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I am a bit confused about the relation between the thermodynamic potentials and

the partition function. Usually one has

[tex]Z = e^{-\beta A}, \qquad A = U-TS[/tex]

right? Here A is the Helmoltz free energy.

Now, when addressing magnetic system I see

[tex]Z = e^{-\beta G}[/tex]

where G is the Gibbs potential, whose "natural variables" are T (temperature, intensive) and H (magnetic field, intensive). That is, G is the analogue of the usual G(T,P) defined for non magnetic system, linked to the internal energy through two Legendre transforms.

What's the justification of this? I mean, G = A - PV. In a magnetic system G = A+MH, where M is the magnetization.

How can Z be the exponential of both A and G?

Also, I read equations like

[tex]U=G-T \frac{\partial G}{\partial T}[/tex]

(K. Huang, Statistical Mechanics pag 393). But this seems to me the relation

between U and A. It seems to me that a PV (or -MH for magnetic systems )

term is missing in the case of G.

Is the second equation for Z obtained by changing some of the assumptions that result in the first equation according to the "usual derivation"?

In some sense a different canonical ensemble?

Thanks a lot for any insight

F