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  • $\begingroup$ Perhaps I need to be a little more explicit. If we couple a system (labeled 1) to a heat bath (labeled 2) we maximize the total entropy $S=S_1+S_2$. The total energy is conserved $E=E_1+E_2$. If we take the heat bath energy to be $E_2=T S_2$, then $S=S_1+E_2/T=S_1+(E-E_1)/T$. Maximizing $S$ is the same as minimizing $E_1-T S_1$, which is the Helmholtz Free Energy. You can do the exact same analysis for $PV$ and $\mu N$, and it doesn't really matter if particles aren't conserved and that $V=\infty$. However, how do you prove that $F-HM$ maximizes global entropy? $\endgroup$
    – ChickenGod
    Commented Mar 27, 2014 at 23:49
  • $\begingroup$ @ChickenGod: I don't know what the first half of your response has to do with the question at the end, or what you set $E_2=TS_2$ for, but let me ask you something in return: How does "$V$" in the derivation of the extremal conditions for the various potential single out that it has to do with volume. If you have a proof for $V$ and $P$, why doesn't it work if you use the other $M$ and $H$ in their place instead? $\endgroup$
    – Nikolaj-K
    Commented Mar 28, 2014 at 0:04
  • $\begingroup$ The first half of my response is a proof that the Helmholtz Free Energy is minimized at equilibrium, and $E_2=T S_2$ comes from integrating the definition of temperature: $1/T=\partial S/\partial E$, modulo a constant. Note that a crucial part of this proof is $E=E_1+E_2$, or $V=V_1+V_2$ or $N=N_1+N_2$, something that we can't write for the magnetic case. It doesn't matter that $V=\infty$ because $dV_1=-dV_2$ is what is really important, and also $\mu=-T\partial S/\partial N$ covers non-conservation of particles. $\endgroup$
    – ChickenGod
    Commented Mar 30, 2014 at 0:02
  • $\begingroup$ @ChickenGod: You can use $\mathrm dV_1=-\mathrm dV_2$ to show equilibration of the intensive variables $P_1,P_2$, but it's not, I think, relevant in the proof to show that $T\mathrm dS\ge \delta Q = \mathrm d(U-\frac{\partial U}{\partial q}\cdot q)$, were $q$ is $V, M, \dots$. $\endgroup$
    – Nikolaj-K
    Commented Mar 30, 2014 at 3:18