A stochastic min-driven coalescence process and its hydrodynamical limit
Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 2, p. 329-357

A stochastic system of particles is considered in which the sizes of the particles increase by successive binary mergers with the constraint that each coagulation event involves a particle with minimal size. Convergence of a suitably renormalized version of this process to a deterministic hydrodynamical limit is shown and the time evolution of the minimal size is studied for both deterministic and stochastic models.

L'évolution d'un système aléatoire de particules est étudiée lorsque la taille des particules croît par coagulation binaire, chaque réaction de coagulation impliquant nécessairement une particule de taille minimale. Nous montrons qu'une version renormalisée du processus stochastique associé converge vers une limite déterministe et étudions l'évolution temporelle de la taille minimale pour les modèles stochastique et déterministe.

DOI : https://doi.org/10.1214/09-AIHP349
Classification:  82C22,  60K35,  60H10,  34A34,  34C11
Keywords: stochastic coalescence, min-driven clustering, hydrodynamical limit
@article{AIHPB_2011__47_2_329_0,
     author = {Basdevant, Anne-Laure and Lauren\c cot, Philippe and Norris, James R. and Rau, Cl\'ement},
     title = {A stochastic min-driven coalescence process and its hydrodynamical limit},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {47},
     number = {2},
     year = {2011},
     pages = {329-357},
     doi = {10.1214/09-AIHP349},
     zbl = {1216.82024},
     mrnumber = {2814413},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2011__47_2_329_0}
}
Basdevant, Anne-Laure; Laurençot, Philippe; Norris, James R.; Rau, Clément. A stochastic min-driven coalescence process and its hydrodynamical limit. Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 2, pp. 329-357. doi : 10.1214/09-AIHP349. http://www.numdam.org/item/AIHPB_2011__47_2_329_0/

[1] D. J. Aldous. Deterministic and stochastic models for coalescence (aggregation, coagulation): A review of the mean-field theory for probabilists. Bernoulli 5 (1999) 3-48. | MR 1673235 | Zbl 0930.60096

[2] J. M. Ball, J. Carr and O. Penrose. The Becker-Döring cluster equations: Basic properties and asymptotic behaviour of solutions. Comm. Math. Phys. 104 (1986) 657-692. | MR 841675 | Zbl 0594.58063

[3] J. Bertoin. Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102. Cambridge Univ. Press, Cambridge, 2006. | MR 2253162 | Zbl 1107.60002

[4] J. Carr and R. L. Pego. Self-similarity in a cut-and-paste model of coarsening. Proc. Roy. Soc. London A 456 (2000) 1281-1290. | MR 1809962 | Zbl 0978.82055

[5] R. W. R. Darling and J. R. Norris. Differential equation approximations for Markov chains. Probab. Surv. 5 (2008) 37-79. | MR 2395153 | Zbl 1189.60152

[6] C. Dellacherie and P.-A. Meyer. Probabilités et Potentiel, Chapters I and IV. Hermann, Paris, 1975. | MR 488194 | Zbl 0138.10402

[7] B. Derrida, C. Godrèche and I. Yekutieli. Scale-invariant regimes in one-dimensional models of growing and coalescing droplets. Phys. Rev. A 44 (1991) 6241-6251.

[8] M. Escobedo, S. Mischler and B. Perthame. Gelation in coagulation and fragmentation models. Comm. Math. Phys. 231 (2002) 157-188. | MR 1947695 | Zbl 1016.82027

[9] T. Gallay and A. Mielke. Convergence results for a coarsening model using global linearization. J. Nonlinear Sci. 13 (2003) 311-346. | MR 1982018 | Zbl 1025.35006

[10] I. Jeon. Existence of gelling solutions for coagulation-fragmentation equations. Comm. Math. Phys. 194 (1998) 541-567. | MR 1631473 | Zbl 0910.60083

[11] I. Jeon. Spouge's conjecture on complete and instantaneous gelation. J. Statist. Phys. 96 (1999) 1049-1070. | MR 1722986 | Zbl 0962.82046

[12] P. Laurençot. The Lifshitz-Slyozov equation with encounters. Math. Models Methods Appl. Sci. 11 (2001) 731-748. | MR 1833001 | Zbl 1013.35054

[13] P. Laurençot and S. Mischler. On coalescence equations and related models. In Modeling and Computational Methods for Kinetic Equations 321-356. P. Degond, L. Pareschi and G. Russo (Eds). Birkhäuser, Boston, 2004. | MR 2068589 | Zbl 1105.82027

[14] Lê Châu-Hoàn. Etude de la classe des opérateurs m-accrétifs de L1(Ω) et accrétifs dans L∞(Ω). Thèse de 3ème cycle, Université de Paris VI, 1977.

[15] F. Leyvraz. Scaling theory and exactly solved models in the kinetics of irreversible aggregation. Phys. Rep. 383 (2003) 95-212.

[16] A. Lushnikov. Coagulation in finite systems. J. Colloid Interface Sci. 65 (1978) 276-285.

[17] A. H. Marcus. Stochastic coalescence. Technometrics 10 (1968) 133-143. | MR 223151

[18] G. Menon, B. Niethammer and R. L. Pego. Dynamics and self-similarity in min-driven clustering. Trans. Amer. Math. Soc. To appear. | MR 2678987 | Zbl 1211.82038

[19] J. R. Norris. Markov Chains. Cambridge Univ. Press, Cambridge, 1997. | Zbl 0938.60058

[20] J. R. Norris. Smoluchowski's coagulation equation: Uniqueness, nonuniqueness and a hydrodynamical limit for the stochastic coalescent. Ann. Appl. Probab. 9 (1999) 78-109. | MR 1682596 | Zbl 0944.60082

[21] M. Smoluchowski. Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen. Physik. Zeitschr. 17 (1916) 557-599.

[22] M. Smoluchowski. Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen. Z. Phys. Chem. 92 (1917) 129-168.

[23] J. A. D. Wattis. An introduction to mathematical models of coagulation-fragmentation processes: A deterministic mean-field approach. Phys. D 222 (2006) 1-20. | MR 2265763 | Zbl 1113.35145