Bernoulli Trial (BT) Collision Operator Scheme Family in the Direct Simulation Monte-Carlo (DSMC): Derivation and Evaluation

The Direct Simulation Monte-Carlo (DSMC) method is based on splitting the rarefied gas process evolution into two ballistic and collision steps within a time step. The collision step is the more complicated, requiring a rigorous theoretical analysis to discover its relation to the fundamental kinetic equations. In this work, we present a systematic derivation and examination of the Bernoulli‑Trial (BT) family of collision schemes. The master Kac equation describes the binary collision interactions as a stochastic process lying in the background of the DSMC probabilistic rules, and it serves as a starting point for our derivation of the BT family. This study is limited to the analysis of four BT members, including Simplified Bernoulli Trials (SBT), Generalized Bernoulli Trials (GBT), Symmetrized Simplified Bernoulli Trials (SSBT), and Symmetrized Generalized Bernoulli Trials (SGBT). These schemes are implemented in the DSMC code, and a simple relaxation problem is simulated to determine optimal values for the pair‑selection parameter N sel .