You are completely right of course. I am merely a professor in theoretical physics who has been teaching this stuff for a number of years now.
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I never said that all the other properties of entropy are now immediately visible. Instead I think it is the only universal starting point of any reasonable discussion or course on the subject.
And lastly I am frankly getting discouraged by all the dismissive responses. So this will be my last comment for the day, and I will leave you in the careful hands of, say, the six other people who are obviously so extremely knowledgeable about this topic. /s
In particular it has always been my starting point whenever I introduce (the entropy of) macro- and micro-states in my statistical physics course.
Entropy is just a number you can associate with a probability distribution. If the distribution is discrete, so you have a set p_i, i = 1..n, which are each positive and sum to 1, then the definition is:
S = - sum_i p_i log( p_i )
Mathematically we say that entropy is a real-valued function on the space of probability distributions. (Elementary exercises: show that S >= 0 and it is maximized on the uniform distribution.)
That is it. I think there is little need for all the mystery.
You probably want to read up a bit on the remarkable life of Lise Meitner.
More worrying is when e.g. job candidates are discussed (often in alphabetical order) and people simply tire out near the end of the meeting. When this happens, be sure to suggest taking a break!
As I wrote before [0], I think that what Aaronson's calls "the theory" is a somewhat bastardized version of quantum mechanics that might suffice for quantum computing but, in my opinion, not for physics.
I discussed the difference here: https://news.ycombinator.com/item?id=38255476
5. This Regulation shall not create any obligation that would require a provider of hosting services or a provider of interpersonal communications services to decrypt data or create access to end-to-end encrypted data, or that would prevent the provision of end-to-end encrypted services.
[0] page 36 onwards of https://www.patrick-breyer.de/wp-content/uploads/2024/04/202...
Higgs was simply among the first to publish an idea that several other people came up with basically simultaneously. He then essentially sat on his hands for the remainder of his academic career, at least scientifically, with a dismal output of less than one not-too-impressive paper per decade - at what was arguably the most exciting time for particle physics.
For all I know he was an admirable teacher and colleague, but I really think that in a world without Peter Higgs we would not have seen a noticeable regression of theoretical particle physics.
There do exist Nobel prize winners from that era who really changed the field, often beyond their prize-winning work: people like Anderson, 't Hooft, Wilson, Weinberg, Gell-Mann, and many others. But Higgs is just not one of them.
[0] https://www.mcmp.philosophie.uni-muenchen.de/people/faculty/...
Or perhaps dr. Hossenfelder's grant proposal was simply bad and the ideas proposed clearly made no sense. Who knows?
It might be hard to judge for the HN crowd, but I just would like to ask why everybody is so keen on trusting the opinions of a contrarian youtuber and blogger instead of, say, anyone with at least a somewhat respectable academic career?
But as a physicist and long-time reader I have come to really appreciate the high quality of the articles by this NYT reporter (Dennis Overbye) on physics and related disciplines.
My ranting is how most of thermo starts with U(S, V, N) whereas I would prefer with S(U, V, N). Either way the differential reads:
dU = T dS - p dV + mu dN
which, if we follow the standard way, is just saying that
T = (dU/dS)_{V, N}
- p = (dU/dV)_{S, N}
mu = (dU/dN)_{S, V}
where the derivatives are really just partial derivatives so I really should have written them with a curly d. Mathematically enthalpy is the Legendre transform where we eliminate V in favor of p:
H(S, p, N) = (U(S, V, N) + p V)_{V = V*(p)}
where V* extremizes the term in parentheses, i.e.
(dU/dV) (S, V*(p), N) + p = 0
Of course this equation is just the above definition of the pressure at constant V, but now we are meant to solve this equation to find V* as a function of p (and S and N), and then plug that back in to get H as a function of p.
The Legendre transform property ensures that:
(dH/dp)_{S,N} = V*(p)
You should note that taking the derivative of H wrt p would in principle also induce a term proportional to dV* / dp since in the definition of H there is V* which depends on p. But that term cancels out because V* is an extremum! So that is why dH/dp gives just V*(p) which is the inverse function of p(V) that you get from (minus) dU/dV. This is the "inverse of derivatives" property of the Legendre transform mentioned in the original post.
