Non-metricity in the continuum limit of randomly-distributed point defects

TL;DR

This paper proves that a dense distribution of point defects in a manifold leads to a smooth limit with a non-metric connection, providing a rigorous foundation for non-metricity in continuum defect models.

Contribution

It introduces a homogenization theorem showing how randomly distributed point defects induce non-metricity in the continuum limit.

Findings

Manifolds with dense point defects converge to smooth Riemannian manifolds.

Levi-Civita connections converge to non-metric connections.

Non-metricity tensor naturally emerges in the continuum limit.

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.