Understand Phase-Space Density: Basics & Concepts

[Shaybay92]

So I think I have the basic idea of what phase-space is... basically a way of representing all possible states of a system in some n dimensional space. So, what then, is phase-space density?

 

[Bill_K]

The individual states of your system lie somewhere in the phase space, but there's a limit to how close together they can be. For example for a system with one degree of freedom, the phase space is spanned by one coordinate and one momentum, x and p. But if you specify x very closely you can't specify p. It's the old Heisenberg uncertainly principle song: Δx Δp ~ h. So according to quantum mechanics each state must occupy a certain volume in phase space all by itself. The density of states ρ is the number of states per element of volume in phase space: dn = ρ dx dp. In this example, ρ will be a constant.

Likewise, for a system with N degrees of freedom you can use 2N variables x_i and p_i. But mechanics doesn't restrict you to Cartesian coordinates - you can use any coordinates you like - polar coordinates for example. If you do that, ρ will not be constant in terms of those coordinates. You'll need to calculate what it is by doing a change of variables.