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Self-similar behavior of the exchange-driven growth model with product kernel. (English) Zbl 1467.82049

Summary: We study the self-similar behavior of the exchange-driven growth model, which describes a process in which pairs of clusters, consisting of an integer number of monomers, interact through the exchange of a single monomer. The rate of exchange is given by an interaction kernel \(K(k, l)\) which depends on the sizes \(k\) and \(l\) of the two interacting clusters and is assumed to be of product form \((k\,l)^\lambda\) for \(\lambda\in[0,2)\) We rigorously establish the coarsening rates and convergence to the self-similar profile found by E. Ben-Naim and P. L. Krapivsky [“Exchange-driven growth”, Phys. Rev. E 68, No. 3, Article ID 031104, 9 p. (2003; doi:10.1103/PhysRevE.68.031104)]. For the explicit kernel, the evolution is linked to a discrete weighted heat equation on the positive integers by a nonlinear time-change. For this equation, we establish a new weighted Nash inequality that yields scaling-invariant decay and continuity estimates. Together with a replacement identity that links the discrete operator to its continuous analog, we derive a discrete-to-continuum scaling limit for the weighted heat equation. Reverting the time-change under the use of additional moment estimates, the analysis of the linear equation yields coarsening rates and self-similar convergence of the exchange-driven growth model.

MSC:

82C22 Interacting particle systems in time-dependent statistical mechanics
82D60 Statistical mechanics of polymers
35C06 Self-similar solutions to PDEs
35R09 Integro-partial differential equations
35Q82 PDEs in connection with statistical mechanics
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