Phonon Monte Carlo: Generating Random Variates for Thermal Transport Simulation

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.