R V~(rR, E)+ o(r, E )~(rR, E)
dE dn' o- (r,0'- OE'4 E) YlI r, O',E')
0 4xr (1-6)
8[S (r, i',E)Y(r, 0',E ')]
+q(r, 1, E)
dE
However, the full Boltzmann-FP equation is extremely challenging to solve
deterministically .
1.4.1 Monte Carlo Method
Monte Carlo techniques are a widely used class of computational algorithms for the
purpose of statistically modeling various physical and mathematical phenomena. In general,
individual probabilistic events are simulated sequentially, where probability distributions
governing particle interactions are statistically sampled to describe the total phenomenon. In
principle, for each case sampled, there are four required inputs; namely, a random number
generator, source characteristics, a simulation phase space, and probability distributions
governing the possible outcomes. Results of individual particle histories are generated using
pseudo-random numbers to sample the probability distributions. This process is repeated many
times, until the statistical precision required for the solution is achieved, based on the law of
Large Numbers and the Central Limit Theorem"l. Therefore, the Monte Carlo technique is an
effective method for indirectly solving the Boltzmann radiation transport equation via statistical
methods, and can readily treat the full physics of coupled photon-electron transport. Simulations
for particle histories are repeated many times for large number of histories, where the computer
code tracks the desired physical quantities, means, and uncertainties (variances) for all of the
simulated histories.