Bernoulli

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

David J. Aldous

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Abstract

Consider N particles, which merge into clusters according to the following rule: a cluster of size x and a cluster of size y merge at stochastic rate K(x,y)/N, where K is a specified rate kernel. This Marcus-Lushnikov model of stochastic coalescence and the underlying deterministic approximation given by the Smoluchowski coagulation equations have an extensive scientific literature. Some mathematical literature (Kingman's coalescent in population genetics; component sizes in random graphs) implicitly studies the special cases K(x,y)=1 and K(x,y)=xy. We attempt a wide-ranging survey. General kernels are only now starting to be studied rigorously; so many interesting open problems appear.

Article information

Source
Bernoulli Volume 5, Number 1 (1999), 3-48.

Dates
First available in Project Euclid: 12 March 2007

Permanent link to this document
https://projecteuclid.org/euclid.bj/1173707093

Mathematical Reviews number (MathSciNet)
MR1673235

Zentralblatt MATH identifier
0930.60096

Keywords
branching process coalescence continuum tree density-dependent Markov process gelation random graph random tree Smoluchowski coagulation equation

Citation

Aldous, David J. Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5 (1999), no. 1, 3--48.https://projecteuclid.org/euclid.bj/1173707093


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References

  • [1] Ackleh, A., Fitzpatrick, B. and Hallam, T. (1984) Approximation and parameter estimation problems for algal aggregation models. Math. Models Methods Appl. Sci., 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. Theory 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. Technical Report 481, Department of Statistics, University of California, Berkeley.
  • [13] Allen, E. and Bastien, P. (1995) On coagulation and the stellar mass function. Astrophys. J., 452, 652-670.
  • [14] Anon. (1996) Economist, 20 April, 74.
  • [15] Barbour, A. (1982) Poisson convergence and random graphs. Math. Proc. Cambridge Phil. Soc., 92, 349-359.
  • [16] Bayewitz, M., Yerushalmi, J., Katz, S. and Shinnar, R. (1974) The extent of correlations in a stochastic coalescence process. J. Atmos. Sci., 31, 1604-1614.
  • [17] Binglin, L. (1987) The exact solution of the coagulation equation with kernel Kij = A(i + j) + B. J. Phys. A: Math. Gen., 20, 2347-2356.
  • [18] Bollobás, B. (1985) Random Graphs. London: Academic Press.
  • [19] Brennan, M. and Durrett, R. (1987) Splitting intervals II: limit laws for lengths. Probab. Theory Related Fields, 75, 109-127.
  • [20] Buffet, E. and Pulé, J. (1990) On Lushnikov's model of gelation. J. Statist. Phys., 58, 1041-1058.
  • [21] Buffet, E. and Pulé, J. (1991) Polymers and random graphs. J. Statist. Phys., 64, 87-110.
  • [22] Carr, J. and da Costa, F. (1992) Instantaneous gelation in coagulation dynamics. Z. Angew. Math. Phys., 43, 974-983.
  • [23] Consul, P. (1989) Generalized Poisson Distributions. New York: Dekker.
  • [24] Daley, D. and Vere-Jones, D. (1988) An Introduction to the Theory of Point Processes. Berlin: Springer-Verlag.
  • [25] 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.
  • [26] 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.
  • [27] Dubovskii, P.B. (1994) Mathematical Theory of Coagulation. Lecture Notes 23. Seoul: Global Analysis Research Center, Seoul National University.
  • [28] Erdös, P. and Rényi, A. (1960) On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci., 5, 17-61.
  • [29] Erdös, P. and Rényi, A. (1961) On the evolution of random graphs. Bull. Inst. Int. Statist., 38, 343- 347.
  • [30] 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.
  • [31] Ernst, M., Ziff, R. and Hendriks, E. (1984) Coagulation processes with a phase transition. J. Colloid Interface Sci., 97, 266-277.
  • [32] Ethier, S. and Kurtz, T.G. (1986) Markov Processes: Characterization and Convergence. New York: Wiley.
  • [33] Evans, S. and Pitman, J. (1996) Construction of Markovian coalescents. To appear in Am. Inst. Henri Poincaré.
  • [34] Flory, P. (1941) Molecular size distribution in three dimensional polymers III. Tetrafunctional branching units. J. Amer. Chem. Soc., 63, 3096-3100.
  • [35] Friedlander, S. and Wang, C. (1966) The self-preserving particle size distribution for coagulation by Brownian motion. J. Colloid Interface Sci., 22, 126-132.
  • [36] Gillespie, D. (1972) The stochastic coalescence model for cloud droplet growth. J. Atmos. Sci., 29, 1496-1510.
