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Sampling the free flight distance

According to the Beer-Lamber law the particle free flight follows an exponential distribution with the following pdf:
$  \wp\left ( d \right )=\mu e^{-\mu d}
A $ \mu where the attenuation coefficient (total macroscopic cross section) values can be taken e.g. from here here(external link).

Let us sample this using the inverse cumulative method:
$ \int_{0}^{X}\mu e^{-\mu y}dy= -e^{-\mu X}+1

Let us equate this to the on (0,1) Uniformly distributed r random number:
$ r= -e^{-\mu X}+1
After reordering and using the fact that the r random number is uniformly distributed on (0,1) therefore in distribution equals to 1-r:
$X_{i}=\frac{\ln \left (r  \right )}{\mu }

In heterogenous material distribution this sampling must be done for each homogenous subdomain separately.

After the flee flight we discuss interaction sampling.

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