Scholar [11] M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi and L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability and collapse,, Phys. Rev. Lett., 96 (2006). doi: 10.1103/PhysRevLett.96.104302. Google Scholar [12] G. Deffuant, D. Neu, F. Amblard and G. Weisbuch, Mixing beliefs among interacting agents,, Adv. Complex Syst., 3 (2000), 87. doi: 10.1142/S0219525900000078. Google Scholar [13] C. Escudero, F. Macià and J. J. L. Velázquez, Two-species coagulation approach to consensus by group level interactions,, Phys. Rev. E, 82 (2010). doi: 10.1103/PhysRevE.82.016113. Google Scholar [14] C. Escudero, C. A. Yates, J. Buhl, I. D. Couzin, R. Erban, I. G. Kevrekidis and P. K. Maini, Ergodic directional switching in mobile insect groups,, Phys. Rev. E, 82 (2010). doi: 10.1103/PhysRevE.82.011926. Google Scholar [15] O. Al Hammal, H. Chaté, I. Dornic and M. A. Muñoz, Langevin description of critical phenomena with two symmetric absorbing states,, Phys. Rev. Lett., 94 (2005). Google Scholar [16] E. Hernández-García and C. López, Clustering, advection and patterns in a model of population dynamics,, Phys. Rev. E, 70 (2004). Google Scholar [17] R. A. Holley and T. M. Liggett, Ergodic theorems for weakly interacting infinite systems and the voter model,, Ann. Probab., 3 (1975), 573. doi: 10.1214/aop/1176996306. Google Scholar [18] C. Huepe and M. Aldana, Intermittency and clustering in a system of self-driven particles,, Phys. Rev. Lett., 92 (2004). doi: 10.1103/PhysRevLett.92.168701. Google Scholar [19] S. Janson, D. E. Knuth, T. Luczak and B. Pittel, The birth of the giant component,, Rand. Struct. Alg., 4 (1993), 233. doi: 10.1002/rsa.3240040303. Google Scholar [20] M. Kreer and O. Penrose, Proof of dynamical scaling in Smoluchowski's coagulation equation with constant kernel,, J. Stat. Phys., 75 (1994), 389. doi: 10.1007/BF02186868. Google Scholar [21] I. M. Lifshitz and V. V. Slyozov, The kinetics of precipitation from supersaturated solid solutions,, J. Phys. Chem. Solids, 19 (1961), 35. doi: 10.1016/0022-3697(61)90054-3. Google Scholar [22] T. M. Liggett, Interacting Particle Systems,, Springer-Verlag, (1985). doi: 10.1007/978-1-4613-8542-4. Google Scholar [23] J. B. McLeod, On the scalar transport equation,, Proc. London Math. Soc., 14 (1964), 445. Google Scholar [24] G. Menon and R. L. Pego, Approach to self-similarity in Smoluchowski's coagulation equations,, Commun. Pure Appl. Math., 57 (2004), 1197. doi: 10.1002/cpa.3048. Google Scholar [25] H. S. Niwa, School size statistics of fish,, J. Theor. Biol., 195 (1998), 351. doi: 10.1006/jtbi.1998.0801. Google Scholar [26] F. Peruani, A. Deutsch and M. Bär, Nonequilibrium clustering of self-propelled rods,, Phys. Rev. E, 74 (2006). doi: 10.1103/PhysRevE.74.030904. Google Scholar [27] M. Pineda, R. Toral and E. Hernández-García, Noisy continuous-opinion dynamics,, J. Stat. Mech., (2009). doi: 10.1088/1742-5468/2009/08/P08001. Google Scholar [28] J. Seinfeld, Atmospheric Chemistry and Physics of Air Polution,, Wiley, (1986). Google Scholar [29] J. Silk and S. D. White, The development of structure in the expanding universe,, Astrophys. J., 223 (1978). doi: 10.1086/182728. Google Scholar [30] T. Sintes, R. Toral and A. Chakrabarti, Reversible aggregation in self-associating polymer systems,, Phys. Rev. E, 50 (1994), 2967. doi: 10.1103/PhysRevE.50.2967. Google Scholar [31] R. Toral and J. Marro, Cluster kinetics in the lattice gas model: The Becker-Doring type of