Finally, we note that, insofar as possible, we shall bypass the physics.
In so doing, we shall concern ourselves with only the matrix of the number
of particles which originate from surface **i** and terminate upon surface
**j**, defined as . The fraction of particles emitted from surface
**i** and absorbed by surface **j** is then , where
is the total number of particles emitted by surface **i**. Specifically, in
the case of radiative transfer, the flux of photons emitted from surface
**i** is equal to . Thus, the one-way rate of radiative heat
transfer from surface **i** to surface **j** is .
Explicitly, we shall concern ourselves only with , as the remainder
of the transport term is application specific and can include the emission
of photons, electrons, ions or molecules, for example. Thus, our problem
reduces to emitting particles from all surfaces **i** in an enclosure and
tracing them to all surfaces **j** of the enclosure, and the problem can be
formulated in terms of either or . ``Enough'' particles
must be emitted from every surface to achieve convergence.
Typically, this is accomplished by scaling up on successive runs
until convergence is attained.

To achieve an accurate simulation, several features are required. First, the physical characteristics of emission must be accurately represented. This includes the outgoing distributions of: direction, energy and spatial location. Secondly, geometry must be accurately represented---both the geometry of the enclosure and the geometry of the particle trajectory. Finally, interactions of the particles with surfaces (i.e. materials) must be accurately represented.

Although it is logical to conceptualize the process chronologically from birth to death, we partition our presentation into the two areas of geometry and material properties. The material properties of the surface determine both the emission distributions and the particle/surface interactions.