Section \[sub:An-atmospheric-TRT\] for an atmospheric TRT example where the pressure and density dependence are included. We note that, for neutron problems, $\sigma^{s}$ can vary rapidly on the micro-scale as well, but we assume her for simplicity that $\sigma^{s}\left(T,E\right)$ smoothly depends on $E$ (i.e., is independent of $\varepsilon$).

The method is based on a rigorous homogenization theory for (\[eq:transport equation, intro\]), and has a computational cost that scales independently of the microscale parameter $\varepsilon$, aside from a pre-computation step analogous to computing Planck-weighted or Rosseland-weighted opacities. In this framework, the fine-scale solution $\psi_{\varepsilon}$ converges to the homogenized solution $\psi_{0}$ in the sense that $$\lim_{\varepsilon_{k}\rightarrow0}\int_{E_{0}}^{E_{1}}\psi_{\varepsilon_{k}}\left(\mathbf{x},\boldsymbol{\Omega},E\right)dE=\int_{E_{0}}^{E_{1}}\psi_{0}\left(\mathbf{x},\boldsymbol{\Omega},E\right)dE,$$ for arbitrary energy values $E_{0}$ and $E_{1}$ and some sequence $\varepsilon_{k}\rightarrow0$. Our derivation assumes that the total opacity $\sigma_{\varepsilon}\left(E,T\right)$ depends on a free parameter $\varepsilon>0$ (governing the characteristic spacing between lines), and that $\left\Vert \sigma_{\varepsilon}\left(\cdot,T\right)\right\Vert _{\infty}$ is uniformly bounded in $\varepsilon>0$ for each $T$. In practice, the final algorithm only makes use of a single opacity $\sigma_{\varepsilon}\left(T,E\right)$ at some fixed characteristic microscale $\varepsilon$ that is much smaller than the macroscopic scale we are interested in capturing.

As a key tool, we use the Young measure $\lambda_{E}^{T}$ associated with $\sigma_{\varepsilon}\left(E,T\right)$ (cf. [@BALL:1989]). Roughly speaking, $\lambda_{E}^{T}$ gives the probability 