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According to the Beer-Lamber law the particle free flight follows an exponential distribution with the following pdf:
{EQUATION(size="70")}$ \wp\left ( d \right )=\mu e^{-\mu d}{EQUATION}
A {EQUATION(size="70")}$ \mu {EQUATION} where the attenuation coefficient (total macroscopic cross section) values can be taken e.g. from here [http://physics.nist.gov/PhysRefData/Xcom/html/xcom1.html|here].
Let us sample this using the inverse cumulative method:
{EQUATION(size="70")}$ \int_{0}^{X}\mu e^{-\mu y}dy= -e^{-\mu X}+1 {EQUATION}
Let us equate this to the on (0,1) Uniformly distributed ''r'' random number:
{EQUATION(size="70")}$ r= -e^{-\mu X}+1 {EQUATION}
After reordering and using the fact that the ''r'' random number is uniformly distributed on (0,1) therefore in distribution equals to 1-r:
{EQUATION(size="70")}$X_{i}=\frac{\ln \left (r \right )}{\mu } {EQUATION}
In heterogenous material distribution this sampling must be done for each homogenous subdomain separately.
After the flee flight we discuss interaction sampling.