# Gibbs free energy in second-order phase transitions

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In chapter 16 of K.Huang's book Statisical Mechanics ("Critical Phenomena") the order parameter $M$ and its conjugate field $H$ are introduced.

The author then claims that if we use $H$ and $T$ as independent thermodynamic variables, we can derive all thermodynamic functions from the Gibbs free energy $G(H,T)$, which is given by

$$Q(H,T)=e^{-\beta G(H,T)}$$

My question is: why is this expression identified with the Gibbs free energy? Since we are in the canonical ensemble, shouldn't the Gibbs free energy be given by the expression

$$Q(V,T)=e^{-\beta (G-PV)}$$

which comes from

$$Q(V,T)=e^{-\beta F(V,T)}$$

and

$$G=F+PV$$

where $F$ is the Helmoltz free energy? Has it something to do with the fact that we are using the variable $H$ and not $P$ or $V$?

I find also confusing the fact that $G$ does not depend on $P$. In Landau's book Statistical Physics, chapter XIV, the Gibbs free energy $\Phi$ is written as a function of $T,P,\eta$, where $\eta$ is the order parameter. But since $\Phi$ must satisfy $\partial \Phi /\partial \eta =0$ at equilibrium, $\Phi$ is actually a function only of $T,P$.

This seems to be in contradiction with the formula used by Huang. What am I missing?

Most magnetic systems can be treated as being incompressible to a good approximation. If you want to account for coupling between mechanical pressure and magnetization, however, then $P$ and $V$ should be included as thermodynamic variables (this would be relevant, for example, in describing the magnetic behavior of a gas of Fermions). As for the interpretation of Gibbs free energy for magnetic systems, it depends somewhat on your convention for internal energy---whether you define it as the energy associated with interactions within the sample, or whether you also include the interaction energy with the external field.

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Sorry, physicists are slack in terminology, and it's not Gibbs energy.

There're many "free energies", connected by Legendre transform, and each doesn't have its name. (You can't name them all.) For magnetism, T and H form a natural set of variables, neglecting pressure. I guess Huang calls it "Gibbs energy" since its all arguments are intensive variables. But it's bad and you shouldn't follow suit.

Sometimes when you take the sum of all values of an extensive variable (say in Monte Carlo simulation) it is called "grand canonical", even if that variable is not $N$. For example in Ising model, when you fix $M$ it is sometimes called "canonical", and when you fix $H$ and let $M$ vary, it is "grand canonical".

I don't mean pressure is not important. In some experiments samples are put under extremely high pressure, but for most solids tripling the pressure from 1 atm hardly changes the material's behavior. (See also a nice comment by an anonymous user.)

BTW: When you ask next time, be sure to state exactly where it is the equation you're referring to. Not only "chap 16", but "sec 16.1" or "in eq 16.2", and also "Huang's 2nd edition". (It's not frequent, but pages can change, e.g. in epub. I don't recommend spcification by the pages.) Remember those who answer (and those who read) spare their precious time for the forum participants. (Each way of communication has its own difficulty, in person, online, etc. ;-)

answered Jul 15, 2016 by (100 points)

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