Z-time.gif | Two bouncing particles: z(t) versus time | Each particle has state (z,p) = (height,momentum) | In the following pictures you will get insights into Liouville's theorem. | Each animation is about a megabyte in size | |

demoE.gif | Two particles state-space view | The left-hand panel shows the heights of the particles; the right-hand one shows their STATES (p,z). When a particle hits the floor, its momentum p is instantly reversed. The dark blue lines show contours in state space of constant energy. | |||

demoL.gif | Many particles starting from one region of state space | Note that the initially-rectangular region evolves into an ever-more-sheared parallelogram, but ITS AREA REMAINS CONSTANT. This is Liouville's theorem. | It is worth viewing this gif in a player that allows you to step through the frames (eg xanim). | ||

demo1.gif | In this animation, in addition to the yellow particles from the previous picture, there are 7000 more particles. The states of all these particles were drawn from the BOLTZMANN DISTRIBUTION: P(state) ~ exp(-beta Energy(state)) | There's a lot to notice in this picture! Notice the flow is like the flow of an incompressible fluid, and that the *density* of particles only depends on the energy. A horizontal section through this picture at any height will yield a Gaussian distribution for velocity that is the same distribution at all heights; A vertical section gives the exponential distribution P(z) ~ exp(-mgz) | |||

demoA.gif | In this animation, which is much the same as the previous one, I tweaked the initial conditions of the particles so that the film loop looks almost continuous | I'm not sure this modification was a success! |