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The Exchange-Driven Growth Model: Basic Properties and Longtime Behavior

  • André SchlichtingEmail author
Article
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Abstract

The exchange-driven growth model describes a process in which pairs of clusters interact through the exchange of single monomers. The rate of exchange is given by an interaction kernel K which depends on the size of the two interacting clusters. Well-posedness of the model is established for kernels growing at most linearly and arbitrary initial data. The longtime behavior is established under a detailed balance condition on the kernel. The total mass density \(\varrho \), determined by the initial data, acts as an order parameter, in which the system shows a phase transition. There is a critical value \(\varrho _c\in (0,\infty ]\) characterized by the rate kernel. For \(\varrho \le \varrho _c\), there exists a unique equilibrium state \(\omega ^\varrho \) and the solution converges strongly to \(\omega ^\varrho \). If \(\varrho > \varrho _c\), the solution converges only weakly to \(\omega ^{\varrho _c}\). In particular, the excess \(\varrho - \varrho _c\) gets lost due to the formation of larger and larger clusters. In this regard, the model behaves similarly to the Becker–Döring equation. The main ingredient for the longtime behavior is the free energy acting as Lyapunov function for the evolution. It is also the driving functional for a gradient flow structure of the system under the detailed balance condition.

Keywords

Convergence to equilibrium Exchange-driven growth Entropy method Mean-field equation Zero-range process 

Mathematics Subject Classification

Primary 34A35 Secondary 34G20 37B25 37D35 82C05 82C26 

Notes

Acknowledgements

The author enjoyed fruitful discussions with Stefan Grosskinsky on the derivation of the system as mean-field limit as well as with Stefan Luckhaus, Barbara Niethammer, and Juan Velázquez on the Becker–Döring system and related topics. The author thanks Emre Esenturk for bringing many references to his attention. Moreover, the author is incredibly grateful for the very constructive and detailed referee reports pointing out missing steps and misprints in an earlier version. The author acknowledges support by the Department of Mathematics I at RWTH Aachen University, where part of this work originates.

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut für Angewandte MathematikUniversität BonnBonnGermany

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