Self-similar behavior of the exchange-driven growth model with product kernel

Abstract

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 ${(kl)}^{\lambda }$ for $\lambda \in [0,2).$ We rigorously establish the coarsening rates and convergence to the self-similar profile found by Ben-Naim and Krapivsky. 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.

Keywords:

• Coarsening
• exchange-driven growth
• Nash continuity
• Nash inequality
• self-similarity

Acknowledgement

The authors are grateful to Emre Esenturk, Stefan Grosskinsky, Barbara Niethammer, and Juan Velazquez for a number of insightful discussions and remarks on this and related topics. The authors thank the Hausdorff Research Institute for Mathematics (Bonn) for the hospitality during the Junior Trimester Program on Kinetic Theory.

Additional information

Funding

The authors are supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044 – 390685587, Mathematics Münster: Dynamics–Geometry–Structure, and EXC 2047 – 390685813, the Hausdorff Center for Mathematics, as well as the Collaborative Research Center 1060 – 211504053, The Mathematics of Emergent Effects at the Universität Bonn.