Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists

Bernoulli

Volumn 5, Issue 1, 1999, Pages 3-48

  Aldous, David J ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Branching process; Coalescence; Continuum tree; Density dependent markov process; Gelation; Random graph; Random tree; Smoluchowski coagulation equation

Indexed keywords

EID: 0001796185     PISSN: 13507265     EISSN: None     Source Type: Journal    
DOI: 10.2307/3318611     Document Type: Article

References (110)

1. Ackleh, A., Fitzpatrick, B. and Hallam, T. (1984) Approximation and parameter estimation problems for algal aggregation models. Math. 'Models Methods Appl. Sei., 4, 291-311.
2. Aldous, D. (1991a) Asymptotic fringe distributions for general families of random trees. Ann. Appl. Probab., 1, 228-266.
3. Aldous, D. (1991b) The continuum random tree I. Ann. Probab., 19, 1-28.
4. Aldous, D. (1991c) The continuum random tree II: An overview. In M. Barlow and N. Bingham (eds), Stochastic Analysis, pp. 23-70. Cambridge, Cambs.: Cambridge University Press.
5. Aldous, D. (1992) Asymptotics in the random assignment problem. Probab. TJieory Related Fields, 93, 507-534.
6. Aldous, D. (1993) The continuum random tree III. Ann. Probab., 21, 248-289.
7. Aldous, D. (1996) Emergence of the giant component in special Marcus-Lushnikov processes. To appear in Random Structures Algorithms.
8. Aldous, D. (1997a) Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab., 25, 812-854
9. Aldous, D. (1997b) Tree-valued Markov chains and Poisson-Galton-Watson distributions. Unpublished.
10. Aldous, D. and Limic, V. (1997) The entrance boundary of the multiplicative coalescent. Unpublished.
11. Aldous, D. and Pitman, J. (1997a) The standard additive coalescent. Technical Report 48a, Dept. of Statistics, U.C. Berkeley.
12. Aldous, D. and Pitman, J. (1997b) Tree-valued Markov chains derived from Galton-Watson processes.
13. Technical Report 481, Department of Statistics, University of California, Berkeley.
14. Allen, E. and Bastien, P. (1995) On coagulation and the stellar mass function. Astrophys. J., 452, 652-670.
15. Anon. (1996) Economist, 20 April, 74.
16. Barbour, A. (1982) Poisson convergence and random graphs. Math. Proc. Cambridge Phil. Soc., 92, 349-359.
17. Bayewitz, M., Yemshalmi, J., Katz, S. and Shinnar, R. (1974) The extent of correlations in a stochastic coalescence process. J. Atmos. Sei., 31, 1604-1614.
18. Binglin, L. (1987) The exact solution of the coagulation equation with kernel Ky = A(i +f) + B. J. Phys. A: Math. Gen., 20, 2347-2356.
19. Bollobâs, B. (1985) Random Graphs. London: Academic Press.
20. Brennan, M. and Durrett, R. (1987) Splitting intervals II: limit laws for lengths. Probab. Tlieory Related Fields, 75, 109-127.
21. Buffet, E. and Pule, J. (1990) On Lushnikov's model of gelation. J. Statist. Phys., 58, 1041-1058.
22. Buffet, E. and Pule, J. (1991) Polymers and random graphs. J. Statist. Phys., 64, 87-110.
23. Carr, J. and da Costa, F. (1992) Instantaneous gelation in coagulation dynamics. Z. Angew. Math. Phys., 43, 974-983
24. Consul, P. (1989) Generalized Poisson Distributions. New York: Dekker.
25. Daley, D. and Vere-Jones, D. (1988) An Introduction to the Tlieory of Point Processes. Berlin: Springer-Verlag.
26. Donnelly, P. and Simons, S. (1993) On the stochastic approach to cluster size distribution during particle coagulation I: Asymptotic expansion in the deterministic limit. J. Phys. A: Math. Gen., 26, 2755-2767.
27. Drake, R. (1972) A general mathematical survey of the coagulation equation. In G. Hidy and J. Brock (eds), Topics in Current Aerosol Research, Part 2. Int. Rev. Aerosol Phys. Chem. 3. pp. 201-376. Oxford: Pergamon.
