In the following, we shall present the formulation for tracing particles from surfaces to surfaces inside an enclosure. A more complete reference is Burns et al., 1990. To illustrate the concepts and computational aspects, we deal with prismatic surfaces, which are infinite in the third dimension. Thus, they can mathematically be described in two dimensions only.
We follow particles from ``birth'' (emission) to ``death'' (absorption or extinction), through possibly many intermediate interactions (reflections) with surfaces. As we directly simulate physical objects (particles), this is often termed the Direct Simulation Monte Carlo method.
Here, between surfaces, particles travel in straight lines and do not interact. Extensions to trajectories other than straight lines are conceptually straightforward, but can be vexing in the difficulty of implementation. Where particles do interact, this may introduce significant additional complexity. Our approach is representative of the transport of photons, neutrons, electrons and molecules in situations where there is no participating medium-where volumetric interactions do not occur. A variety of application areas including thermal radiative transport, neutron damage and molecular sputter are well modelled under these restrictive assumptions.
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 
and terminate upon surface
, defined as 
. The fraction of particles emitted from surface
and absorbed by surface is then 
, where 
is the total number of particles emitted by surface . Specifically, in
the case of radiative transfer, the flux of photons emitted from surface
is equal to 
. Thus, the one-way rate of radiative heat
transfer from surface to surface 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 in an enclosure and
tracing them to all surfaces 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.