Inspired, as usual, by Leonid’s recent post, I decided to first write a script that would mimic his. After that, since I had all the numbers worked out, I wrote two more MATLAB programs: one that mimicked elastic collisions in 2-dimensions, and one that mimics them in 3.
In theory, you can specify the number of particles and their radius, as well as the mass, position, and initial velocity for each (I didn’t vectorize radius for some reason, so I cannot model balls of different sizes bouncing around). However, in practice I just generate random vectors for each of these numbers. The final aspect is that the domain I put the balls in was a pool table of 9 x 4.5 units, or 9 x 4.5 x 4.5 for the 3D version. This was just to make calculating the reflecting angle easier when a ball hit the wall.
As with Leonid’s code, mine works by checking whether the next step will cause any collisions, then adjusting the velocity vector so that the collision didn’t happen (using conservation of momentum and kinetic energy). This algorithm is not “smart” in the sense that by avoiding one collision, it might get pushed into a second collision which it does not detect, and if a particle gets going fast enough, it can reflect off a wall from a large distance (my time step is just 0.01). You can spot this in some of the figures below.
Anyways, here are some of the outputs. I did not go through the trouble of turning these into .gifs, but they play fairly smoothly. What happens is I simulate N particles of varying masses and velocities bouncing around in a 2- or 3- dimensional box for T seconds, then plot the path of one of the particles. The end position of all the particles, plus this path, is in each picture below (with the “tracked” particle colored in red).