# Introduction to differential forms in thermodynamics

I've studied differential geometry just enough to be confident with differential forms. Now I want to see application of this formalism in thermodynamics.

I'm looking for a small reference, to learn familiar concepts of (equilibrium ?) thermodynamics formulated through differential forms.

Once again, it shouldn't be a complete book, a chapter at max, or an article.

UPD Although I've accepted David's answer, have a look at the Nick's one and my comment on it.

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This is the second time somebody has confused the differential forms of algebraic geometry with the infinitesimal displacements in thermodynamics. They are only related because mathematicians decided to purge infinitesimals from math, only to have Abraham Robinson reintroduce them with a vengeance. Just because it has a d in it, doesn't make it a differential form. –  Ron Maimon Jul 18 '12 at 18:11
@RonMaimon: It seems that you are not familiar with the well-known fact that one can give the infinitesimals in analysis a perfectly valid interpretation in terms of differential geometry. See, e.g., the book ''Applied differential geometry'' by Burke. From p.xiii of the preface: ''Here we will be able to turn most of the infinitesimals commonly seen in physics into the appropriate geomentric objects, usually into either rates (tangent vectors) or gradients (differential forms).'' –  Arnold Neumaier Jul 18 '12 at 18:53
@ArnoldNeumaier: Of course I am familiar with it, it works for a very special case--- smooth analysis. The key word in the quote is "most". It used to be "most" but in modern physics it's only "often" and "fewer and fewer". The infinitesimal analysis of nonsmooth objects took over with the path integral. The derivative of $\phi$ appearing in the scalar path integral is a nonsmooth infinitesimal change. It also puts a layer of obfuscation on top of infinitesimals, which are rigorous as they stand, and Leibnitz's definition was essentially ok, as shown and extended by Robinson. –  Ron Maimon Jul 18 '12 at 21:11
@RonMaimon: Show me any physically useful thing done with Robinson-stylew infinitesimals in thermodynamics that cannot be done with differential forms. Differential forms give very naturally and with little technical overhead all the transformations that physicists need. On the other hand, Robinson needs already a lot of work to even define infinitesimals and get to the point where they can be used in analysis. And hardly anyone is using it; in physics nobody I know of. –  Arnold Neumaier Jul 19 '12 at 11:02
@ArnoldNeumaier: I don't call them "Robinson style infinitesimals", I call them "physicist's infinitesimals". Robinson's stuff is just the way to force this on mathematicians. Consider the long-wavelength thermodynamics of the magnetization of the 3d Ising model. Consider only the long-wavelength fluctuations m_\sigma(x) over a ball of infinite radius $\sigma$ centered at x. This infinite wavelength magnetization is described by 3d self-interacting scalar with infinitesimal couplings, so to talk about the spatial derivatives of m is a full path integral. This appears in Landau somewhere. –  Ron Maimon Jul 19 '12 at 16:08
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There are two articles by S.G. Rajeev: Quantization of Contact Manifolds and Thermodynamics and A Hamilton-Jacobi Formalism for Thermodynamics in which he reviews the formulation of thermodynamics in terms of contact geometry and explains a number of examples such as van der Waals gases and the thermodynamics of black holes in this picture.

Contact geometry is intended primarily to applications of mechanical systems with time varying Hamiltonians by adding time to the phase space coordinates. The dimension of contact manifolds is thus odd. Contact geometry is formulated in terms of a basic one form, the contact one form:

$\alpha = dq^0 -p_i dq^i$

($q^0$ is the time coordinate). The key observation in Rajeev's formulation is that one can identify the contact structure with the first law:

$\alpha = dU -TdS + PdV$

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This is a nice answer, but I don't think it is good to mislead students about this--- the formalism of differential forms is not the proper way to talk about infinitesimal ideas, they are too poor a concept. The student is just seeing "dT" and "dP" and thinks this is something to do with the wedge product and submanifolds, when it doesn't. –  Ron Maimon Jul 18 '12 at 18:13
@Ron: it doesn't? how do you interpret the legendre-transformation in terms of infinitesimals? –  Christoph Jul 18 '12 at 18:56
@RonMaimon: one doesn't need wedge products to define the 1-forms used in thermodynamics. And the manifolds are 1-dimensional, so everythign is very natural and simple. –  Arnold Neumaier Jul 18 '12 at 19:04
@ArnoldNeumaier: My objection is that it is too simple a special case--- it is a trivial case of the glory of antisymmetric high dimensional forms, and therefore the formalism is a bad fit to the domain. One dimensional forms and curves are just gradients and the fundamental theorem of calculus. This is equivalent to intuitive infinitesimal thinking (or to rigorous infinitesimal thinking), but the latter generalizes immediately to cases like random walks and stochastic calculus, where the functions are distributions. This is what everyone secretly thinks inside, and one should not hide it. –  Ron Maimon Jul 18 '12 at 21:06
@Christoph: The way Legendre thought about it, his work predates Cauchy and Weierstrauss. The infinitesimal change in (U-TS) is dU - TdS - SdT. It happens to be equivalent to form calculus, which is why form notation is chosen to coincide with infinitesimal notation, but forms are not infinitesimal, they are differential, and they have a limited set of operations defined on them, those appropriate for smooth calculus. It's a subset of the infinitesimal intuition which is identical in the smooth case, but the infinitesimal intuition gives you the nonsmooth analogs, like in Ito calculus. –  Ron Maimon Jul 18 '12 at 21:08
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I'm afraid that from the aesthetic side, there is not too much differential geometry to discover in (equilibrium) thermodynamics (at least on a undergrad level and if you don't want to bother with the conceptual question how to properly define the idea of heat for the most abstract situations). I suppose any book on thermodynamics has some sections, which makes use of the mathematical properties, which come from holding on parameter constant and so on.

