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Continuum Limits of Particles Interacting via Diffusion
Nicholas Alikakos, Giorgio Fusco and Georgia Karali
We consider a two phase system mainly in 3 dimensions and we examine the coarsening of the spatial distribution, driven by the reduction of interface energy and limited by diffusion as described by the quasi static Stefan free boundary problem. Under the appropriate scaling we pass rigorously to the limit by taking into account the motion of the centers and the deformation of the spherical shape. We distinguish between two different cases and we derive the classical mean field model and another continuum limit corresponding to critical density which can be related to a continuity equation obtained recently by Niethammer and Otto.
So, the theory of Lifschitz-Slyosov and Wagner is improved by taking into account the geometry of the spatial distribution.