Clearly the enthalpy is the nicer gadget to have if you work at constant pressure since then one of its arguments is simply held constant.
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
First, I was explicitly talking about the handful of brilliant people who literally no longer have to apply to grants, or, if they do so, they are all but guaranteed to get funding for whatever topic they want to work on.
And in your second paragraph you repeat the usual talking point. But why do you think it did not convince those people?
Witten has given quite a lot of interviews over the years, but his more mainstream views are unlikely to make it to the front page.
I am pretty sure that all these people simply kept thinking that string theory was the most promising way to understand quantum gravity and even (to some extent) 'just' quantum field theory.
So why do people decide to trust the opinions of Peter Woit over those of, say, Edward Witten?
No, or at least I did not mean to: I said "know how to compute", not "compute". One typically uses the Schrodinger equation to do so (although Pauli did not need it), but this starting point is nowhere to be found here.
> I agree that the Heisenberg uncertainty principle is important, but it certainly be derived from Aaronson's point of view (e.g. the standard Robertson-Schrödinger inequality is easily obtained).
Robertson-Schrödinger is fairly trivial, at least in Aaronson's finite-dimensional world. But you conveniently forgot how to "derive" the part of QM that actually gives you the value of the commutator sitting on the right-hand side. So will you just postulate it? That sounds pretty terrible pedagogically, and it might be better to provide at least some general discussion. And that discussion is exactly what I am advocating as a necessary ingredient in any self-respecting introduction to (let alone derivation of) QM.
> I also think that self-adjoint operators and the correspondence principle are a fairly terrible way to think about observables in quantum mechanics.
No teacher of QM should introduce POVMs before talking about positions and momenta.
Aaronson claims that QM "can be derived" (quote from the original comment) are followed by an introduction to some aspects of QM that leads to strictly none of these things. That is why I am unhappy with it, and I still do not see why I am "somewhat mistaken" (quote from you).
In fact, I can go even further and say that (from a quick glance at least) in his whole book positions and momenta make no appearance, let alone the correspondence principle. (I do not even see hermitian operators!) Without them I just do not see any reasonable "derivation" of (or, more properly, argumentation for) what I call QM.
For example, the paper you cite is entirely understandable for anyone with some training in hermitian QM. In contrast, good luck trying to understand elementary concepts like the spectrum of the hydrogen atom or interference of matter waves from unitary QM.
Of course the field of quantum info has progressed enormously and has its own interesting challenges, for which hermitian QM is all but useless.
Heisenberg's matrix mechanics was the first, and formulated entirely in terms of linear algebra. See for example the introduction of basic linear algebra techniques in the famous Bohr-Jordan paper from 1925:
I discussed the difference earlier: https://news.ycombinator.com/item?id=38255476
I am afraid that they may not have an equally compelling explanation for the same problem in the rest of the world.
Of course I can imagine that, say, constitutional law might currently be a bit more difficult to teach. But I still do not see what concretely the Yale administration is called upon to stop doing.
First, the authors make a general claim about "most physics textbooks" without providing a single example. I think one will often encounter more nuanced statements in the better and more widely used textbooks.
And I think the paper sorely lacks evidence for the general claim in the concluding sentence: "The idea that physics progress through a series of crisis, is hard to defend." Not only do they present only a single example, but even in that case one could claim that the "crisis" started after the discovery of Planck's formula! After all, it fitted the data supremely well but required this mystery constant: h.
It took physicist a quarter century to resolve the deeper meaning of Planck's constant. If that was not a crisis in physics then I do not know what would qualify as one.
https://en.wikipedia.org/wiki/Deutsche_Physik
I think it is remarkable that a racist argument against a description of nature could even gain a foothold in the physics community. Fortunately, people like Planck and Heisenberg could see the insanity of it all.