Monte Carlo method for radiative transfer

Definition

Let $D \subseteq \R^3$ be a region filled with a medium with absorption coefficient $\kappa \ge 0$ and scattering coefficient $\sigma \ge 0$, where $\kappa + \sigma > 0$. Let $p$ be a scattering phase function, normalized as in the radiative transfer equation. Assume for simplicity that $\kappa$ and $\sigma$ are constant, and put $\beta = \kappa + \sigma$. The Monte Carlo method for radiative transfer estimates radiative quantities by simulating $N$ independent bundle histories, each as follows.

1. Draw the emission point and the initial direction from the distribution of the emitted radiation.
2. Draw the free path length $s = -\ln(\xi)/\beta$ with $\xi$ uniform on $(0, 1)$, and move the bundle a distance $s$ along its direction.
3. If the bundle leaves $D$, record where it hits the boundary and stop.
4. Otherwise, with probability $\kappa / \beta$, record an absorption at that point and stop. With probability $\sigma / \beta$, draw a new direction $\Omega$ with density $p(\Omega' \cdot \Omega)/(4\pi)$, where $\Omega'$ is the old direction, and go to step 2.

A quantity of interest is, for example, the energy absorbed in a subregion or reaching a part of the boundary. It is estimated by the fraction of the $N$ histories that end there, times the total emitted power. This is an instance of the Monte Carlo method.