Kolokoltsov, V. N. (Vasiliĭ Nikitich). (2004) Hydrodynamic limit of coagulation-fragmentation type models of k-nary interacting particles. Journal of Statistical Physics, Vol.115 (No.5/6). pp. 1621-1653. ISSN 0022-4715

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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 non-deterministic measure-valued limit (diffusion approximation) is discussed.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH):	Particles -- Dynamics, Particles -- Mathematical models, Mass transfer -- Mathematical models
Journal or Publication Title:	Journal of Statistical Physics
Publisher:	Springer
ISSN:	0022-4715
Date:	2004
Volume:	Vol.115
Number:	No.5/6
Page Range:	pp. 1621-1653
Identification Number:	10.1023/B:JOSS.0000028071.96950.12
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
References:	[Al] D.J. Aldous. Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5:2 (1999), 3-48. [Am] H. Amann. Coagulation-Fragmentation processes. Arch. Rational Mech. Anal. 151 (2000), 339-366. [BC] J.M. Ball, J. Carr. The discrete coagulation-fragmentation equations: existence, uniqueness and density conservation. J. Stat. Phys. 61 (1990), 203-234. [Be] V.P. Belavkin. Quantum branching processes and nonlinear dynamics of multi- quantum systems. Dokl. Acad. Nauk SSSR 301:6 (1988), 1348-1352. Engl. Tansl. in Sov. Phys. Dokl. 33:8 (1988), 581-582. [BK] V. Belavkin, V. Kolokoltsov. On general kinetic equation for many particle systems with interaction, fragmentation and coagulation. Proc. R. Soc. Lond. A 459 (2002), 1-22. [BW] Ph. Benian, D. Wrzosek. On infinite system of reaction-diffusion equations. Adv. Math. Sci. Appl. 7 (1997), 349-364. [Br1] J. Bertoin. Homogeneous fragmentation processes. Prob. Theory Relat. Fields 121 (2001), 301-318. [Br2] J. Bertoin. Self-similar fragmentations. Ann. Inst. H. Poincare Probab. Stat. 38 (2002), 319-340. [BL] J. Bertoin, J.-F. LeGall. Stochastic flows associated to coalescent processes. Preprint 2002 Available via http://felix.proba.jussieu.fr/mathdoc/preprints/ [CC] J. Carr, F. da Costa. Instantaneous gelation in coagulation dynamics. Z. Angew. Math. Phys. 43 (1992), 974-983. [CP] J.F. Collet, F. Poupaud. Existence of solutions to coagulation-fragmentation systems with diffusion. Transport Theory Statist. Phys. 25 (1996), 503-513. [Da] D. Dawson. Measure-Valued Markov Processes. Hennequin P.L. (ed.) Ecole d'Ete de probabilit¶es de Saint-Flour XXI-1991. Springer Lect. Notes Math. 1541 (1993), 1-260. [DKW] S.C. Davies, J.R. King and J.A. Wattis. The Smoluchovski coagulation equations with continuous injection. J. Phys. A: Math. Gen. 32 (1999), 7745-7763. [Du] P.B. Dubovski. A "triangle" of interconnected coagulation models. J. Phys. A: Math.Gen. 32 (1999), 781-793. [Dy] E. Dynkin. An Introduction to Branching Measure-Valued Processes. CRM monograph series 6, AMS, Providence RI, 1994. [EK] S.N. Ethier, T.G. Kurtz. Markov Processes. Characterization and convergence. John Wiley Sons 1986. [Ja] N. Jacob. Pseudo-differential operators and Markov processes. Vol. II: Generators and their potential theory. London: Imperial College Press, 2002. [Je] I. Jeon. Existence of Gelling Solutions for Coagulation-Fragmentation Equations. Comm. Math. Phys. 194 (1998), 541-567. [Ko1] V. Kolokoltsov. Measure-valued limits of interacting particle systems with k-nary interactions I. One-dimensional limits. Probab. Theory Relat. Fields 126 (2003), 364-394. [Ko2] V. Kolokoltsov. Measure-valued limits of interacting particle systems with k-nary interactions II. Preprint (2002). To appear in Stochastics and Stochastics Reports. [Ko3] V. Kolokoltsov. On Extensions of Mollified Boltzmann and Smoluchovski Equations to Particle Systems with a k-nary Interaction. Russian Journal Math. Phys. 10:3 (2003), 268-295. [Ko4] V. Kolokoltsov. Symmetric stable laws and stable-like jump-diffusions. Proc. London Math. Soc. 3:80 (2000), 725-768. [Ko5] V. Kolokoltsov. On Markov processes with decomposable pseudo-differential generators. Preprint (2002). To appear in Stochastics and Stochastics Reports. [KK] M. Kostoglou, A.J. Karabelas. A study of nonlinear breakage equation: Analytical and asymptotic solutions. J. Phys. A: Math.Gen. 33 (2000), 1221-1232. [LLW] M. Lachowicz, Ph. Laurencot, D. Wrzosek. On the Oort-Hulst-Savronov coagulation equation and its relation to the Smoluchowski equation. SIAM J. Math. Anal. 34 (2003), 1399-1421. [LM] Ph. Laurencot, S. Mischler. The Continuous Coagulation-Fragmentation Equations with Diffusion. Arch. Rational Mech. Anal. 162 (2002), 45-99. [LW] P. Laurencot, D. Wrzosek. The Discrete Coagulation Equations with Collisional Breakage. J. Stat. Phys. 104, 1/2 (2001), 193-220. [MS] M. MÄohle, S. Sagitov. A classification of coalescent processes for haploid exchangeable population models. Ann. Prob. 29 (2001), 1547-1562. [No1] J. Norris. Smoluchovski's Coagulation Equation: Uniqueness, Nonuniqueness and a Hydrodynamic Limit for the Stochastic Coalescent. The Annals of Applied Probability 9:1 (1999), 78-109. [No2] J. Norris. Cluster Coagulation. Comm. Math. Phys. 209 (2000), 407-435. [Ok] A. Okubo. Dynamical aspects of animal grouping: swarms, schools, flocks and herds. Adv. Biophys 22 (1986), 1-94. [Pi] J.Pitman. Coalescents with multiple collisions. Ann. Prob. 27 (1999), 1870-1902. [Sa] V.S. Safronov. Evolution of the Pre-Planetary Cloud and the Formation of the Earth and Planets. Moscow, Nauka, 1969 (in Russian). Engl. transl.: Israel Program for Scientific Translations, Jerusalem, 1972. [Sc] J. Schweinsberg. Coalescents with simultaneous multiple collisions. Electr. J. Prob. 5 (2000), 1-50. [Sr] R.C. Srivastava. A simple model of particle coalescence and breaup. J. Atmos. Sci. 39 (1982), 1317-1322. [Sz] A. Sznitman. Topics in Propagation of Chaos. In: Ecole d'Ete de Probabilites de Saint-Flour XIX-1989. Springer Lecture Notes Math. 1464 (1991), 167-255. [Wi] D. Wilkins. A geometrical interpretation of the coagulation equation. J. Phys. A: Math. Gen. 15 (1982), 1175-1178. [Wr] D. Wrzosek. Mass-conservation solutions to the discrete coagulation-fragmentation model with diffusion. Nonlinear Anal. 49 (2002), 297-314.
URI:	http://wrap.warwick.ac.uk/id/eprint/39383

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