T05.1(b,c)

Using the next partition function obtained in a) :

b)

In the termodynamic limit :

Differential of work, from the lecture notes, know we can find the differential of work using the partial derivate of the logaritmic function of omega with respect to the mechanical control variables (in this case the oscllation frecuencies) :

 Internal energy:

 

Entropy, in the canonical ensemble we get the termodynamic potential Helmholtz energy so, the entropy can be obtained using the relation F=U-TS:

 

c)

For the microcanonical ensemble, i find the entropy:

And the differential of work and energy:

both results are the same as the ones found using the canonical ensemble.