  • [37] Golovin, A. (1963) The solution of the coagulating equation for cloud droplets in a rising air current. Izv. Geophys. Ser., 5, 482-487.
  • [38] Grimmett, G.R. (1980) Random labelled trees and their branching networks. J. Austral. Math. Soc. (Ser. A), 30, 229-237.
  • [39] Heilmann, O.J. (1992) Analytical solutions of Smoluchowski's coagulation equation. J. Phys. A: Math. Gen., 25, 3763-3771.
  • [40] Hendriks, E., Spouge, J., Eibl, M. and Shreckenberg, M. (1985) Exact solutions for random coagulation processes. Z. Phys. B, 58, 219-227.
  • [41] Hendriks, E. and Ziff, R. (1985) Coagulation in a continuously stirred tank reactor. J. Colloid Interface Sci., 105, 247-256.
  • [42] Jaffe, A. and Quinn, F. (1993) Theoretical mathematics: towards a cultural synthesis of mathematics and theoretical physics. Bull. Amer. Math. Soc. 29, 1-13.
  • [43] Jaffe, A. and Quinn, F. (1994) Bull. Amer. Math. Soc., 30, 159-211.
  • [44] Janson, S., Knuth, D.E., Luczak, T. and Pittel, B. (1993) The birth of the giant component. Random Struct. Algorithms, 4, 233-358.
  • [45] Jeon, I. (1996) Gelation Phenomena, Ph.D. Thesis, Ohio State University.
  • [46] Jeon, I. (1997) Existence of gelling solutions for coagulation-fragmentation equations. To appear in Commun. Math. Phys.
  • [47] Jiang, Y. and Leyvraz, F. (1993) Scaling theory for ballistic aggregation. J. Phys. A: Math. Gen., 26, L176-L186.
  • [48] Kingman, J. (1982) The coalescent. Stochastic Process. Applic., 13, 235-248.
  • [49] Knight, F. (1971) Some condensation processes of McKean type. J. Appl. Probab., 8, 399-406.
  • [50] Koutzenogii, K., Levykin, A. and Sabelfeld, K. (1996) Kinetics of aerosol formation in the free molecule regime in presence of condensable vapor. J. Aerosol Sci., 27, 665-679.
  • [51] Kreer, M. and Penrose, O. (1994) Proof of dynamical scaling in Smoluchowski's coagulation equation with constant kernel. J. Statist. Phys., 75, 389-407.
  • [52] Krivitsky, D. (1995) Numerical solution of the Smoluchowski kinetic equation and asymptotics of the distribution function. J. Phys. A: Math. Gen., 28, 2025-2039.
  • [53] Lang, R. and Nguyen, X. (1980) Smoluchowski's theory of coagulation in colloids holds rigorously in the Boltzmann-Grad limit. Z. Wahrscheinhchkeitstheorie Verw. Geb., 54, 227-280.
  • [54] Lushnikov, A. (1973) Evolution of coagulating systems. J. Colloid Interface Sci., 45, 549-556.
  • [55] Lushnikov, A. (1978a) Certain new aspects of the coagulation theory. Izv. Atmos. Ocean Phys., 14, 738-743.
  • [56] Lushnikov, A. (1978b) Coagulation in finite systems. J. Colloid Interface Sci., 65, 276-285.
  • [57] Marcus, A. (1968) Stochastic coalescence. Technometrics, 10, 133-143.
  • [58] McLeod, J. (1962) On an infinite set of nonlinear differential equations. Quart. J. Math. Oxford, 13, 119-128.
  • [59] McLeod, J. (1964) On the scalar transport equation. Proc. London Math. Soc., 14, 445-458.
  • [60] Muralidar, R. and Ramkrishna, D. (1985) An inverse problem in agglomeration kinetics. J. Colloid Interface Sci., 112, 348-361.
  • [61] Nerman, O. and Jagers, P. (1984) The stable doubly infinite pedigree process of supercritical branching populations. Z. Wahrscheinhchkeitstheorie Verw. Geb., 65, 445-460.
  • [62] Norris, J.R. (1997) Smoluchowski's coagulation equation: uniqueness, non-uniqueness and a hydrodynamic limit for the stochastic coalescent. Unpublished.
  • [63] Olivier, B., Sorensen, C. and Taylor, T. (1992) Scaling dynamics of aerosol coagulation. Phys. Rev. A, 45, 5614-5623.
  • [64] 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.
  • [65] Pitman, J. (1996) Coalescent random forests. Technical Report 457, Department of Statistics, University of California, Berkeley.
  • [66] Pittel, B. (1990) On tree census and the giant component in sparse random graphs. Random Struct. Algorithms, 1, 311-342.
  • [67] Pyke, R. (1965) Spacings. J. Roy. Statist. Soc., Ser. B, 27, 395-449.
  • [68] 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.
  • [69] Schumann, T. (1940) Theoretical aspects of the size distribution of fog droplets. Quart. J. Roy. Meteorol. Soc., 66, 195-207.
  • [70] Scott, W. (1968) Analytic studies of cloud droplet coalescence. J. Atmos. Sci., 25, 54-65.
  • [71] Seinfeld, J. (1986) Atmospheric Chemistry and Physics of Air Polution. New York: Wiley.
  • [72] Sheth, R. and Pitman, J. (1997) Coagulation and branching process models of gravitational clustering. Mon. Not. Roy. Astron. Soc., 289, 66-80.
  • [73] Shirvani, M. and Roessel, H.V. (1992) The mass-conserving solutions of Smoluchowski's coagulation equation: the general bilinear kernel. Z. Angew. Math. Phys., 43, 526-535.
