Diffuse Interface Energies with Microscopic Heterogeneities: Homogenization and Rare Events

TL;DR

This paper studies the homogenization of Allen-Cahn energies with microscopic heterogeneities, showing how the ratio of heterogeneity scale to interface width determines whether homogenized limits or rare events dominate.

Contribution

It establishes precise conditions under which homogenization applies and identifies the rare events regime where atypical configurations influence the energy limit.

Findings

Homogenization holds when the heterogeneity scale decays fast enough compared to interface width.

Rare events can dominate the energy limit if the heterogeneity scale decays too slowly.

The results apply to both random and almost periodic media.

Abstract

We analyze Allen-Cahn functionals with stationary ergodic coefficients in the regime where the length scale $\delta$ of the heterogeneities is much smaller (microscopic) than the interface width $\epsilon$ (mesoscopic). In the main result, we prove that if the ratio $\delta \epsilon^{-1}$ decays fast enough compared to $\epsilon$, then homogenization effects dominate, and the $\Gamma$-limit of the energy is the same as if the coefficients had been replaced by their homogenized values. As a byproduct of the proof, this implies that homogenization holds in the periodic setting whenever $\delta \epsilon^{-1}$ vanishes with $\epsilon$, no matter how slowly. Via explicit examples, we prove this is sharp: if $\delta \epsilon^{-1}$ decays too slowly, then improbable or atypical local configurations of the medium begin to play a role, and the $\Gamma$-limit may be smaller than the one predicted by homogenization theory. We refer to this as the rare events regime, and we prove that it can occur in both random and almost periodic media.