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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.

Qmechanic
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Yrogirg
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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
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    @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
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    @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
  • @RonMaimon: And now make this logically impeccable. You'll have far more difficulties with the infinitesimal logic than with the logic of standard analysis. But I see you don't care about rigor; so its useless to discuss this further. – Arnold Neumaier Jul 19 '12 at 17:54
  • @ArnoldNeumaier: I do care about rigor. Infinitesimals are rigorous now, Robinson made them so. They are a way of shoving a few epsilons and deltas under the rug in an automatic fasion. So you say: f is continuous if f(x+dx) and f(x) are only infinitesimally different, and you can translate to epsilon-delta without difficulty. But the infinitesimal form makes it easier to prove theorems, because you don't have as many quantifier alternations, so you can keep the thing in your head, that's all. The physicists just assumed that a rigorous version exist, and Robinson justified this belief. – Ron Maimon Jul 22 '12 at 09:35
  • @RonMaimon Your tone here and on your "ideal gas" answer is at times strident and insulting. It makes you sound like you are defending an indefensible position. Chill -- people will take you more seriously. – garyp Mar 09 '14 at 16:23
  • Differential forms in thermodynamics: https://www.av8n.com/physics/thermo-forms.htm , https://www.av8n.com/physics/thermo-forms.htm (by J. Denker, https://www.av8n.com/physics/ ). – Quillo Mar 20 '23 at 19:49

4 Answers4

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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:

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

Urb
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David Bar Moshe
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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
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    @Ron: it doesn't? how do you interpret the legendre-transformation in terms of infinitesimals? – Christoph Jul 18 '12 at 18:56
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    @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
  • @RonMaimon: This debate made the thread somewhat interesting to read. I think it's always good to take a look an isolize concepts and work on them to see where they get. I don't think many see "what really the point is" about th infinitesimal stuff in your approach - at least I don't. The relation to Ito calculus (and the need for not thinking in terms of maps on the tangent space) and the like is probably not obvious to most - at least to me. – Nikolaj-K Jul 18 '12 at 22:19
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    @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 01:50
  • @Ron: re Legendre transformation, I was thinking along the lines of section 8 of http://numdam.org/item?id=AIHPA_1977__27_1_101_0 – Christoph Jul 19 '12 at 08:05
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    @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
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    @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
  • I do agree that nonstandard calculus has not been given a great formulation, since the current standardized nonstandard models are full of superficial axiom of choice (infinitesimals introduced Robinson's way are manifestly constructive). The main purpose is explained by Terrance Tao somewhere on his blog, it is to automate certain limit arguments which involve several quantifier alternations, and thus reduce the complexity to the intuitive level by reducing the quantifier alternation complexity. This is useful. But for a physicist, it just tells you not to substitute differentials everywhere. – Ron Maimon Jul 19 '12 at 15:53
  • @Ron: I don't believe that article is paywalled (I just accessed the PDF from home and via a free proxy); in section 8, it is shown that the equations of state of the ideal gas fix a lagragian submanifold of a symplectic space as an example of the interpretation of the legendre transformation in context of symplectic geometry – Christoph Jul 19 '12 at 17:01
  • @Christoph: It's not paywalled, I was just stupid. I looked at the paper, most of it is saying normal stuff in mathese, and this irritates me. The interesting thing in section 8 is that he is identifying a symplectic structure in the phase space of an ideal gas, so there is a nontrivial symplectic form (which is a nontrivial differential form--- a 2 form) and using it for identifying special surfaces. This is indeed a nontrivial application of forms to thermodynamics, and this might constitute an answer, but it is different from the other answers here. I'll read it more. – Ron Maimon Jul 19 '12 at 17:12
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    @RonMaimon: ''I am just irate that a perfectly sensible notion, that of an infinitesimal differential, ... has been excized from the curriculum and is made taboo.'' It is not made taboo; it is just not found useful enough on the rigorous level to be pursued by more than a few afficionados. Differential geometry is far more useful than nonstandard analysis. – Arnold Neumaier Jul 19 '12 at 17:51
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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 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)

Urb
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Nikolaj-K
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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
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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.

