Thermodynamics equivalence between entropy in two ensembles

[nashed]

So while practicing statistical mechanics problems I was faced with the following problem : calculate the entropy as function of energy for an ensemble of harmonic oscillators ( the Hamiltonian is ##\sum_{i=1}^N \frac {p_i^2} {2m} + \frac {m\cdot\omega\cdot q_i^2} 2##) ).

Now the official solution is given in the micro-canonical ensemble and turns out to be ## S \approx N\cdot K_b \ln {\frac {2\pi\cdot E} {Nh\omega}} ##

While using the canonical ensemble I get ## S=N\cdot K_b \ln {\frac {2\cdot\pi E} {Nh\omega}}+ 2NK_b - NKb\ln{N} ##

I'm not sure how to argue the equivalence under the thermodynamic limit, any suggestion on how to do that?

 

[Future Bruno]

The thermodynamic limit is essentially the limit of large system size, where the macroscopic properties of the system remain unchanged. In this case, the entropy of the system should not depend on the size of the system, since the total energy is fixed. This means that the entropy in the canonical ensemble should be equal to the entropy in the micro-canonical ensemble in the thermodynamic limit. To show this, we can take the limit of large N, and compare the two expressions for the entropy. As N goes to infinity, the second term in the canonical expression will go to zero, and the two expressions will be equal.