• Support PF! Buy your school textbooks, materials and every day products Here!

Microcanoncal partition function

  • Thread starter Pacopag
  • Start date
  • #1
197
4
1. Homework Statement
Does anyone know the mathematical definition of the microcanonical partition function?
I've seen
[tex]\Omega = {E_0\over{N!h^{3n}}}\int d^{3N}q d^{3N}p \delta(H - E)[/tex]
where H=H(p,q) is the Hamiltonian. This looks like a useful definition.
Only thing is I don't know what [tex]E_0[/tex] is.


2. Homework Equations



3. The Attempt at a Solution
 
Last edited:

Answers and Replies

  • #2
979
1
The microcanonical partition function is just a count of the number of states that satisfy extensive constraints on volume, energy, etc. The probability of each state is then trivially one over the partition function.
 
  • #3
197
4
But in the case of a classical system the number of states is uncountable because the position and momenta are continuous.
 
  • #4
979
1
In which case you can still calculate the phase space volume and the probability distribution is uniform over that volume --- it's the obvious generalisation.
 
  • #5
197
4
Ok. Good. Now I see why my "hint" was to bring the constant energy surface in phase space into a sphere (because I know how to find the volume of a sphere).

Thank you very much genneth.
 

Related Threads for: Microcanoncal partition function

  • Last Post
Replies
4
Views
3K
  • Last Post
Replies
2
Views
906
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
5
Views
3K
  • Last Post
Replies
3
Views
7K
  • Last Post
Replies
0
Views
2K
  • Last Post
Replies
0
Views
1K
  • Last Post
Replies
0
Views
2K
Top