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, i and j , is proportional to their relative speed, P c o l l [ i , j ] = | v i − − v j | ∑ ∑ m = 1 N c ∑ ∑ n = 1 m − − 1 | v m − − v n | where N c 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: