Abstract
We present a homogenization theorem for isotropically-distributed point defects, by considering a sequence of manifolds with increasingly dense point defects. The loci of the defects are chosen randomly according to a weighted Poisson point process, making it a continuous version of the first passage percolation model. We show that the sequence of manifolds converges to a smooth Riemannian manifold, while the Levi-Civita connections converge to a non-metric connection on the limit manifold. Thus, we obtain rigorously the emergence of a non-metricity tensor, which was postulated in the literature to represent continuous distribution of point defects.
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Kupferman, R., Maor, C. & Rosenthal, R. Non-metricity in the continuum limit of randomly-distributed point defects. Isr. J. Math. 223, 75–139 (2018). https://doi.org/10.1007/s11856-017-1620-x
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DOI: https://doi.org/10.1007/s11856-017-1620-x