Phase-Space (density?)

  • Thread starter Shaybay92
  • Start date
  • #1
124
0
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?
 

Answers and Replies

  • #2
Bill_K
Science Advisor
Insights Author
4,157
201
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 xi and pi. 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.
 

Related Threads on Phase-Space (density?)

  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
1
Views
624
  • Last Post
2
Replies
34
Views
4K
  • Last Post
2
Replies
29
Views
15K
  • Last Post
Replies
2
Views
2K
Replies
2
Views
3K
Replies
17
Views
1K
Replies
2
Views
892
Top