stafusa
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Arnold Neumaier
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    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
  • @Arnold: I started to read your book today (just the first chapter for now) and kept a list of typos/missing symbols, statements I find questionable etc; are you interested in that (ie should I keep adding to my list or stop now before investing any real work)? – Christoph Jul 18 '12 at 19:10
  • @Christoph: ''the same'' can be used in a literal sense and in a figurative sense; I used it in the latter sense. To state precisely what I meant to convey: For purposes of interpreting the thermodynamic formalism, infinitesimals and differential forms are the same thing. - Moreover, Robinson infinitesimals are conceptually not really easier to grasp than differential 1-forms (which in fact need much less machinery than the full Cartan calculus). Yes, please send me your list of typos (and other comments on things worth improving). – Arnold Neumaier Jul 18 '12 at 19:12
  • @Arnold: I'll send you my list once I've done some further reading... – Christoph Jul 18 '12 at 19:16
  • @ArnoldNeumaier: I agree with you that 1-forms play a role, and integration along arcs is important, and hence 1-forms are used. But these are simply gradients. The nontrivial formalism of forms is the higher dimensional forms and the wedge product, and this never appears in thermodynamics. In statistics there are cases where infinitesimals are not differentials. In the classical cases, smooth functions of T,P,U, I agree that you can shoehorn infinitesimals into differentials, this is a pity, because the infinitesimals are source of intuition, and you secretly think this way underneath. – Ron Maimon Jul 18 '12 at 21:01
  • @RonMaimon: The OP asked for applications of differential geometry to thermodynamics, and there you need 1-forms and no higher order forms. But one needs 1-forms that are not gradients, as in general the differentials are not exact. – Arnold Neumaier Jul 19 '12 at 10:57
  • @ArnoldNeumaier: Yes, so what. That doesn't change anything. The calculus of forms, when restricted to 1-forms, is a pale uninspired less intuitive shadow of the calculus of infinitesimal displacements. The calculus of infinitesimals is what everyone has in their heads, and the forms just happen to coincide with this concept formally. But the infinitesimal concept is rich and fertile, while the form concept is closed and finite. – Ron Maimon Jul 19 '12 at 15:41
  • @RonMaimon: The literature shows that differentials as discussed in differential geometry are much more fertile than infinitesimals. – Arnold Neumaier Jul 19 '12 at 17:45
  • @ArnoldNeumaier: They are "fertile" because the differentials repackaged the concept of infinitesimals, after passing it through a formal machine of tangent spaces whose only purpose is to erect a barrier to entry for students. It's the same concept. – Ron Maimon Jul 19 '12 at 18:42
  • @RonMaimon: When properly introduced, differentials in $R^n$ are no barrier for students. I introduce them in my introductory analysis courses for first year students. They are far easier to introduce in a logically impeccable manner than infinitesimals. – Arnold Neumaier Jul 19 '12 at 20:46
  • My opinion is that differential 1-forms are much more intuitive than infinitesimals. – timur Apr 27 '17 at 21:42
  • Differential 1-forms and the d operation are better than the gradient. Gradients make it seem like the operations depend on an inner product, which they don't. The reason why people have gradients in their head is historical. The definition of gradient is $\nabla f(x) \cdot v = \lim ...$ and whenever we use a gradient we always inner product it with something: $\nabla f(x) \cdot v$. So the inner product is purely notational, and it's better to define the gradient as a covector in the first place. That's all that 1-forms are. – Jules Mar 09 '18 at 13:27
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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.

Urb
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hyportnex
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  • Hi Hyportnex, how do you interpret the anti commutavity of the wedge in the first equation? – tryst with freedom Jul 09 '21 at 11:28
  • @trystwithfreedom the anti-commutativity is the mathematical expression of the directionality as to who does work on whom. The contour integral $I=\oint ydx$ represents the enclosed area but the contour itself has a sign that flips if you reverse the path. Within the contour the infinitesimal area elements are $\tilde dx\wedge \tilde dy=- \tilde dy\wedge \tilde dx$ and $I=\int\int\tilde dx\wedge \tilde dy$ – hyportnex Oct 28 '23 at 19:56