Abstract
Hydrodynamic limit of general k-nary mass exchange processes with discrete mass distribution is described by a system of kinetic equations that generalize classical Smoluchovski's coagulation equations and many other models that are intensively studied in the current mathematical and physical literature. Existence and uniqueness theorems for these equations are proved. At last, for k-nary mass exchange processes with k>2 an alternative nondeterministic measure-valued limit (diffusion approximation) is discussed.
Similar content being viewed by others
References
D. J. Aldous, Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists, Bernoulli 5:3–48 (1999).
H. Amann, Coagulation-fragmentation processes, Arch. Rational Mech. Anal. 151: 339–366 (2000).
J. M. Ball and J. Carr, The discrete coagulation-fragmentation equations: Existence, uniqueness, and density conservation, J. Stat. Phys. 61:203–234 (1990).
V. P. Belavkin, Quantum branching processes and nonlinear dynamics of multi-quantum systems, Dokl. Acad. Nauk SSSR 301:1348–1352 (1988). V. P. BelavkinEngl. Transl. in Sov. Phys. Dokl. 33:581–582 (1988).
V. Belavkin and V. Kolokoltsov, On general kinetic equation for many particle systems with interaction, fragmentation, and coagulation, Proc. Roy. Soc. London Ser. A 459:1–22 (2002).
Ph. Benian and D. Wrzosek, On infinite system of reaction-diffusion equations, Adv. Math. Sci. Appl. 7:349–364 (1997).
J. Bertoin, Homogeneous fragmentation processes, Probab. Theory Related Fields 121: 301–318 (2001).
J. Bertoin, Self-similar fragmentations, Ann. Inst. H. Poincaré Probab. Stat. 38:319–340 (2002).
J. Bertoin and J.-F. LeGall, Stochastic flows associated to coalescent processes, preprint 2002, available via http://felix.proba.jussieu.fr/mathdoc/preprints/
J. Carr and F. da Costa, Instantaneous gelation in coagulation dynamics, Z. Angew. Math. Phys. 43:974–983 (1992).
J. F. Collet and F. Poupaud, Existence of solutions to coagulation-fragmentation systems with diffusion, Transport Theory Statist. Phys. 25:503–513 (1996).
D. Dawson, Measure-valued Markov processes, in École d'été de probabilités de Saint-Flour XXI-1991, P. L. Hennequin, ed., Springer Lect. Notes Math. Vol. 1541 (Springer, 1993), pp. 1–260.
S. C. Davies, J. R. King, and J. A. Wattis, The Smoluchovski coagulation equations with continuous injection, J. Phys. A: Math. Gen. 32:7745–7763 (1999).
P. B. Dubovski, A “triangle” of interconnected coagulation models, J. Phys. A: Math. Gen. 32:781–793 (1999).
E. Dynkin, An Introduction to Branching Measure-Valued Processes, CRM Monograph Series Vol. 6 (AMS, Providence, RI, 1994).
S. N. Ethier and T. G. Kurtz, Markov Processes. Characterization and Convergence (Wiley, 1986).
N. Jacob, Pseudo-Differential Operators and Markov Processes. Vol. II: Generators and Their Potential Theory (Imperial College Press, London, 2002).
I. Jeon, Existence of gelling solutions for coagulation-fragmentation equations, Comm. Math. Phys. 194:541–567 (1998).
V. Kolokoltsov, Measure-valued limits of interacting particle systems with k-nary interactions I. One-dimensional limits, Probab. Theory Related Fields 126:364–394 (2003).
V. Kolokoltsov, Measure-valued limits of interacting particle systems with k-nary interactions II, Preprint (2002), to appear in Stochastics and Stochastics Reports.
V. Kolokoltsov, On Extensions of mollified Boltzmann and Smoluchovski equations to particle systems with a k-nary interaction, Russian J. Math. Phys. 10:268–295 (2003).
V. Kolokoltsov, Symmetric stable laws and stable-like jump-diffusions, Proc. London Math. Soc. 3:725–768 (2000).
V. Kolokoltsov, On Markov processes with decomposable pseudo-differential generators, Preprint (2002), to appear in Stochastics and Stochastics Reports.
M. Kostoglou and A. J. Karabelas, A study of nonlinear breakage equation: Analytical and asymptotic solutions, J. Phys. A: Math. Gen. 33:1221–1232 (2000).
M. Lachowicz, Ph. Laurencot, and D. Wrzosek, On the Oort–Hulst–Savronov coagulation equation and its relation to the Smoluchowski equation, SIAM J. Math. Anal. 34:1399–1421 (2003).
Ph. Laurencot and S. Mischler, The continuous coagulation-fragmentation equations with diffusion, Arch. Rational Mech. Anal. 162:45–99 (2002).
P. Laurencot and D. Wrzosek, The discrete coagulation equations with collisional breakage, J. Stat. Phys. 104:193–220 (2001).
M. Möhle and S. Sagitov, A classification of coalescent processes for haploid exchangeable population models, Ann. Probab. 29:1547–1562 (2001).
J. Norris, Smoluchovski's coagulation equation: Uniqueness, nonuniqueness, and a hydrodynamic limit for the stochastic coalescent, Ann. Appl. Probab. 9:78–109 (1999).
J. Norris, Cluster coagulation, Comm. Math. Phys. 209:407–435 (2000).
A. Okubo, Dynamical aspects of animal grouping: Swarms, schools, flocks, and herds, Adv. Biophys. 22:1–94 (1986).
J. Pitman, Coalescents with multiple collisions, Ann. Probab. 27:1870–1902 (1999).
V. S. Safronov, Evolution of the Pre-Planetary Cloud and the Formation of the Earth and Planets (Nauka, Moscow, 1969) (in Russian). V. S. SafronovEngl. transl.: Israel Program for Scientific Translations, Jerusalem, 1972.
J. Schweinsberg, Coalescents with simultaneous multiple collisions, Electr. J. Prob. 5:1–50 (2000).
R. C. Srivastava, A simple model of particle coalescence and breaup, J. Atmos. Sci. 39:1317–1322 (1982).
A. Sznitman, Topics in propagation of chaos, in École d'été de Probabilités de Saint-Flour XIX-1989, Springer Lecture Notes Math. Vol. 1464 (Springer, 1991), pp. 167–255.
D. Wilkins, A geometrical interpretation of the coagulation equation, J. Phys. A: Math. Gen. 15:1175–1178 (1982).
D. Wrzosek, Mass-conservation solutions to the discrete coagulation-fragmentation model with diffusion, Nonlinear Anal. 49:297–314 (2002).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kolokoltsov, V.N. Hydrodynamic Limit of Coagulation-Fragmentation Type Models of k-Nary Interacting Particles. Journal of Statistical Physics 115, 1621–1653 (2004). https://doi.org/10.1023/B:JOSS.0000028071.96950.12
Issue Date:
DOI: https://doi.org/10.1023/B:JOSS.0000028071.96950.12