# Microcanoncal partition function

1. Apr 11, 2008

### Pacopag

1. The problem statement, all variables and given/known data
Does anyone know the mathematical definition of the microcanonical partition function?
I've seen
$$\Omega = {E_0\over{N!h^{3n}}}\int d^{3N}q d^{3N}p \delta(H - E)$$
where H=H(p,q) is the Hamiltonian. This looks like a useful definition.
Only thing is I don't know what $$E_0$$ is.

2. Relevant equations

3. The attempt at a solution

Last edited: Apr 11, 2008
2. Apr 12, 2008

### genneth

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. Apr 12, 2008

### Pacopag

But in the case of a classical system the number of states is uncountable because the position and momenta are continuous.

4. Apr 12, 2008

### genneth

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. Apr 12, 2008

### Pacopag

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.

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