  Phonon Monte Carlo (PMC) is a versatile stochasic technique for solving the
Boltzmann transport equation for phonons. It is particularly well suited for
analyzing thermal transport in structures that have real-space roughness or are
too large to simulate directly using atomistic techniques. PMC hinges on the
generation and use of \textit{random variates} -- specific values of the random
variables that correspond to physical observables -- in a way that accurately
and efficiently captures the appropriate distribution functions. We present the
relative merits of the inversion and rejection techniques for generating random
variates on several examples relevant in thermal transport: drawing phonons
from a thermal distribution and with full or isotropic dispersion, randomizing
outgoing momentum upon diffuse boundary scattering, implementing contacts
(boundary and internal), and conserving energy in the simulation. We also
identify common themes in phonon generation and scattering that are helpful for
reusing code in the simulation (generating thermal-phonon attributes vs
internal contacts; diffuse surface scattering vs boundary contacts). We hope
these examples will inform the reader about the mechanics of random-variate
generation and how to choose a good approach for whatever problem is at hand,
and aid in the more widespread use of PMC for thermal transport simulation.
