# Gibbs vs Helmoltz potential in magnetic systems

• FranzDiCoccio
In summary: I'm starting to see the light now. In summary, the thermodynamic potentials and the partition function are related in a way that's not intuitive. The magnetic work term is ignored, and the equation linking U and G still seems weird to me.

#### FranzDiCoccio

Hi all,

I am a bit confused about the relation between the thermodynamic potentials and
the partition function. Usually one has

$$Z = e^{-\beta A}, \qquad A = U-TS$$

right? Here A is the Helmoltz free energy.

Now, when addressing magnetic system I see

$$Z = e^{-\beta G}$$

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?

$$U=G-T \frac{\partial G}{\partial T}$$

(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

FranzDiCoccio said:
I mean, G = A - PV. In a magnetic system G = A+MH, where M is the magnetization.

Shouldn't this be $G = A + PV$, $G' = A + PV - MH$, where $G'$ is the revised Gibbs potential for magnetic systems where $H$ is constant? After all, $U = TS - PV + MH + \sum_i \mu_i N_i$.

Could the sometimes-missing $PV$ term indicate that the authors consider $P-V$ work (i.e., thermal expansion work) to be negligible compared to magnetic work? (Or alternatively, that they assume the system to be in a vacuum?)

FranzDiCoccio said:
In some sense a different canonical ensemble?

Yes. There's a discussion about the canonical/Gibbs canonical ensembles in L13/14 of Kardar's http://ocw.mit.edu/OcwWeb/Physics/8-333Fall-2007/LectureNotes/index.htm [Broken].

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Hey, thanks a lot for your time

Mapes said:
Shouldn't this be $G = A + PV$, $G' = A + PV - MH$, where $G'$ is the revised Gibbs potential for magnetic systems where $H$ is constant? After all, $U = TS - PV + MH + \sum_i \mu_i N_i$.

Yes, you're right. I've got the signs wrong.

Could the sometimes-missing $PV$ term indicate that the authors consider $P-V$ work (i.e., thermal expansion work) to be negligible compared to magnetic work? (Or alternatively, that they assume the system to be in a vacuum?)

I agree on this as well, but the disappearance of the $P-V$ term does not bother me very much. My problem is with the partition function.
Shouldn't that be

$$Z = e^{-\beta (G+MH)}$$

Why the magnetic work term seems to be ignored?

Also, the equation linking $U$ and $G$ still seems weird to me.
I mean, ignoring the PV terms as much as one usually ignores the MH terms, it seems to me
that one should have

$$G - T \frac{\partial G}{\partial T} = G + TS = A - MH + TS = U - MH$$

i.e. once again one work term too much (K. Huang has U instead of U-MH).

Could this be related to the fact that $H$ is constant?

Thanks a lot again,

F

atyy said:
Yes. There's a discussion about the canonical/Gibbs canonical ensembles in L13/14 of Kardar's http://ocw.mit.edu/OcwWeb/Physics/8-333Fall-2007/LectureNotes/index.htm [Broken].

Nice suggestion, thanks. These notes are a bit condensed, but I think I'm seeing some light now. Man, I'm really slow.

Let me try to tell you what I am getting from all of this.

So when deriving the "usual" canonical ensemble one assumes that T is fixed and the
system exchanges heat with a heat bath. The volume is fixed in that case, so the system does not exchange other forms of work.
If the volume is allowed to change while keeping fixed the pressure, mechanical work is exchanged as well.

Now I can believe that by going along the same lines of the derivation of the "usual canonical ensemble" one gets an additional work term in the exponential giving the probability of a microstate, other than the internal energy term.

There are some signs and coefficients that I do not see right now, but now everything
makes more sense (perhaps).
I just see that the signs for the mechanic and magnetic work terms must be opposite,
essentially because pressure reduces the volume while the magnetic field increases the
magnetization.
I'll try to work out the details.

So the whole point seems to be that one wants to fix not only the temperature but also the external force (e.g. pressure or magnetic field).

Could we say that Gibbs Canonical Ensemble is doubly canonical?

