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I, too, spent a long time staring at expressions like “half-invert p(x, v) to get v(x, p) s.t. p(x, v(x, q)) = q then the Legendre transform is

I, too, spent a long time staring at expressions like “half-invert p(x, v) to get v(x, p) s.t. p(x, v(x, q)) = q then the Legendre transform is H(x, p) = p v(x, p) – L(x, v(x, p))” And I did come to one of the same conclusions as this article, which is that if we're talking pure mathematics, these “thermodynamic” expressions like (∂L/∂v)_x, (∂L/∂p)_x are deeply easy to get confused about and in fact you should just say “the derivative of the function with respect to its first argument holding the other arguments constant” and therefore introduce different functions which compute the same value under different symbols, say Λ(x, p) = L(x, v(x, p)) ∂₂Λ = ∂₂L ∂₂v so that you're not scratching your head about “why is the derivative of L with respect to v showing up here, v is now a function isn't it?” The formulation of first f derivatives as inverse functions is new to me but makes sense. However, I do think that we do even worse with linear algebra. I believe I could walk up to any college senior in physics and they wouldn't know that “the determinant is the product of the eigenvalues,” but this should be as well-known as “the mitochondria are the powerhouse of the cell.” I think this is because we introduce a complicated way to calculate determinants and then we use determinants to calculate the eigenvalues?

