- #1

nashed

- 58

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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?

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?

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