Classical Phase Space Densities for One or a Few Particles

This is the first time in several years that I've done a post about physics that didn't have to do with my research. This came about from thinking about applying techniques in statistical physics to game theory; although I still have a lot more to learn about that and need to do more to flesh out those ideas, it occurred to me in the process that I never had such a good intuition for the phase space density in classical mechanics, and notes that I've found online focus almost exclusively on the phase space density of a large number of particles in an explicitly statistical treatment. I intend to use this post to shed light on why this may be the case, help build intuition for how things like the Liouville equation work for simple systems of one or a few particles, and reinforce the notion that there is no classical analogue to the phenomenon of a multi-particle entangled quantum state yielding a mixed single-particle state under a partial trace. Follow the jump to see more.