28. Dubovskii, P.D. (1994) Mathematical Tlieory of Coagulation. Lecture Notes 23. Seoul: Global Analysis Research Center, Seoul National University.
29. Erdös, P. and Rényi, A. (1960) On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sei., 5, 17-61.
30. Erdös, P. and Rényi, A. (1961) On the evolution of random graphs. Bull. Inst. Int. Statist., 38, 343347.
31. Ernst, M. (1983) Exact solutions of the nonlinear Boltzmann equation and related equations. In J. Lebowitz and E. Montrell (eds), Nonequilibrium Phenomena I, pp. 51-119. Amsterdam: North-Holland.
32. Ernst, M., Ziff, R. and Hendriks, E. (1984) Coagulation processes with a phase transition. J. Colloid Interface Sei., 97, 266-277.
33. Ethier, S. and Kurtz, T.G. (1986) Markov Processes: Characterization and Convergence. New York: Wiley.
34. Evans, S. and Pitman, J. (1996) Construction of Markovian coalescents. To appear in Am. Inst. Henri Poincaré.
35. Flory, P. (1941) Molecular size distribution in three dimensional polymers III. Tetrafunctional branching units. J. Amer. Chein. Soc., 63, 3096-3100.
36. Friedlander, S. and Wang, C. (1966) The self-preserving particle size distribution for coagulation by Brownian motion. J. Colloid Interface Sei., 22, 126-132.
37. Gillespie, D. (1972) The stochastic coalescence model for cloud droplet growth. J. Atmos. Sei., 29, 1496-1510.
38. Golovin, A. (1963) The solution of the coagulating equation for cloud droplets in a rising air current. Izv. Geophys. Ser., 5, 482-487.
39. Grimmett, G.R. (1980) Random labelled trees and their branching networks. J. Austral. Math. Soc. (Ser. A), 30, 229-237.
40. Heilmann, O.J. (1992) Analytical solutions of Smoluchowski's coagulation equation. J. Phys. A: Math. Gen., 25, 3763-3771.
41. Hendriks, E., Spouge, J., Eibl, M. and Shreckenberg, M. (1985) Exact solutions for random coagulation processes. Z Phys. B, 58, 219-227.
42. Hendriks, E. and Ziff, R. (1985) Coagulation in a continuously stirred tank reactor. J. Colloid Interface Sei., 105, 247-256.
43. Jaffe, A. and Quinn, F. (1993) Theoretical mathematics: towards a cultural synthesis of mathematics and theoretical physics. Bull. Amer. Math. Soc. 29, 1-13.
44. Jaffe, A. and Quinn, F. (1994) Bull. Amer. Math. Soc., 30, 159-211.
45. Janson, S., Knuth, D.E., Luczak, T. and Pittel, B. (1993) The birth of the giant component. Random Stnict. Algorithms, 4, 233-358.
46. Jeon, I. (1996) Gelation Phenomena, Ph.D. Thesis, Ohio State University.
47. Jeon, I. (1997) Existence of gelling solutions for coagulation-fragmentation equations. To appear in Commun. Math. Phys.
48. Jiang, Y. and Leyvraz, F. (1993) Scaling theory for ballistic aggregation. J. Phys. A: Math. Gen., 26, L176-L186.
49. Kingman, J. (1982) The coalescent. Stochastic Process. Applic., 13, 235-248.
50. Knight, F. (1971) Some condensation processes of McKean type. J. Appl. Probab., 8, 399-406.
51. Koutzenogii, K., Levykin, A. and Sabelfeld, K. (1996) Kinetics of aerosol formation in the free molecule regime in presence of condensable vapor. J. Aerosol Sei., 27, 665-679.
52. Kreer, M. and Penrose, O. (1994) Proof of dynamical scaling in Smoluchowski's coagulation equation with constant kernel. J. Statist. Phys., 75, 389-407.
53. Krivitsky, D. (1995) Numerical solution of the Smoluchowski kinetic equation and asymptotics of the distribution function. J. Phys. A: Math. Gen., 28, 2025-2039.
54. Lang, R. and Nguyen, X. (1980) Smoluchowski's theory of coagulation in colloids holds rigorously in the Boltzmann-Grad limit. Z. Wahrscheinlichkeitstheorie l'env. Geb., 54, 227-280.
55. Lushnikov, A. (1973) Evolution of coagulating systems. J. Colloid Interface Sei., 45, 549-556.
