Surface Tension and $\mathrm{\Gamma }$-Convergence of Van der Waals-Cahn-Hilliard Phase Transitions in Stationary Ergodic Media

We study the large scale equilibrium behavior of Van der Waals-Cahn-Hilliard phase transitions in stationary ergodic media. Specifically, we are interested in free energy functionals of the following form \begin{equation*} $\mathcal{F}$^{\omega}(u) = \int_{$\mathbb{R}$^{d}} \left(\frac{1}{2} \varphi^{\omega}(x,Du(x))^{2} + W(u(x)) \right) , dx, \end{equation*} where $W$ is a double-well potential and ${\phi }^{\omega }(x,\cdot )$ is a stationary ergodic Finsler metric. We show that, at large scales, the random energy ${\mathcal{F}}^{\omega }$ can be approximated by the anisotropic perimeter associated with a deterministic Finsler norm $\tilde{\phi }$. To find $\tilde{\phi }$, we build on existing work of Alberti, Bellettini, and Presutti, showing, in particular, that there is a natural sub-additive quantity in this context.