Something I do not grasp completely is: why don't we include the additional term in the energy of the microstate? I'm thinking to particles with a magnetic moment, for instance...
But perhaps I should understand completely the isobaric ensemble and draw an analogy with the magnetic system.

Thanks again

F

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FranzDiCoccio said:
Hi all,

I am a bit confused about the relation between the thermodynamic potentials and
the partition function. Usually one has

$$Z = e^{-\beta A}, \qquad A = U-TS$$

right? Here A is the Helmoltz free energy.

Now, when addressing magnetic system I see

$$Z = e^{-\beta G}$$

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?
...

It can't. This is the story : source of your ( and not only yours! ) confusion is that some authors insist that internal energy U must have all extensive variables ! The price they pay for such IMHO unreasonable approach is that in presence of field B they "redefine" internal energy as U' = U + M*B ! End result is that whenever magnetic field appears in thermodynamics of your problem, all formulas must be changed to preserve this "deep principle about natural variables of U", so you got two sets of formulas : magnetic ones and nonmagnetic ones. Difference between those is what confuses you.

In my opinion, the best thing you can do is avoid this by accepting that field B ( H ) is "natural" variable of internal energy U ( named spectroscopic internal energy ) and only then will you have consistent relations where one doesn't have to worry is Z exponential of A or G this Monday =)

xlines said:
It can't. This is the story : source of your ( and not only yours! ) confusion is that some authors insist that internal energy U must have all extensive variables ! The price they pay for such IMHO unreasonable approach is that in presence of field B they "redefine" internal energy as U' = U + M*B ! End result is that whenever magnetic field appears in thermodynamics of your problem, all formulas must be changed to preserve this "deep principle about natural variables of U", so you got two sets of formulas : magnetic ones and nonmagnetic ones. Difference between those is what confuses you.

In my opinion, the best thing you can do is avoid this by accepting that field B ( H ) is "natural" variable of internal energy U ( named spectroscopic internal energy ) and only then will you have consistent relations where one doesn't have to worry is Z exponential of A or G this Monday =)

Hi xlines,

thanks for your reply. I'm sort of glad of knowing that I'm not the only one who gets confused by this.

Unfortunately at the moment I'm still struggling with all of this stuff. I sort of see the issues at stake, but I'm not able to "tie the loose ends". Meaning I think I have not really understood the problem yet.
A part of the story seems to be changing the name of things. What people seems to call energy here is what one would usually call an enthalpy. Or is it?
I think this has to do with your comment about the "natural variables" of the internal energy being extensive quantities.
I see that they are all forms of energy, but this name-changing makes me loose my bearings with thermodynamic functions. Doesn't all this mess up the "thermodynamic squares"?

Could you suggest a book or website where these things are discussed? So far the only place I've found is the MIT lecture notes suggested by atyy, but I'd like to see some examples, or perhaps a derivation...

Anyway thanks again

F

## 1. What is the Gibbs vs Helmoltz potential in magnetic systems?

The Gibbs and Helmoltz potentials are thermodynamic potentials that describe the energy and entropy of a magnetic system, respectively. They are used to analyze the equilibrium states of a system and determine the conditions under which a system will spontaneously change its state.

## 2. How do these potentials differ from each other?

The main difference between the Gibbs and Helmoltz potentials is the variable they hold constant. The Gibbs potential holds temperature and pressure constant, while the Helmoltz potential holds temperature and volume constant.

## 3. Which potential is more commonly used in analyzing magnetic systems?

In most cases, the Gibbs potential is the more commonly used potential in analyzing magnetic systems. This is because it takes into account both temperature and pressure, which are important factors in determining the equilibrium state of a system.

## 4. What is the significance of these potentials in magnetic systems?

The Gibbs and Helmoltz potentials play a crucial role in understanding the behavior of magnetic systems. They help determine the stability, phase transitions, and magnetic properties of a system.

## 5. Are there any limitations to using these potentials in magnetic systems?

While the Gibbs and Helmoltz potentials provide valuable insights into the behavior of magnetic systems, they do have some limitations. These potentials assume a closed and isolated system, and do not take into account external factors such as magnetic fields and non-equilibrium conditions.