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:
[1]

G. Arfken, Mathematical Methods for Physicists,, 3rd edition, (1985).

[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.

[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.

[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.

[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.

[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.

[7]

P. Clifford and A. Sudbury, A model for spatial conflict,, Biometrika, 60 (1973), 581. doi: 10.1093/biomet/60.3.581.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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).

[16]

E. Hernández-García and C. López, Clustering, advection and patterns in a model of population dynamics,, Phys. Rev. E, 70 (2004).

[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.

[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.

[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.

[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.

[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.

[22]

T. M. Liggett, Interacting Particle Systems,, Springer-Verlag, (1985). doi: 10.1007/978-1-4613-8542-4.

[23]

J. B. McLeod, On the scalar transport equation,, Proc. London Math. Soc., 14 (1964), 445.

[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.

[25]

H. S. Niwa, School size statistics of fish,, J. Theor. Biol., 195 (1998), 351. doi: 10.1006/jtbi.1998.0801.

[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.

[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.

[28]

J. Seinfeld, Atmospheric Chemistry and Physics of Air Polution,, Wiley, (1986).

[29]

J. Silk and S. D. White, The development of structure in the expanding universe,, Astrophys. J., 223 (1978). doi: 10.1086/182728.

[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.

[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.

[32]

R. Toral and C. J. Tessone, Finite size effects in the dynamics of opinion formation,, Commun. Comput. Phys., 2 (2007), 177.

[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).

[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.

[35]

R. M. Ziff, Kinetics of polymerization,, J. Stat. Phys., 23 (1980), 241. doi: 10.1007/BF01012594.

show all references

References:
[1]

G. Arfken, Mathematical Methods for Physicists,, 3rd edition, (1985).

[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.

[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.

[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.

[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.

[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.

[7]

P. Clifford and A. Sudbury, A model for spatial conflict,, Biometrika, 60 (1973), 581. doi: 10.1093/biomet/60.3.581.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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).

[16]

E. Hernández-García and C. López, Clustering, advection and patterns in a model of population dynamics,, Phys. Rev. E, 70 (2004).

[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.

[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.

[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.

[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.

[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.

[22]

T. M. Liggett, Interacting Particle Systems,, Springer-Verlag, (1985). doi: 10.1007/978-1-4613-8542-4.

[23]

J. B. McLeod, On the scalar transport equation,, Proc. London Math. Soc., 14 (1964), 445.

[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.

[25]

H. S. Niwa, School size statistics of fish,, J. Theor. Biol., 195 (1998), 351. doi: 10.1006/jtbi.1998.0801.

[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.

[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.

[28]

J. Seinfeld, Atmospheric Chemistry and Physics of Air Polution,, Wiley, (1986).

[29]

J. Silk and S. D. White, The development of structure in the expanding universe,, Astrophys. J., 223 (1978). doi: 10.1086/182728.

[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.

[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.

[32]

R. Toral and C. J. Tessone, Finite size effects in the dynamics of opinion formation,, Commun. Comput. Phys., 2 (2007), 177.

[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).

[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.

[35]

R. M. Ziff, Kinetics of polymerization,, J. Stat. Phys., 23 (1980), 241. doi: 10.1007/BF01012594.

[1]

Qiaolin He. Numerical simulation and self-similar analysis of singular solutions of Prandtl equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 101-116. doi: 10.3934/dcdsb.2010.13.101

[2]

Weronika Biedrzycka, Marta Tyran-Kamińska. Self-similar solutions of fragmentation equations revisited. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 13-27. doi: 10.3934/dcdsb.2018002

[3]

F. Berezovskaya, G. Karev. Bifurcations of self-similar solutions of the Fokker-Plank equations. Conference Publications, 2005, 2005 (Special) : 91-99. doi: 10.3934/proc.2005.2005.91

[4]

Hyungjin Huh. Self-similar solutions to nonlinear Dirac equations and an application to nonuniqueness. Evolution Equations & Control Theory, 2018, 7 (1) : 53-60. doi: 10.3934/eect.2018003

[5]

Marco Cannone, Grzegorz Karch. On self-similar solutions to the homogeneous Boltzmann equation. Kinetic & Related Models, 2013, 6 (4) : 801-808. doi: 10.3934/krm.2013.6.801

[6]

Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637

[7]

Dongho Chae, Kyungkeun Kang, Jihoon Lee. Notes on the asymptotically self-similar singularities in the Euler and the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1181-1193. doi: 10.3934/dcds.2009.25.1181

[8]

Jochen Merker, Aleš Matas. Positivity of self-similar solutions of doubly nonlinear reaction-diffusion equations. Conference Publications, 2015, 2015 (special) : 817-825. doi: 10.3934/proc.2015.0817

[9]

Hideo Kubo, Kotaro Tsugawa. Global solutions and self-similar solutions of the coupled system of semilinear wave equations in three space dimensions. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 471-482. doi: 10.3934/dcds.2003.9.471

[10]

Zoran Grujić. Regularity of forward-in-time self-similar solutions to the 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 837-843. doi: 10.3934/dcds.2006.14.837

[11]

Rostislav Grigorchuk, Volodymyr Nekrashevych. Self-similar groups, operator algebras and Schur complement. Journal of Modern Dynamics, 2007, 1 (3) : 323-370. doi: 10.3934/jmd.2007.1.323

[12]

Christoph Bandt, Helena PeÑa. Polynomial approximation of self-similar measures and the spectrum of the transfer operator. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4611-4623. doi: 10.3934/dcds.2017198

[13]

Anna Chiara Lai, Paola Loreti. Self-similar control systems and applications to zygodactyl bird's foot. Networks & Heterogeneous Media, 2015, 10 (2) : 401-419. doi: 10.3934/nhm.2015.10.401

[14]

D. G. Aronson. Self-similar focusing in porous media: An explicit calculation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1685-1691. doi: 10.3934/dcdsb.2012.17.1685

[15]

G. A. Braga, Frederico Furtado, Vincenzo Isaia. Renormalization group calculation of asymptotically self-similar dynamics. Conference Publications, 2005, 2005 (Special) : 131-141. doi: 10.3934/proc.2005.2005.131

[16]

Bendong Lou. Self-similar solutions in a sector for a quasilinear parabolic equation. Networks & Heterogeneous Media, 2012, 7 (4) : 857-879. doi: 10.3934/nhm.2012.7.857

[17]

Shota Sato, Eiji Yanagida. Singular backward self-similar solutions of a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 897-906. doi: 10.3934/dcdss.2011.4.897

[18]

Shota Sato, Eiji Yanagida. Forward self-similar solution with a moving singularity for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 313-331. doi: 10.3934/dcds.2010.26.313

[19]

L. Olsen. Rates of convergence towards the boundary of a self-similar set. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 799-811. doi: 10.3934/dcds.2007.19.799

[20]

Marek Fila, Michael Winkler, Eiji Yanagida. Convergence to self-similar solutions for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 703-716. doi: 10.3934/dcds.2008.21.703

2017 Impact Factor: 1.219

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (1)

[Back to Top]