# Homework Help: Prove the 3 definitions of entropy are equivalent (stat. mechanics)

1. Jul 9, 2014

### Tosh5457

1. The problem statement, all variables and given/known data

$$S(E,V) = kln(\Gamma(E) )\\ S(E,V) = kln(\omega(E) )\\ S(E,V) = kln(\Sigma(E) )\\$$

S entropy, k Boltzmann's constant. Prove these 3 are equivalent up to an additive constant.

2. Relevant equations

$$\Gamma(E) = \int_{E<H<E+\Delta}^{'}dpdq\\ \Gamma(E)=\omega\Delta \\ \Delta << E\\ \Sigma(E) = \int_{H<E}^{'}dpdq\\ \omega = \frac{\partial \Sigma}{\partial E}\\$$

H is the system's Hamiltonian and E is an arbitrary energy. These are integrations over all the p and q's, I wrote them like that to abbreviate.

3. The attempt at a solution

Using the 1st definition I can get to the 2nd one, but I can't reach at sigma's definition.
$$kln(\Gamma(E)) = kln(\omega\Delta) = kln(\omega) + kln(\Delta)\\ ln(\Delta) << ln(\omega) => S = kln(\omega)\\$$

Last edited: Jul 9, 2014
2. Jul 12, 2014

### Greg Bernhardt

I would say $\Gamma (E)=\Sigma (E+\Delta) -\Sigma (E)=\Delta \frac{\partial \Sigma}{\partial E}+...$