So I suggest that starting with the axioms and the potentials, you involve yourself with the following basic statements, which make "heavy use" of the formalism:

(The articles all contain the derivations too)

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That's indeed a nice point, it doesn't really worth introducing differential forms just for themselves. It would be worthy if such a formalism naturally incorporated in its structure the duality, conjugacy of thermodynamic variables. David has provided an example of this approach, but it is higher than an undergrad level and frankly speaking by this time I've never been actually meditating on this conjugacy. I really need to think it over, especially in the view of classical non-equllibrium thermodynamics. –  Yrogirg Jul 21 '12 at 15:34

Bernard Schutz. Geometrical Methods in Physics.

Differentials Forms with Applications to the Physical Sciences

Each has a chapter or a section on this, but I don't know what they are offhand.

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You are confusing the "dT" "dP" in thermodynamics with differential forms. This is unfortunate--- there are two different concepts, differential forms and infinitesimals. Differential forms are antisymmetric algebra of antisymmetric tensors useful for integration over different dimension surfaces. The formalism is useful for expressing topological integrals, for cohomology.

The "differentials" you see in thermodynamics are really infinitesimal displacements, not differentials. You must keep the concepts distinct, as you can sensibly talk about $\sqrt{dT}$ or $dT^2$ in thermodynamics, which make no sense as differential forms.

I explained this important distinction in this answer: Differentiating the ideal gas law . It has a lot of downvotes, but I am not confused on this.

Try to not get confused either. Differential forms are not a good substitude for infinitesimals, and neither are the abstract differentials in smooth calculus. The only rigorous substitute for infinitesimals are rigorous infinitesimals, and the only good formal version is Abraham Robinson's nonstandard analysis. This allows for any manipulation on real numbers to be extended logically to infinitesimal numbers, including taking square-roots, and taking infinitesimal powers, and whatever you want to do.

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I expected ignorant downvotes on this, given the stupid downvotes on the linked answer to the other question, but for the OP, please remember that when learning differential forms, they are not the rigorous version of infinitesimals, and people are using infinitesimals in thermodynamics, not forms. Forms are only the formalism for Stoke's theorem and differentials on smooth manifolds and submanifolds. –  Ron Maimon Jul 18 '12 at 18:06
Differential 1-forms are useful for integration over curves (1-dimensional manifolds), as they appear in even the most elementaty discussions of thermodynamics applied to heat engines. –  Arnold Neumaier Jul 18 '12 at 19:02
Ron, with reference to your comment to another answer, I learnt one-forms as the elements of the cotangent space to a manifold (ie dual vectors) and p-forms as the tensor product thereof. Why do they need to be integrated to have meaning? –  James Jul 18 '12 at 20:21
@James: I meant only that their purpose is that they are ready to be integrated over lower dimensional surfaces, one forms over curves, two forms over surfaces. That's what gives the calculus meaning--- the wedge product and the restriction to antisymmetry. Of course the components have meaning independently. I learned them the same way--- they are the antisymmetric projection of the tensor product of p-forms (the antisymmetric projection is what makes any form higher than a 1 form useless for thermodynamics, and important for integration over subsurfaces). We have no disagreements over forms. –  Ron Maimon Jul 18 '12 at 20:54
@ArnoldNeumaier: I agree that the 1-dimensional integrals of 1-forms appear in thermodynamics. That's the extent of it. These appear because this is the integrability condition of a function to be a gradient. The point of the form calculus is to extend this to integrability of higher dimensional objects, and I don't know a single instance where you use a true p-form with p>1 in thermodynamics. The p=1 case is just a trivial case, and can be understood more simply in other ways. –  Ron Maimon Jul 18 '12 at 20:55