the particles are moved and then a random set of pairs collide. In the absence of external fields (e.g., gravity) the particles move ballistically as . Any particle that reaches a boundary or a surface has its position and velocity reset accordingly (e.g., periodic boundary conditions). After all the particles have moved, they are sorted into cells and some are randomly selected to collide. based on probabilities and collision rates obtained from the kinetic theory of gases. After the velocities of all colliding particles have been reset, statistical sampling is performed and then the process is repeated for the next time step. Collisions On each timestep the particles are sorted into spatial cells and only particles in the same cell are allowed to collide. Typically the dimension of a cell is no larger than a mean free path. All pairs of particles in a cell are candidate collision partners, regardless of their actual trajectories. The details of how collisions are calculated in DSMC depend on the molecular interaction model; here we take the hard spheres model, which is the simplest. In the hard spheres model, the collision probability for the pair of particles, and , is proportional to their relative speed, where is the number of particles in the cell and the summations are over particles within the cell. Because of the double sum in the denominator it can be computationally expensive to use this collision probability directly. Instead, the following rejection sampling scheme can be used to select collision pairs: A pair of candidate particles, and , is chosen at random and their relative speed, , is computed. The pair is accepted as collision partners if , where is the maximum relative speed in the cell and is a uniform deviate in [0, 1). If the pair is accepted, the collision is processed; the velocities of the particles are reset but positions are unchanged. After the collision is processed or if the pair is rejected, return to step 1. This procedure is correct even if the value of is overestimated, although it is less efficient in the sense that more candidates are rejected. After the collision pair is chosen, their post-collision velocities, and , are evaluated. Writing the relative velocity in terms of spherical angles, and these angles are selected by a Monte Carlo process with distributions given by the collision model. For the hard spheres model these angles