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Existence of gelling solutions for coagulation-fragmentation equations. (English) Zbl 0910.60083

Summary: We study the Smoluchowski coagulation-fragmentation equation, which is an infinite set of nonlinear ordinary differential equations describing the evolution of a mono-disperse system of particles in a well stirred solution. Approximating the solutions of the Smoluchowski equations by a sequence of finite Markov chains, we investigate the qualitative behavior of the solutions. We determine a device on the finite chains which can detect the gelation phenomena – the density dropping phenomena. It shows how the gelation phenomena are reflected on the sequence of finite Markov chains. Using this device, we determine various types of gelation kernels and get the bounds of gelation times.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C22 Interacting particle systems in time-dependent statistical mechanics
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