Exchange Driven Growth

E. Ben-Naim and P.L. Krapivsky

We study a class of growth processes in which clusters evolve via exchange of particles. We show that depending on the rate of exchange there are three possibilities: I) Growth: Clusters grow indefinitely; II) Gelation: All mass is transformed into an infinite gel in a finite time; and III) Instant Gelation. In regimes I and II, the cluster size distribution attains a self-similar form. The large size tail of the scaling distribution is $\Phi(x)\sim\exp(-x^{2-\nu})$, where $\nu$ is a homogeneity degree of the rate of exchange. At the borderline case $\nu=2$, the distribution exhibits a generic algebraic tail, $\Phi(x)\sim x^{-5}$. In regime III, the gel nucleates immediately and consumes the entire system. For finite systems, the gelation time vanishes logarithmically, $T\sim [\ln N]^{-(\nu-2)}$, in the large system size limit $N\to\infty$. The theory is applied to coarsening in the infinite range Ising-Kawasaki model and in electrostatically driven granular layers.


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