56. Lushnikov, A. (1978a) Certain new aspects of the coagulation theory. Izv. Atmos. Ocean Phys., 14, 738-743.
57. Lushnikov, A. (1978b) Coagulation in finite systems. J. Colloid Interface Sei., 65, 276-285.
58. Marcus, A. (1968) Stochastic coalescence. Technometrics, 10, 133-143.
59. McLeod, J. (1962) On an infinite set of nonlinear differential equations. Quart. J. Math. Oxford, 13, 119-128.
60. McLeod, J. (1964) On the scalar transport equation. Proc. London Math. Soc., 14, 445-458.
61. Muralidar, R. and Ramkrishna, D. (1985) An inverse problem in agglomeration kinetics. J. Colloid Interface Sei., 112, 348-361.
62. Nerman, O. and Jagers, P. (1984) The stable doubly infinite pedigree process of supercritical branching populations. Z. Wahrscheinlichkeitstheorie Ver\v. Geb., 65, 445-460.
63. Morris, J.R. (1997) Smoluchowski's coagulation equation: uniqueness, non-uniqueness and a hydrodynamic limit for the stochastic coalescent. Unpublished.
64. Olivier, B., Sorensen, C. and Taylor, T. (1992) Scaling dynamics of aerosol coagulation. Phys. Rev. A, 45, 5614-5623.
65. Pavlov, Y.L. (1977) Limit theorems for the number of trees of a given size in a random forest. Math. USSR Sbornik, 32, 335-345.
66. Pitman, J. (1996) Coalescent random forests. Technical Report 457, Department of Statistics, University of California, Berkeley.
67. Pittel, B. (1990) On tree census and the giant component in sparse random graphs. Random Struct. Algorithms, 1, 311-342.
68. Pyke, R. (1965) Spacings. J. Roy. Statist. Soc., Ser. B, 27, 395-449.
69. Sabelfeld, K., Rogasinsky, S., Kolodko, A. and Levykin, A. (1996) Stochastic algorithms for solving Smolouchovsky coagulation equation and applications to aerosol growth simulation. Monte Carlo Methods Applic., 2, 41-87.
70. Schumann, T. (1940) Theoretical aspects of the size distribution of fog droplets. Quart. J. Roy. Meteorol. 'Soc., 66, 195-207
71. Scott, W. (1968) Analytic studies of cloud droplet coalescence. J. Atmos. Sei. 25, 54-65.
72. Seinfeld, J. (1986) Atmospheric Chemistry and Physics of Air Pohttion. New York: Wiley.
73. Sheth, R. and Pitman, J. (1997) Coagulation and branching process models of gravitational clustering. Mon. Not. Roy. Astron. Soc., 289, 66-80.
74. Shirvani, M. and Roessel, U.V. (1992) The mass-conserving solutions of Smoluchowski's coagulation equation: the general bilinear kernel. Z. Angew. Math. Phys., 43, 526-535.
75. Silk, J. and Takahashi, T. (1979) A statistical model for the initial stellar mass function. Astrophys. J., 229, 242-256.
76. Silk, J. and White, S. (1978) The development of structure in the expanding universe. Astrophys. J., 223, L59-L62.
77. Smit, D., Hounslow, M. and Paterson, W. (1994) Aggregation and gelation 1: Analytical solutions for CST and batch operation. Chem. Engng Sei., 49, 1025-1035.
78. Smit, D., Hounslow, M. and Paterson, W. (1995) Aggregation and gelation III: Numerical classification of kernels and case studies of aggregation and growth. Chem. Engng Sei., 50, 849-862.
79. Smoluchowski, M. (1916) Drei Vorträge über Diffusion, Brownsche Bewegung und Koagulation von Kolloidteilchen. Phys. Z., 17, 557-585.
80. Spouge, J.L. (1983a) Solutions and critical times for the monodisperse coagulation equation when a(i,f) = A + B(i + j)+Cij. J. Phys. A: Math. Gen., 16, 767-773.
81. Spouge, J.L. (1983b) Solutions and critical times for the polydisperse coagulation equation when a(x,y) = A + B(x + y) + Cxy. J. Phys. A: Math. Gen., 16, 3127-3132.
82. Stewart, C., Crowe, C. and Saunders, S. (1993) A model for simultaneous coalescence of bubble clusters. Chem. Engng Sei., 48, 3347-3354.
83. Stockmayer, W. (1943) Theory of molecular size distribution and gel forrnation in branched chain polymers. J. Chem. Phys., 11, 45-55.
