# State-space diagram of vertically-bouncing particles

 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!
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