prof-dr-ir 3y ago
Agreed, the way thermodynamics is often taught is such a mess. My personal and controversial [0] take is that the free energy should really be seen as the Legendre transform of the entropy, not of the energy. I know it is ultimately semantics, but this viewpoint makes the passage from the micro-canonical to the canonical ensemble so much nicer. In particular, the saddle point approximation for the canonical partition function makes it natural that the ensembles are equivalent in the thermodynamic limit... through a Legendre transform! Bonus corollary: the statement mentioned in the blog about the derivatives being each other's inverses is just saying that T(E) and E(T) in respectively the micro-canonical and the canonical ensemble define the same relation between E and T. [0] Proof of controversiality: even Wikipedia disagrees with me here, see https://en.wikipedia.org/wiki/Thermodynamic_free_energy https://en.wikipedia.org/wiki/Thermodynamic_free_energy
- shiandow 3y ago
  I'm not even sure if it makes sense to view it as a Legendre transform. Or well, it is one, I'm just not sure if it's a good definition. You get the free energy for 'free' if you use a Lagrange multiplier to maximize entropy while keeping the energy fixed (temperature is the inverse of that Lagrange parameter). In one fell swoop this shows why temperature is a thing and why minimizing the free energy is important. The Legendre transform just returns the value of the constraint from the minimized function, but at that point why bother? I do agree that it makes more sense to see the fee energy as a Legendre transform of the entropy, that's kind of what you end up doing if you minimize entropy in this way.
  - amluto 3y ago
    I’ve taken an excellent graduate class in thermodynamics, and I’ve never seen a definition of enthalpy that is both coherent and involves Legendre transforms. Here’s my definition: the internal energy of the stuff in a box is a useful quantity, and one can call it E. But E is the energy needed to assemble the stuff in the box if you start with an empty box of the appropriate volume. This makes physical sense, and it’s perfectly fine for calculating things related to, say, anything that happens in a vacuum. Or anything that happens in a rigid box. But we live in a very large atmosphere, we mostly do experiments at constant pressure. If you take a flexible baggy and assemble its contents, you need the energy to make the contents (that’s E) and also some extra energy to displace air to make room for the contents, and the latter part requires extra energy equal to P (the constant atmospheric pressure) times V (the volume of the bag). So we give E + PV the fancy name “enthalpy”, and it turns out to be useful. Maybe there’s a Legendre transform somewhere, but I’ve never seen any use for it other than to try, poorly, to convince someone of its existence. And it’s genuinely awkward — energy and enthalpy are fairly general physical quantities that one could, in principle, measure, and one already needs to start constraining the system to think of them as functions of anything sensible. And then the HVAC industry seems to have borrowed the term “enthalpy” to mean, roughly, “temperature and humidity”. And “energy” means “temperature” or maybe “heat” but probably actually means “enthalpy, but only the thermal part and not the chemical part”. You’ll be lucky to find any math at all, let alone a Legendre transformation. Don’t get me started on “pressure”.
shiandow 3y ago
I know programmers like to blame mathematicians for writing functions with lots of one letter variable names, but it's the physicists who insist on doing so without defining any of them. You want to know what V is? It's clearly the potential, we've defined it six papers ago! Oh you were wondering what it's type was, well it's usually a scalar field. No, don't write the parameter as t that changes the whole meaning!
- Phiwise_ 3y ago
  Sussman has a great guest lecture that mentions exactly these sorts of issues, and that he found it much easier to verify work he and his grad students did in mathematical physics after developing the "mechanics programming" notation he explains in Structure and Interpretarion of Classical Mechanics and Functional Differential Geometry.
archgoon 3y ago
> I believe I could walk up to any college senior in physics and they wouldn't know that “the determinant is the product of the eigenvalues," Unless things have gotten significantly worse in physics education in the past decade, I'd be happy to take the other side of that bet. I will also be willing to bet they could prove it. The problem you'd have with physicists is convincing them that there are matrices that aren't diagonalizable.
mydogcanpurr 3y ago
> I think this is because we introduce a complicated way to calculate determinants and then we use determinants to calculate the eigenvalues? Yes, the determinant should be taught and defined as the volume of the parallelepiped in n-dimensions defined by the columns of the given square matrix. This perspective makes it immediately obvious that the eigenvalues scale the parallelepiped in each of its dimensions (a basis of eigenvectors makes it even simpler). Of course the volume (determinant) must be the product of these scaling factors (eigenvalues)! Since algebra is too convenient for solving problems, this geometric intuition is often an afterthought if it's even taught at all.
- lupire 3y ago
  What trash math classes were you all in that didn't teach all of this?
- programjames 3y ago
  I think you first need to define "volume" as a bunch of simplices put together, or the (-1)^{...} term is unmotivated.
- defrost 3y ago
  As anecdata this was taught in first year university mathematics for math, engineering, physics, chemistry, etc. students in 1981 in all three universities in Perth Western Australia aka "the most isolated city in the world" [1] It never occurred to me that geometric parallels would not be given in linear algebra courses. [1] https://about.soar.earth/blog-pages/how-the-worlds-most-isolated-city-became-the-city-of-light https://about.soar.earth/blog-pages/how-the-worlds-most-isol...

prof-dr-ir 3y ago

Agreed, the way thermodynamics is often taught is such a mess. My personal and controversial [0] take is that the free energy should really be seen as the Legendre transform of the entropy, not of the energy. I know it is ultimately semantics, but this viewpoint makes the passage from the micro-canonical to the canonical ensemble so much nicer. In particular, the saddle point approximation for the canonical partition function makes it natural that the ensembles are equivalent in the thermodynamic limit... through a Legendre transform! Bonus corollary: the statement mentioned in the blog about the derivatives being each other's inverses is just saying that T(E) and E(T) in respectively the micro-canonical and the canonical ensemble define the same relation between E and T. [0] Proof of controversiality: even Wikipedia disagrees with me here, see https://en.wikipedia.org/wiki/Thermodynamic_free_energy https://en.wikipedia.org/wiki/Thermodynamic_free_energy

