are uniformly distributed over the unit sphere. The azimuthal angle is uniformly distributed between 0 and , so it is selected as where is a uniform deviate in [0, 1). The polar angle is distributed according to the probability density, Using the change of variable , we have so The post-collision velocities are set as Note that by conservation of linear momentum and energy the center of mass velocity and the relative speed are unchanged in a collision. That is, and This process is repeated for every pair of colliding particles. From the collision frequency, , given by kinetic theory the total number of hard sphere collisions in a cell during a time is where is the particle diameter and is the volume of the cell. Since collision candidates go through a rejection sampling procedure the ratio of total accepted to total candidates for hard sphere particles is The number of collision candidates selected in a cell over a time step is This approach for determining the number of collisions is known as the No-Time-Counter (NTC) method. If is set excessively high then the algorithm processes the same number of collisions (on average) but the simulation is inefficient because many candidates are rejected. References External links Direct Simulation Monte Carlo Method: Visual Simulation Programs created by GA Bird. DSMC Demo Applet by Greg Khanlarov Course material on DSMC (part of Computational Physics tutorial by Franz J. Vesely, University of Vienna) Course material on DSMC and recent developments (given at IPAM UCLA by Lorenzo Pareschi, University of Ferrara) Monte Carlo methods Statistical mechanics