  • [74] Silk, J. and Takahashi, T. (1979) A statistical model for the initial stellar mass function. Astrophys. J., 229, 242-256.
  • [75] Silk, J. and White, S. (1978) The development of structure in the expanding universe. Astrophys. J., 223, 59-62.
  • [76] Smit, D., Hounslow, M. and Paterson, W. (1994) Aggregation and gelation I: Analytical solutions for CST and batch operation. Chem. Engng Sci., 49, 1025-1035.
  • [77] 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 Sci., 50, 849-862.
  • [78] Smoluchowski, M. (1916) Drei Vorträge über Diffusion, Brownsche Bewegung und Koagulation von Kolloidteilchen. Phys. Z., 17, 557-585.
  • [79] Spouge, J.L. (1983a) Solutions and critical times for the monodisperse coagulation equation when a(i,j) = A + B(i+j) + Cij. J. Phys. A: Math. Gen., 16, 767-773.
  • [80] 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.
  • [81] Stewart, C., Crowe, C. and Saunders, S. (1993) A model for simultaneous coalescence of bubble clusters. Chem. Engng Sci., 48, 3347-3354.
  • [82] Stockmayer, W. (1943) Theory of molecular size distribution and gel formation in branched chain polymers. J. Chem. Phys., 11, 45-55.
  • [83] Tanaka, H. and Nakazawa, K. (1993) Stochastic coagulation equation and validity of the statistical coagulation equation. J. Geomagn. Geoelectr., 45, 361-381.
  • [84] Tanaka, H. and Nakazawa, K. (1994) Validity of the stochastic coagulation equation and runaway growth of protoplanets. Icarus, 107, 404-412.
  • [85] Tavare, S. (1984) Line-of-descent and genealogical processes and their applications in population genetics models. Theor. Population Biol., 26, 119-164.
  • [86] 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.
  • [87] Trizac, E. and Hansen, J.-P. (1996) Dynamics and growth of particles undergoing ballistic coalescence. J. Statist. Phys., 82, 1345-1370.
  • [88] 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.
  • [89] van Dongen, P. (1987a) Fluctuations in coagulating systems II. J. Statist. Phys., 49, 927-975.
  • [90] van Dongen, P. (1987b) On the possible occurrence of instantaneous gelation in Smoluchowski's coagulation equation. J. Phys. A: Math. Gen., 20, 1889-1904.
  • [91] van Dongen, P. (1987c) Solutions of Smoluchowski's coagulation equation at large cluster sizes. Physica A, 145, 15-66.
  • [92] van Dongen, P. and Ernst, M. (1984) Size distribution in the polymerization model Af RBg. J. Phys. A: Math. Gen., 17, 2281-2297.
  • [93] van Dongen, P. and Ernst, M. (1985) Cluster size distribution in irreversible aggregation at large times. J. Phys. A: Math. Gen., 18, 2779-2793.
  • [94] 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.
  • [95] van Dongen, P. and Ernst, M. (1987a) Fluctuations in coagulating systems. J. Statist. Phys., 49, 879- 926.
  • [96] van Dongen, P. and Ernst, M. (1987b) Tail distribution of large clusters from the coagulation equation. J. Colloid Interface Sci., 115, 27-35.
  • [97] van Dongen, P. and Ernst, M. (1988) Scaling solutions of Smoluchowski's coagulation equation. J. Statist. Phys., 50, 295-329.
  • [98] van Kampen, N. (1981) Stochastic Processes in Physics and Chemistry. Amsterdam: North-Holland.
  • [99] van Lint, J. and Wilson, R. (1992) A Course in Combinatorics. Cambridge, Cambs.: Cambridge University Press.
  • [100] Vemury, S., Kusters, K. and Pratsinis, S. (1994) Time-lag for attainment of the self-preserving particle size distribution by coagulation. J. Colloid Interface Sci., 165, 53-59.
  • [101] Vicsek, T. (1992) Fractal Growth Phenomena, 2nd edn. Singapore: World Scienti®c.
  • [102] Wang, C.-S. (1966) A mathematical study of the particle size distribution of coagulating disperse systems. Ph.D. Thesis, California Institute of Technology.
  • [103] Wetherill, G. (1990) Comparison of analytical and physical modeling of planetisimal accumulation. Icarus, 88, 336-354.
  • [104] White, W. (1980) A global existence theorem for Smoluchowski's coagulation equation. Proc. Amer. Math. Soc., 80, 273-276.
  • [105] White, W. and Wiltzius, P. (1995) Real space measurement of structure in phase separating binary fluid mixtures. Phys. Rev. Lett., 75, 3012-3015.
  • [106] Whittle, P. (1986) Systems in Stochastic Equilibrium. New York: Wiley.
  • [107] 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.
  • [108] Ziff, R.M. (1980) Kinetics of polymerization. J. Statist. Phys., 23, 241-263.
  • [109] Ziff, R., Ernst, M. and Hendriks, E. (1983) Kinetics of gelation and universality. J. Phys. A: Math. Gen., 16, 2293-2320.