shiandow 3y ago
I'm not even sure if it makes sense to view it as a Legendre transform. Or well, it is one, I'm just not sure if it's a good definition. You get the free energy for 'free' if you use a Lagrange multiplier to maximize entropy while keeping the energy fixed (temperature is the inverse of that Lagrange parameter). In one fell swoop this shows why temperature is a thing and why minimizing the free energy is important. The Legendre transform just returns the value of the constraint from the minimized function, but at that point why bother? I do agree that it makes more sense to see the fee energy as a Legendre transform of the entropy, that's kind of what you end up doing if you minimize entropy in this way.
- amluto 3y ago
  I’ve taken an excellent graduate class in thermodynamics, and I’ve never seen a definition of enthalpy that is both coherent and involves Legendre transforms. Here’s my definition: the internal energy of the stuff in a box is a useful quantity, and one can call it E. But E is the energy needed to assemble the stuff in the box if you start with an empty box of the appropriate volume. This makes physical sense, and it’s perfectly fine for calculating things related to, say, anything that happens in a vacuum. Or anything that happens in a rigid box. But we live in a very large atmosphere, we mostly do experiments at constant pressure. If you take a flexible baggy and assemble its contents, you need the energy to make the contents (that’s E) and also some extra energy to displace air to make room for the contents, and the latter part requires extra energy equal to P (the constant atmospheric pressure) times V (the volume of the bag). So we give E + PV the fancy name “enthalpy”, and it turns out to be useful. Maybe there’s a Legendre transform somewhere, but I’ve never seen any use for it other than to try, poorly, to convince someone of its existence. And it’s genuinely awkward — energy and enthalpy are fairly general physical quantities that one could, in principle, measure, and one already needs to start constraining the system to think of them as functions of anything sensible. And then the HVAC industry seems to have borrowed the term “enthalpy” to mean, roughly, “temperature and humidity”. And “energy” means “temperature” or maybe “heat” but probably actually means “enthalpy, but only the thermal part and not the chemical part”. You’ll be lucky to find any math at all, let alone a Legendre transformation. Don’t get me started on “pressure”.

shiandow 3y ago

I know programmers like to blame mathematicians for writing functions with lots of one letter variable names, but it's the physicists who insist on doing so without defining any of them. You want to know what V is? It's clearly the potential, we've defined it six papers ago! Oh you were wondering what it's type was, well it's usually a scalar field. No, don't write the parameter as t that changes the whole meaning!

Phiwise_ 3y ago
Sussman has a great guest lecture that mentions exactly these sorts of issues, and that he found it much easier to verify work he and his grad students did in mathematical physics after developing the "mechanics programming" notation he explains in Structure and Interpretarion of Classical Mechanics and Functional Differential Geometry.

archgoon 3y ago

> I believe I could walk up to any college senior in physics and they wouldn't know that “the determinant is the product of the eigenvalues," Unless things have gotten significantly worse in physics education in the past decade, I'd be happy to take the other side of that bet. I will also be willing to bet they could prove it. The problem you'd have with physicists is convincing them that there are matrices that aren't diagonalizable.

mydogcanpurr 3y ago

> I think this is because we introduce a complicated way to calculate determinants and then we use determinants to calculate the eigenvalues? Yes, the determinant should be taught and defined as the volume of the parallelepiped in n-dimensions defined by the columns of the given square matrix. This perspective makes it immediately obvious that the eigenvalues scale the parallelepiped in each of its dimensions (a basis of eigenvectors makes it even simpler). Of course the volume (determinant) must be the product of these scaling factors (eigenvalues)! Since algebra is too convenient for solving problems, this geometric intuition is often an afterthought if it's even taught at all.

lupire 3y ago
What trash math classes were you all in that didn't teach all of this?
programjames 3y ago
I think you first need to define "volume" as a bunch of simplices put together, or the (-1)^{...} term is unmotivated.
defrost 3y ago
As anecdata this was taught in first year university mathematics for math, engineering, physics, chemistry, etc. students in 1981 in all three universities in Perth Western Australia aka "the most isolated city in the world" [1] It never occurred to me that geometric parallels would not be given in linear algebra courses. [1] https://about.soar.earth/blog-pages/how-the-worlds-most-isolated-city-became-the-city-of-light https://about.soar.earth/blog-pages/how-the-worlds-most-isol...