# 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.

• 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

## 6 Answers

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:

$$\alpha = dU -TdS + PdV$$

• @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
• @NickKidman: I was trying today to think of a case where the distinction between infinitesimal and differential would be useful just in standard thermodynamics (it's not easy). For example, consider a case where S(U) has a random component, like the thermodynamics of electrons occupying localized states. Perhaps you can find a case where the temperature is distributional. Really, I am just irate that a perfectly sensible notion, that of an infinitesimal differential, which is a source of intuition for hundreds of years of mathematics, has been excized from the curriculum and is made taboo. – Ron Maimon Jul 19 '12 at 1:50
• @RonMaimon: ''This is what everyone secretly thinks inside, and one should not hide it.'' You claim to know the sectrets of everyone, although you know it only of yourself. I spent months studying nonstandard analysis and its applications, and was very disappointed. It is far inferior to ordinary analysis, including diffferential geometry. therefore you don't find it seriously employed except by those coming from the nonstandard analysis school. Differential geometry, in contrast, is indispensable in many applications. – Arnold Neumaier Jul 19 '12 at 11:06
• @ArnoldNeumaier: This is what everyone secretly thinks, that's not an opinion, that's a statement of fact. You can tell, because they sometimes write stuff like "Near the critical temperature $T_0$, the magnetization of the 2d Ising model m(T) obeys $m(T_0+dT)= C(dT)^{1\over 8}$...". Especially someone who studied infinitesimals. How could infinitesimal analysis be inferior? It is at least equivalent to formal analysis, and the main point is that the proofs of Leibnitz are given 100% unimpeachable rigorous formulation which are not very different from Leibnitz's original presentation. – Ron Maimon Jul 19 '12 at 15:47

Chapter 7 of my online book Classical and quantum mechanics via Lie algebras derives in 17 pages (pp. 161-177) the main concepts of equilibrium thermodynamics in a physically elementary and mathematically rigorous form. Differential forms appear on p.167 where reversible transformations are defined, and are applied on p.168 to the Gibbs-Duhem equation and the first law of thermodynamics.

Note that Chapter 7 is completely self-contained can be read independent from the earlier chapters.

• Those are not really differential forms. Differential forms are not forms unless they are supposed to be integrated over a surface to give them meaning. The dT's and dP's in thermo are infinitesimals, not differentials, nor differential forms. What you are doing is shoehorning infinitesimal thinking into form language, which stunts one and misinterprets the other. I can't think of a real application of differential forms proper to thermodynamics, perhaps there is some index theorem for phase transitions somewhere. – Ron Maimon Jul 18 '12 at 17:30
• @RonMaimon: First of all, a differential form is a differential form even if it is never integrated. Integration of differential forms is possible but not part of the definition of a form. - However, you can indeed integrate all my formulas involving differentials over any path along which a reversible transformation is realized, and get meaningful and consistent results. - Infinitesimals and differential forms are the same thing, something well-known to mathematicians. – Arnold Neumaier Jul 18 '12 at 18:45
• @Arnold: infinitesimals and differential forms are not the same thing; the framework of non-standard analysis got a rigorous set-theoretic formulation in the 70s and there are results which are indeed more easily formulated using infinitesimals; I'm not sure if non-standard analysis adds anything to thermodynamics in particular, though – Christoph Jul 18 '12 at 18:55
• @Christoph: I didn't mean that they are formally the same thing; formally they are very different things. But the formulas of thermodynamics make perfect sense when interpreted as differentials in the sense of differential geometry rather than as infinitesimal sin the sense of Robinson. Indeed, Thermodynamics is not about infinitesimal changes (which are unobservable) but about finite reversible changes along arcs, for which you need to integrate a differential = differential 1-form along the arc. All integrals that one finds in discussions of Carnot cycles are integrals over 1-forms! – Arnold Neumaier Jul 18 '12 at 18:58
• @Arnold: your comment contains the quote "Infinitesimals and differential forms are the same thing, something well-known to mathematicians"; that seems to have been a typo, tough, if your argument is actually that differential forms and not infinitesimals are the correct framework for thermodynamics (which might indeed be the case - I don't know enough about non-standard analysis to be a judge of that) – Christoph Jul 18 '12 at 19:04

I'm afraid that from the aesthetic side, there is not too much differential geometry to discover in (equilibrium) thermodynamics (at least on an 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)

• 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

Professor Hannay wrote a very interesting article "Carnot and the fields formulation of elementary thermodynamics," Am. J. Phys. 74 2, February 2006, pp134-140. Putting aside the rather strange and unusual notation he shows that how Carnot's efficiency formula can be written using differential forms, specifically with wedge product. Rewritten in a more conventional form than is in the article Carnot's efficiency equation is written as $$\frac {1}{T} \tilde q \wedge \tilde dT = \tilde d \tilde w ,$$ where the ~ denotes a differential form, $$\tilde d$$ is the exterior derivative, $$\tilde q$$ and $$\tilde w$$ are the heat and work 1-forms. Hannay also writes the 1st and 2nd laws as: $$\tilde d \tilde q + \tilde d \tilde w = 0$$ and $$\tilde d \left( \frac {\tilde q}{T}\right) =0$$

So there is use for higher order forms than just 1-forms in thermostatics.

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.

• 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

I've got a big surprise about the seemingly extensive uncertainty on the physical interpretation of differential forms, differentials, and infinitesimals. My understanding is this: infinitesimal quantities can take any numbers that are "small without limit", but they represent always finite quantities (numbers). This is a kind of definition, what made possible to replace the "limes" in the definition of the derivative of a function, and define as the "quotient of differentials"! There is no difference between infinitesimals and differentials in this sense. If the definition of the derivative is different, do not use the same term as that in the classical meaning. This is what leads to misunderstanding. The terminology should be cleaned. In thermodynamics, the inner energy $$E$$ of a (thermodynamical) body is a homogeneous function of $$n$$ variables. $$E=E(V,r_1,...,r_{n-1})$$. It is quite absurd to maintain, that e.g. $$dE$$ is not a total differential of the infinitesimal variation of the independent variables (see Euler-formulas for zero-order and first-order homogeneous functions. I cope with the definition of densities as the derivatives of integral functions for more than one variable. I've thought, differential k-form will give the necessary insight this discussion revealed, the differential form is not the adequate mathematical tool for defining densities. Think for instance of the definition radiance (publication of Lessig).