Loading Relativistic Maxwell Distributions in Particle Simulations

Numerical algorithms to load relativistic Maxwell distributions in particle-in-cell (PIC) and Monte-Carlo simulations are presented. For stationary relativistic Maxwellian, the inverse transform method and the Sobol algorithm are reviewed. To boost particles to obtain relativistic shifted-Maxwellian, two rejection methods are proposed in a physically transparent manner. Their acceptance efficiencies are ${\approx}50\%$ for generic cases and $100\%$ for symmetric distributions. They can be combined with arbitrary base algorithms.