84. Tanaka, H. and Nakazawa, K. (1993) Stochastic coagulation equation and validity of the statistical coagulation equation. J. Geomagn. Geoclcctr., 45, 361-381.
85. Tanaka, H. and Nakazawa, K. (1994) Validity of the stochastic coagulation equation and runaway growth of protoplanets. Icanis, 107, 404-412.
86. Tavare, S. (1984) Line-of-descent and genealogical processes and their applications in population genetics models. Tlieor. Population Biol, 26, 119-164.
87. Treat, R. (1990) An exact solution of the Smoluchowski equation and its correspondence to the solution of the continuous equation. J. Phys. A: Math. Gen., 23, 3003-3016.
88. Trizac, E. and Hansen, J.-P. (1996) Dynamics and growth of particles undergoing ballistic coalescence. J. Statist. Phys., 82, 1345-1370.
89. Trubnikov, B. (1971) Solution of the coagulation equation in the case of a bilinear coefficient of adhesion of particles. Soviet Phys.-Dokl., 16, 124-125.
90. van Dongen, P. (1987a) Fluctuations in coagulating systems II. J. Statist. Phys., 49, 927-975.
91. van Dongen, P. (1987b) On the possible occurrence of instantaneous gelation in Smoluchowski's coagulation equation. J. Phys. A: Math. Gen., 20, 1889-1904.
92. van Dongen, P. (1987c) Solutions of Smoluchowski's coagulation equation at large cluster sizes. Physica A, 145, 15-66.
93. van Dongen, P. and Ernst, M. (1984) Size distribution in the polymerization model AfRBg. J. Phys. A: Math. Gen., 17, 2281-2297.
94. van Dongen, P. and Ernst, M. (1985) Cluster size distribution in irreversible aggregation at large times. J. Phys. A: Math. Gen., 18, 2779-2793.
95. van Dongen, P. and Ernst, M. (1986) On the occurrence of a gelation transition in Smoluchowski's coagulation equation. J. Statist. Phys., 44, 785-792.
96. van Dongen, P. and Ernst, M. (1987a) Fluctuations in coagulating systems. J. Statist. Phys., 49, 879926.
97. van Dongen, P. and Ernst, M. (1987b) Tail distribution of large clusters from the coagulation equation. J. Colloid Interface Sei., 115, 27-35.
98. van Dongen, P. and Ernst, M. (1988) Scaling solutions of Smoluchowski's coagulation equation. J. Statist. Phys., 50, 295-329.
99. van Kämpen, N. (1981) Stochastic Processes in Physics and Chemistry. Amsterdam: North-Holland
100. van Lint, J. and Wilson, R. (1992) A Course in Combinatorics. Cambridge, Cambs.: Cambridge University Press.
101. Vemury, S., Küsters, K. and Pratsinis, S. (1994) Time-lag for attainment of the self-preserving particle size distribution by coagulation. J. Colloid Interface Sei., 165, 53-59.
102. Vicsek, T. (1992) Fractal Growth Phenomena, 2nd edn. Singapore: World Scientific.
103. Wang, C.-S. (1966) A mathematical study of the particle size distribution of coagulating disperse systems. Ph.D. Thesis, California Institute of Technology.
104. Wetherill, G. (1990) Comparison of analytical and physical modeling of planetisimal accumulation. Icarus, 88, 336-354.
105. White, W. (1980) A global existence theorem for Smoluchowski's coagulation equation. Proc. Amer. Math. Soc., 80, 273-276.
106. White, W. and Wiltzius, P. (1995) Real space measurement of structure in phase separating binary fluid mixtures. Phys. Rev. Lett., 75, 3012-3015
107. Whittle, P. (1986) Systems in Stochastic Equilibrium. New York: Wiley.
108. Yao, A. (1976) On the average behavior of set merging algorithms. In Proceedings of the Eighth ACM Symposium on Theory of Computing, pp. 192-195. ACM.
109. Ziff, R.M. (1980) Kinetics of polymerization. J. Statist. Phys., 23, 241-263.
110. Ziff, R., Ernst, M. and Hendriks, E. (1983) Kinetics of gelation and universality. J. Phys. A: Math. Gen., 16, 2293-2320.