equations,, J. Phys. C: Solid State Phys., 20 (1987), 2491. doi: 10.1088/0022-3719/20/17/004. Google Scholar [32] R. Toral and C. J. Tessone, Finite size effects in the dynamics of opinion formation,, Commun. Comput. Phys., 2 (2007), 177. Google Scholar [33] F. Vázquez and C. López, Systems with two symmetric absorbing states: relating the microscopic dynamics with the macroscopic behavior,, Phys. Rev. E, 78 (2008). Google Scholar [34] C. A. Yates, R. Erban, C. Escudero, I. D. Couzin, J. Buhl, I. G. Kevrekidis, P. K. Maini and D. J. T. Sumpter, Inherent noise can facilitate coherence in collective swarm motion,, Proc. Nat. Acad. Sci. USA, 106 (2009), 5464. doi: 10.1073/pnas.0811195106. Google Scholar [35] R. M. Ziff, Kinetics of polymerization,, J. Stat. Phys., 23 (1980), 241. doi: 10.1007/BF01012594. Google Scholar

show all references

##### References:
 [1] G. Arfken, Mathematical Methods for Physicists,, 3rd edition, (1985). Google Scholar [2] J. M. Ball and J. Carr, The discrete coagulation-fragmentation equations: Existence, uniqueness, and density conservation,, J. Stat. Phys., 61 (1990), 203. doi: 10.1007/BF01013961. Google Scholar [3] M. Bodnar and J. J. L. Velázquez, An integro-differential equation arising as a limit of individual cell-based models,, J. Diff. Eqs., 222 (2006), 341. doi: 10.1016/j.jde.2005.07.025. Google Scholar [4] J. Buhl, D. J. T. Sumpter, I. D. Couzin, J. J. Hale, E. Despland, E. R. Miller and S. J. Simpson, From disorder to order in marching locusts,, Science, 312 (2006), 1402. doi: 10.1126/science.1125142. Google Scholar [5] J. A. Carrillo, M. R. D'Orsogna and V. Panferov, Double milling in self-propelled swarms from kinetic theory,, Kin. Rel. Mod., 2 (2009), 363. doi: 10.3934/krm.2009.2.363. Google Scholar [6] C. Castellano, S. Fortunato and V. Loreto, Statistical physics of social dynamics,, Rev. Mod. Phys., 81 (2009), 591. doi: 10.1103/RevModPhys.81.591. Google Scholar [7] P. Clifford and A. Sudbury, A model for spatial conflict,, Biometrika, 60 (1973), 581. doi: 10.1093/biomet/60.3.581. Google Scholar [8] M. Conti, B. Meerson, A. Peleg and P. V. Sasorov, Phase ordering with a global conservation law: Ostwald ripening and coalescence,, Phys. Rev. E, 65 (2002). doi: 10.1103/PhysRevE.65.046117. Google Scholar [9] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the Lambert $W$ function,, Adv. Comput. Math., 5 (1996), 329. doi: 10.1007/BF02124750. Google Scholar [10] A. Czirók, A.-L. Barabási and T. Vicsek, Collective motion of self-propelled particles: Kinetic phase transition in one dimension,, Phys. Rev. Lett., 82 (1999), 209. Google Scholar [11] M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi and L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability and collapse,, Phys. Rev. Lett., 96 (2006). doi: 10.1103/PhysRevLett.96.104302. Google Scholar [12] G. Deffuant, D. Neu, F. Amblard and G. Weisbuch, Mixing beliefs among interacting agents,, Adv. Complex Syst., 3 (2000), 87. doi: 10.1142/S0219525900000078. Google Scholar [13] C. Escudero, F. Macià and J. J. L. Velázquez, Two-species coagulation approach to consensus by group level interactions,, Phys. Rev. E, 82 (2010). doi: 10.1103/PhysRevE.82.016113. Google Scholar [14] C. Escudero, C. A. Yates, J. Buhl, I. D. Couzin, R. Erban, I. G. Kevrekidis and P. K. Maini, Ergodic directional switching in mobile insect groups,, Phys. Rev. E, 82 (2010). doi: 10.1103/PhysRevE.82.011926. Google Scholar [15] O. Al Hammal, H. Chaté, I. Dornic and