system assumed that the correlated assumption (\[eq:basic opacity assumption\]) is an equality for appropriate functions $\chi\left(x,E,T\right)$ and $\sigma_{\varepsilon}\left(E\right)$. In this section, we assume instead that $$\sigma_{\varepsilon}\left(E,T\right)\approx\chi\left(\sigma_{\varepsilon}\left(E,T_{0}\right),E,T\right),\label{eq:correlated assumption, relaxation section}$$ *approximately* holds, where $T_{0}$ is some appropriate reference temperature. That is, we approximate the opacity at a general temperature $T$ as functionally related to the opacity as some reference temperature $T_{0}$. As discussed below, we also include the possibility of slow energy variation in the functional relationship (\[eq:correlated assumption, relaxation section\]), which arises naturally in our following discussion. In general, the correlated assumption (\[eq:correlated assumption, relaxation section\]) does not hold. We therefore discuss in this section how to compute a function $\chi\left(x,E,T\right)$ that best approximates $\sigma_{\varepsilon}\left(E,T\right)\approx\chi\left(\sigma_{\varepsilon}\left(E,T_{0}\right),E,T\right)$.

To motivate the basic idea, consider two random variables $X\geq0$ and $Y\geq0$ and the associated joint probability density $p\left(x,y\right)=\mathbb{P}\left(X=x,Y=y\right)$. Then it is a well-known fact that the conditional expected value, $$\chi\left(x\right)\equiv\int_{0}^{\infty}yp\left(x,y\right)dy,\label{eq:conditional expected value}$$ minimizes the mean squared error $$\mathbb{E}\left(Y-\chi\left(X\right)\right)^{2}=\min_{g}\mathbb{E}\left(Y-g\left(X\right)\right)^{2}\equiv\int_{0}^{\infty}\int_{0}^{\infty}\left(y-g\left(x\right)\right)^{2}p\left(x,y\right)dxdy,$$ among all ($X$-measurable) functions $g$. In other words, $\chi\left(X\right)$ is the best functional fit to $Y$ in the sense of minimizing the average mean squared error.

This naturally leads us to consider the joint Young measure $\lambda_{E,T}\left(\xi_{1},\xi_{2}\right)$ associated with the pair of functions $\left(\sigma_{\varepsilon}\left(E,T_{0}\right),\sigma_{\varepsilon}\left(E,T\right)\right)$ (we explicitly 