The evolution of the system is integrated in time steps, τ τ , which are typically on the order of the mean collision time for a particle. At each time step all 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 r i ( t + τ τ ) = r i ( t ) + v i ( t ) τ τ . 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.