M. A. Muñoz, Langevin description of critical phenomena with two symmetric absorbing states,, Phys. Rev. Lett., 94 (2005). Google Scholar [16] E. Hernández-García and C. López, Clustering, advection and patterns in a model of population dynamics,, Phys. Rev. E, 70 (2004). Google Scholar [17] R. A. Holley and T. M. Liggett, Ergodic theorems for weakly interacting infinite systems and the voter model,, Ann. Probab., 3 (1975), 573. doi: 10.1214/aop/1176996306. Google Scholar [18] C. Huepe and M. Aldana, Intermittency and clustering in a system of self-driven particles,, Phys. Rev. Lett., 92 (2004). doi: 10.1103/PhysRevLett.92.168701. Google Scholar [19] S. Janson, D. E. Knuth, T. Luczak and B. Pittel, The birth of the giant component,, Rand. Struct. Alg., 4 (1993), 233. doi: 10.1002/rsa.3240040303. Google Scholar [20] M. Kreer and O. Penrose, Proof of dynamical scaling in Smoluchowski's coagulation equation with constant kernel,, J. Stat. Phys., 75 (1994), 389. doi: 10.1007/BF02186868. Google Scholar [21] I. M. Lifshitz and V. V. Slyozov, The kinetics of precipitation from supersaturated solid solutions,, J. Phys. Chem. Solids, 19 (1961), 35. doi: 10.1016/0022-3697(61)90054-3. Google Scholar [22] T. M. Liggett, Interacting Particle Systems,, Springer-Verlag, (1985). doi: 10.1007/978-1-4613-8542-4. Google Scholar [23] J. B. McLeod, On the scalar transport equation,, Proc. London Math. Soc., 14 (1964), 445. Google Scholar [24] G. Menon and R. L. Pego, Approach to self-similarity in Smoluchowski's coagulation equations,, Commun. Pure Appl. Math., 57 (2004), 1197. doi: 10.1002/cpa.3048. Google Scholar [25] H. S. Niwa, School size statistics of fish,, J. Theor. Biol., 195 (1998), 351. doi: 10.1006/jtbi.1998.0801. Google Scholar [26] F. Peruani, A. Deutsch and M. Bär, Nonequilibrium clustering of self-propelled rods,, Phys. Rev. E, 74 (2006). doi: 10.1103/PhysRevE.74.030904. Google Scholar [27] M. Pineda, R. Toral and E. Hernández-García, Noisy continuous-opinion dynamics,, J. Stat. Mech., (2009). doi: 10.1088/1742-5468/2009/08/P08001. Google Scholar [28] J. Seinfeld, Atmospheric Chemistry and Physics of Air Polution,, Wiley, (1986). Google Scholar [29] J. Silk and S. D. White, The development of structure in the expanding universe,, Astrophys. J., 223 (1978). doi: 10.1086/182728. Google Scholar [30] T. Sintes, R. Toral and A. Chakrabarti, Reversible aggregation in self-associating polymer systems,, Phys. Rev. E, 50 (1994), 2967. doi: 10.1103/PhysRevE.50.2967. Google Scholar [31] R. Toral and J. Marro, Cluster kinetics in the lattice gas model: The Becker-Doring type of equations,, J. Phys. C: Solid State Phys., 20 (1987), 2491. doi: 10.1088/0022-3719/20/17/004. Google Scholar [32] R. Toral and C. J. Tessone, Finite size effects in the dynamics of opinion formation,, Commun. Comput. Phys., 2 (2007), 177. Google Scholar [33] F. Vázquez and C. López, Systems with two symmetric absorbing states: relating the microscopic dynamics with the macroscopic behavior,, Phys. Rev. E, 78 (2008). Google Scholar [34] C. A. Yates, R. Erban, C. Escudero, I. D. Couzin, J. Buhl, I. G. Kevrekidis, P. K. Maini and D. J. T. Sumpter, Inherent noise can facilitate coherence in collective swarm motion,, Proc. Nat. Acad. Sci. USA, 106 (2009), 5464. doi: 10.1073/pnas.0811195106. Google Scholar [35] R. M. Ziff, Kinetics of polymerization,, J. Stat. Phys., 23 (1980), 241. doi: 10.1007/BF01012594. Google Scholar
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