distribution of values $\sigma_{\varepsilon}\left(E,T\right)$ in a vanishingly small neighborhood of $E$ as $\varepsilon\rightarrow0$. The key property that the Young measure $\lambda_{E}^{T}$ satisfies (see [@BALL:1989] for a proof) is that, for any continuous function $F\left(E,\xi\right)$ defined for $E\geq0$ and $\xi\geq0$, there is a sequence $\varepsilon_{k}\rightarrow0$ such that $$\lim_{\varepsilon_{k}\rightarrow0}\int_{E_{0}}^{E_{1}}F\left(E,\sigma_{\varepsilon_{k}}\left(E,T\right)\right)dE=\int_{E_{0}}^{E_{1}}\left(\int_{0}^{\infty}F\left(E,\xi\right)\lambda_{E}^{T}\left(d\xi\right)\right)dE.\label{eq:Young measure, key rep-1}$$ Intuitively, for small $\varepsilon$ and for all $E'\in\left[E-\Delta E/2,E+\Delta E/2\right]$ in a neighborhood of $E$ with $0<\varepsilon\ll\Delta E$, the values $F\left(E',\sigma_{\varepsilon}\left(E'\right)\right)\approx F\left(E,\sigma_{\varepsilon}\left(E'\right)\right)$ can be wildly varying since $\sigma_{\varepsilon}\left(E'\right)$ can rapidly oscillate for $E'\in\left[E-\Delta E/2,E+\Delta E/2\right]$; however, the average value of $F\left(E,\sigma_{\varepsilon}\left(E'\right)\right)$ for $E'\in\left[E-\Delta E/2,E+\Delta E/2\right]$ is given by weighting $F\left(E,\xi\right)$ against the probability of $\sigma_{\varepsilon}\left(E\right)\in\left[\xi-d\xi/2,\xi+d\xi/2\right]$, $$\int_{0}^{\infty}F\left(E,\xi\right)\lambda_{E}^{T}\left(d\xi\right).\label{eq:expected value}$$ A simple but instructive example is the Elsasser band model [@ELSAS:1938], $$\sigma_{\varepsilon}\left(E\right)=\frac{\cosh\left(\beta\right)+1}{\cosh\left(\beta\right)-\cos\left(2\pi E/\varepsilon\right)},\label{eq:Elsasser band model, intro}$$ $\beta>0$, which models an infinite number of Lorenz lines with equal spacing of $\varepsilon$ and uniform strength; in Section \[sub:Example-1:-the\], we compute $\lambda_{E}\left(\xi\right)$ for (\[eq:Elsasser band model, intro\]) analytically. The Young measure for $\sigma_{\varepsilon}\left(E,T\right)$–and the associated homogenized solution—exists if $\left\Vert \sigma_{\varepsilon}\left(\cdot,T\right)\right\Vert _{\infty}$ is uniformly bounded in $\varepsilon>0$ for each $T$ [@BALL:1989]. Our derivation will be a direct application of (\[eq:Young measure, key rep-1\]).

In order to derive the homogenized equations from (\[eq:Young measure, key rep-1\]), we assume 