Complutense University Library

Remarks on accessible steady states for some coagulation-fragmentation systems


Herrero, Miguel A. and Rodrigo, Marianito R. (2007) Remarks on accessible steady states for some coagulation-fragmentation systems. Discrete and Continuous Dynamical Systems. Series A., 17 (3). pp. 541-552. ISSN 1078-0947

[img] PDF
Restringido a Repository staff only hasta 31 December 2020.


Official URL:


In this paper we consider some systems of ordinary differential equations which are related to coagulation-fragmentation processes. In particular, we obtain explicit solutions {c(k)(t)} of such systems which involve certain coefficients obtained by solving a suitable algebraic recurrence relation. The coefficients are derived in two relevant cases: the high-functionality limit and the Flory-Stockmayer model. The solutions thus obtained are polydisperse (that is, c(k)(0) is different from zero for all k >= 1) and may exhibit monotonically increasing or decreasing total mass. We also solve a monodisperse case (where c(1)(0) is different from zero but c(k)(0)is equal to zero for all k >= 2) in the high-functionality limit. In contrast to the previous result, the corresponding solution is now shown to display a sol-gel transition when the total initial mass is larger than one, but not when such mass is less than or equal to one.

Item Type:Article
Uncontrolled Keywords:Coagulation-fragmentation; exact solutions; gelation; sol-gel transition; molecular-size distribution; diffusion; equations; kinetics; polymerization; aggregation; equilibrium; existence; polymers
Subjects:Sciences > Mathematics > Differential equations
Medical sciences > Medicine > Hematology
ID Code:16325

D. J. Aldous, Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists, Bernoulli, 5 (1999), 3–48.

H. Amann, Coagulation-fragmentation processes, Arch. Rat. Mech. Anal., 151 (2000), 339–366.

Ph. Benilan and D. Wrzosek, On an infinite system of reaction-diffusion equations, Adv. Math. Sci. Appl., 7 (1997), 351–366.

K. Binder, Theory for the dynamics of clusters, II. Critical diffusion in binary systems for phase separation, Phys. Rev. B, 15 (1977), 4425–4447.

S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Mod. Phys., 15 (1943), 1–91.

R. J. Cohen and G. B. Benedek, Equilibrium and kinetic theory of polymerization and the sol-gel transition, J. Phys. Chem., 86 (1982), 3696–3714.

F. P. da Costa, A finite-dimensional dynamical model for gelation in coagulation processes, J. Nonlinear Sci., 8 (1998), 619–653.

P. van Dongen and M. H. Ernst, Kinetics of reversible polymerization, J. Stat. Phys., 37 (1984), 301–329.

M. Escobedo, Ph. Lauren¸cot, S.Mischler, and B. Perthame, Gelation and mass conservation in coagulation-fragmentation models, J. Diff. Eq., 195 (2003), 143–174.

S. K. Friedlander, “Smoke, Dust and Haze: Fundamentals of Aerosol Dynamics,” Oxford University Press, 2000.

P. J. Flory, Molecular size distribution in three dimensional polymers I. Gelation, J. Am. Chem. Soc., 63 (1941), 3038–3090.

P. J. Flory, Molecular size distribution in three dimensional polymers II. Trifunctional branching units, J. Am. Chem. Soc., 63 (1941), 3091–3096.

P. J. Flory, Molecular size distribution in three dimensional polymers III. Trifunctional branching units, J. Am. Chem. Soc., 63 (1941), 3096–3100.

A. Fasano and F. Rosso, Dynamics of dispersions with multiple breakage and volume scattering, in “Computational Methods and Multiphase Flow”, (eds. H. Power and C. Brebbia), Advances in Fluid Mechanics, WIT Press (2001).

G. Grimmett and D. Stirzaker, “Probability and Random Processes,” Oxford University Press, 2000.

M. A. Herrero, J. J. L. Velázquez, and D.Wrzosek, Sol-gel transition in a coagulation-diffusion model, Physica D, 141 (2000), 221–247.

Ph. Laurençot and S. Mischler, The continuous coagulation-fragmentation equations with diffusion, Arch. Rat. Mech. Anal., 162 (2002), 45–99.

F. Leyvraz, Existence and properties of post-gel solutions for the kinetic equation of coagulation, J. Phys. A, 16 (1983), 2861–2873.

F. Leyvraz and H. R. Tschudi, Singularities in the kinetics of coagulation processes, J. Phys. A., 14 (1981), 3389–3405.

J. B. Mc Leod, On an infinite set of non-linear differential equations, Quart. J. Math. Oxford, 2 (1962), 119–128.

H. G. Rotstein, A. Novick-Cohen, and R. Tannenbaum, Gelation and cluster growth with cluster-wall interactions, J. Stat. Phys., 90 (1998), 1–24.

P. Sandkühler, J. Sefcik, and M. Morbidelli, Kinetics of gel formation in dilute dispersions with strong attractive particle interactions, Adv. in Colloid and Interface Science, 108-109 (2004), 133–143.

M. Slemrod, Coagulation-difussion systems: derivation and existence of solutions for the diffuse interface structure equations, Physica D, 46 (1990), 351–366.

M. von Smoluchowski, Drei Vorträge über Diffusion, Brownische Bewegung und Koagulation von Kolloidteilchen, Physik Z, 17 (1916), 557–585.

J. L. Spouge, Equilibrium polymer size distributions, Macromolecules, 16 (1983), 121–127.

W. H. Stockmayer, Theory of molecular size distribution and gel formation in branched-chain polymers, J. Chem. Phys., 11 (1943), 45–55.

P. Whittle, Statistical processes of aggregation and polymerization, Proc. Cambridge Phil. Soc., 61 (1965), 475–495.

Deposited On:12 Sep 2012 11:10
Last Modified:07 Feb 2014 09:27

Repository Staff Only: item control page