Dynamics and self-similarity in min-driven clustering

Menon, Govind; Niethammer, Barbara; Pego, Robert

doi:10.1090/S0002-9947-2010-05085-8

Dynamics and self-similarity in min-driven clustering
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by Govind Menon, Barbara Niethammer and Robert L. Pego PDF

Trans. Amer. Math. Soc. 362 (2010), 6591-6618 Request permission

Abstract:

We study a mean-field model for a clustering process that may be described informally as follows. At each step a random integer $k$ is chosen with probability $p_k$, and the smallest cluster merges with $k$ randomly chosen clusters. We prove that the model determines a continuous dynamical system on the space of probability measures supported in $(0,\infty )$, and we establish necessary and sufficient conditions for the approach to self-similar form. We also characterize eternal solutions for this model via a Lévy-Khintchine formula. The analysis is based on an explicit solution formula discovered by Gallay and Mielke, extended using a careful choice of time scale.

References

Jean Bertoin, Eternal solutions to Smoluchowski’s coagulation equation with additive kernel and their probabilistic interpretations, Ann. Appl. Probab. 12 (2002), no. 2, 547–564. MR 1910639, DOI 10.1214/aoap/1026915615
Jean Bertoin and Jean-Francois Le Gall, Stochastic flows associated to coalescent processes. III. Limit theorems, Illinois J. Math. 50 (2006), no. 1-4, 147–181. MR 2247827
S. Minakshi Sundaram, On non-linear partial differential equations of the parabolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 479–494. MR 0000088
Jack Carr and Robert Pego, Self-similarity in a coarsening model in one dimension, Proc. Roy. Soc. London Ser. A 436 (1992), no. 1898, 569–583. MR 1177577, DOI 10.1098/rspa.1992.0035
Laurens de Haan, An Abel-Tauber theorem for Laplace transforms, J. London Math. Soc. (2) 13 (1976), no. 3, 537–542. MR 407542, DOI 10.1112/jlms/s2-13.3.537
B. Derrida, C. Godrèche, and I. Yekutieli, Scale-invariant regimes in one-dimensional models of growing and coalescing droplets, Phys. Rev. A 44 (1991), no. 10, 6241–6251.
Julian Bonder, Über die Darstellung gewisser, in der Theorie der Flügelschwingungen auftretender Integrale durch Zylinderfunktionen, Z. Angew. Math. Mech. 19 (1939), 251–252 (German). MR 42, DOI 10.1002/zamm.19390190409
T. Gallay and A. Mielke, Convergence results for a coarsening model using global linearization, J. Nonlinear Sci. 13 (2003), no. 3, 311–346. MR 1982018, DOI 10.1007/s00332-002-0543-8
I. Ispolatov, P. L. Krapivsky, and S. Redner, War: The dynamics of vicious civilizations, Phys. Rev. E 54 (1996), no. 2, 1274–1289.
Govind Menon and Robert L. Pego, Approach to self-similarity in Smoluchowski’s coagulation equations, Comm. Pure Appl. Math. 57 (2004), no. 9, 1197–1232. MR 2059679, DOI 10.1002/cpa.3048
Govind Menon and Robert L. Pego, The scaling attractor and ultimate dynamics for Smoluchowski’s coagulation equations, J. Nonlinear Sci. 18 (2008), no. 2, 143–190. MR 2386718, DOI 10.1007/s00332-007-9007-5
Tatsuzo Nagai and Kyozi Kawasaki, Statistical dynamics of interacting kinks II, Physica A 134 (1986), 482–521.
P. Erdös, On the distribution of normal point groups, Proc. Nat. Acad. Sci. U.S.A. 26 (1940), 294–297. MR 2000, DOI 10.1073/pnas.26.4.294
F. W. J. Olver, Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0435697