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Subcritical, critical and supercritical size distributions in random coagulation-fragmentation processes

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Abstract

We consider the asymptotic probability distribution of the size of a reversible random coagula-tion-fragmentation process in the thermodynamic limit. We prove that the distributions of small, medium and the largest clusters converge to Gaussian, Poisson and 0–1 distributions in the supercritical stage (post-gelation), respectively. We show also that the mutually dependent distributions of clusters will become independent after the occurrence of a gelation transition. Furthermore, it is proved that all the number distributions of clusters are mutually independent at the critical stage (gelation), but the distributions of medium and the largest clusters are mutually dependent with positive correlation coefficient in the supercritical stage. When the fragmentation strength goes to zero, there will exist only two types of clusters in the process, one type consists of the smallest clusters, the other is the largest one which has a size nearly equal to the volume (total number of units).

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Authors and Affiliations

  1. Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200030, P. R. China

    Dong Han

  2. Department of Statistics, Fudan University, 200433, Shanghai, P. R. China

    Xin Sheng Zhang

  3. Department of Mathematics, University of California, Irvine, CA, USA

    Wei An Zheng

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  1. Dong Han
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  2. Xin Sheng Zhang
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  3. Wei An Zheng
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Correspondence to Dong Han.

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This research is supported by National Science Foundation of China (10371074)

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Han, D., Zhang, X.S. & Zheng, W.A. Subcritical, critical and supercritical size distributions in random coagulation-fragmentation processes. Acta. Math. Sin.-English Ser. 24, 121–138 (2008). https://doi.org/10.1007/s10114-007-0978-9

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  • DOI: https://doi.org/10.1007/s10114-007-0978-9

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