Helmholtz free energy in the canonical ensemble

[amedeo_fisi]

Hello everybody :D
My question is: given the distribution of the canonical ensemble, how do we get the helmoltz free energy?
I think we can't use A = U-TS because we don't know how to write S. So what's the solution? Thanks

 

[DrClaude]

The simplest way is to use
$$
A = - k T \ln Z
$$

 

[amedeo_fisi]

Thanks DrClaude, but I already know the result! How do you get that formula?

 

[DrClaude]

You should be able to find the derivation in any statistical physics textbook. You can also find a derivation on Wikipedia.