Radiative transfer equation

Example

Let $D \subseteq \R^3$ be a region filled with a medium, and let $S^2$ be the unit sphere of directions. The unknown is the specific intensity $I(x, \Omega)$ of radiation at a point $x \in D$ travelling in the direction $\Omega \in S^2$, at one fixed frequency. Let $\kappa(x) \ge 0$ and $\sigma(x) \ge 0$ be the absorption and scattering coefficients, $B(x)$ the emission source, and $p(\Omega' \cdot \Omega) \ge 0$ the phase function, normalized by $\frac{1}{4\pi} \int_{S^2} p(\Omega' \cdot \Omega) \, d\Omega' = 1$. The stationary radiative transfer equation is $$\Omega \cdot \nabla_x I(x, \Omega) + \big(\kappa(x) + \sigma(x)\big) I(x, \Omega) = \kappa(x) B(x) + \frac{\sigma(x)}{4\pi} \int_{S^2} p(\Omega' \cdot \Omega) \, I(x, \Omega') \, d\Omega'.$$ It contains a derivative of $I$ along the direction $\Omega$ and an integral of $I$ over all directions. So it is an integro-partial differential equation.

Check: if $\kappa$, $\sigma$ and $B$ are constant, the constant intensity $I = B$ is a solution. The left side is $(\kappa + \sigma) B$, and by the normalization of $p$ the right side is $\kappa B + \sigma B$.