Monte Carlo Methods for Solving the Boltzmann Transport Equation

Abstract

We review Monte Carlo methods for solving the Boltzmann equation for applications to small-scale transport processes, with particular emphasis on nanoscale heat transport as mediated by phonons. Our discussion reviews the numerical foundations of Monte Carlo algorithms, basic simulation methodology, as well as recent developments in the field. Examples of the latter include formulations for calculating the effective thermal conductivity of periodically nanostructured materials and variance-reduction methodologies for reducing the computational cost associated with statistical sampling of field properties of interest, such as the temperature and heat flux. Recent developments are presented in the context of applications of current practical interest, including multiscale problems that have motivated some of the most recent developments.

Simulation of Heat Transport in Graphene Nanoribbons Using the Ab-Initio Scattering Operator

“…As shown previously [5,6,9], deviational methods are significantly more efficient than traditional Monte Carlo methods because they simulate only the deviation from equilibrium. This results in drastically reduced statistical uncertainty, but also the ability to automatically and adaptively concentrate the computational effort in regions where kinetic effects are important, which is of great importance in the simulation of multiscale phenomena [6,9,12].…”

“…This results in drastically reduced statistical uncertainty, but also the ability to automatically and adaptively concentrate the computational effort in regions where kinetic effects are important, which is of great importance in the simulation of multiscale phenomena [6,9,12].…”

“…Here, n 0 λ = n 0 (ω λ ; T 0 ) is an appropriately chosen Bose-Einstein (equilibrium) distribution; the control temperature T 0 may be chosen as a function of space as a means of improving variance reduction or imposing temperature gradients on the material [6,7,9].…”

“…Equation (7) highlights the fact that this work uses a discrete representation of the reciprocal space required by the scattering model, but a continuous representation of physical space. The dynamics of these "deviational" particles follow [6,7,9] from…”

“…The technique proposed here is based on the low variance deviational simulation Monte Carlo (LVDSMC) [5][6][7][8][9], an efficient stochastic method for obtaining solutions of the BTE. In this paper, we present and discuss the algorithmic developments necessary for treating the ab initio collision operator and its discrete representation in reciprocal space.…”