Compositional Thermostatics (Part 2)

7 February, 2022

guest post by Owen Lynch

In Part 1, John talked about a paper that we wrote recently:

• John Baez, Owen Lynch and Joe Moeller, Compositional thermostatics.

and he gave an overview of what a ‘thermostatic system’ is.

In this post, I want to talk about how to compose thermostatic systems. We will not yet use category theory, saving that for another post; instead we will give a ‘nuts-and-bolts’ approach, based on examples.

Suppose that we have two thermostatic systems and we put them in thermal contact, so that they can exchange heat energy. Then we predict that their temperatures should equalize. What does this mean precisely, and how do we derive this result?

Recall that a thermostatic system is given by a convex space X and a concave entropy function S \colon X \to [-\infty,\infty]. A ‘tank’ of constant heat capacity, whose state is solely determined by its energy, has state space X = \mathbb{R}_{> 0} and entropy function S(U) = C \log(U), where C is the heat capacity.

Now suppose that we have two tanks of heat capacity C_1 and C_2 respectively. As thermostatic systems, the state of both tanks is described by two energy variables, U_1 and U_2, and we have entropy functions

S_1(U_1) = C_1 \log(U_1)

S_2(U_2) = C_2 \log(U_2)

By conservation of energy, the total energy of both tanks must remain constant, so

U_1 + U_2 = U

for some U; equivalently

U_2 = U - U_1

The equilibrium state then has maximal total entropy subject to this constraint. That is, an equilibrium state (U_1^{\mathrm{eq}},U_2^{\mathrm{eq}}) must satisfy

S_1(U_1^{\mathrm{eq}}) + S_2(U_2^{\mathrm{eq}}) = \max_{U_1+U_2=U} S_1(U_1) + S_2(U_2)

We can now derive the condition of equal temperature from this condition. In thermodynamics, temperature is defined by

\displaystyle{ \frac{1}{T} = \frac{\partial S}{\partial U} }

The interested reader should calculate this for our entropy functions, and in doing this, see why we identify C with the heat capacity. Now, manipulating the condition of equilibrium, we get

\max_{U_1+U_2=U} S_1(U_1) + S_2(U_2) = \max_{U_1} S_1(U_1) + S_2(U-U_1)

As a function of U_1, the right hand side of this equation must have derivative equal to 0. Thus,

\displaystyle{ \frac{\partial}{\partial U_1} (S_1(U_1) + S_2(U-U_1)) = 0 }

Now, note that if U_2 = U - U_1, then

\displaystyle{  \frac{\partial}{\partial U_1} S(U-U_1) = -\frac{\partial}{\partial U_2} S(U_2) }

Thus, the condition of equilibrium is

\displaystyle{  \frac{\partial}{\partial U_1} S_1(U_1) = \frac{\partial}{\partial U_2} S_2(U_2) }

Using the fact that

\displaystyle{ \frac{1}{T_1} = \frac{\partial}{\partial U_1} S_1(U_1) , \qquad \frac{1}{T_2} = \frac{\partial}{\partial U_2} S_2(U_2) }

the above equation reduces to

\displaystyle{ \frac{1}{T_1} = \frac{1}{T_2} }

so we have our expected condition of temperature equilibriation!

The result of composing several thermostatic systems should be a new thermostatic system. In the case above, the new thermostatic system is described by a single variable: the total energy of the system U = U_1 + U_2. The entropy function of this new thermostatic system is given by the constrained supremum:

S(U) = \max_{U = U_1 + U_2} S_1(U_1) + S_2(U_2)

The reader should verify that this ends up being the same as a system with heat capacity C_1 + C_2, i.e. with entropy function given by

S(U) = (C_1 + C_2) \log(U)

A very similar argument goes through when one has two systems that can exchange both heat and volume; both temperature and pressure are equalized as a consequence of entropy maximization. We end up with a system that is parameterized by total energy and total volume, and has an entropy function that is a function of those quantities.

The general procedure is the following. Suppose that we have n thermostatic systems, (X_1,S_1),\ldots,(X_n,S_n). Let Y be a convex space, that we think of as describing the quantities that are conserved when we compose the n thermostatic systems (i.e., total energy, total volume, etc.). Each value of the conserved quantities y \in Y corresponds to many different possible values for x_1 \in X_1, \ldots x_n \in X_n. We represent this with a relation

R \subseteq X_1 \times \cdots \times X_n \times Y

We then turn Y into a thermostatic system by using the entropy function

S(y) = \max_{R(x_1,\ldots,x_n,y)} S_1(x_1) + \ldots + S_n(x_n)

It turns out that if we require R to be a convex relation (that is, a convex subspace of X_1 \times \cdots \times X_n \times Y) then S as defined above ends up being a concave function, so (Y,S) is a true thermostatic system.

We will have to wait until a later post in the series to see exactly how we describe this procedure using category theory. For now, however, I want to talk about why this procedure makes sense.

In the statistical mechanical interpretation, entropy is related to the probability of observing a specific macrostate. As we scale the system, the theory of large deviations tells us that seeing any macrostate other than the most probable macrostate is highly unlikely. Thus, we can find the macrostate that we will observe in practice by finding the entropy maxima. For an exposition of this point of view, see this paper:

• Jeffrey Commons, Ying-Jen Yang and Hong Qian, Duality symmetry, two entropy functions, and an eigenvalue problem in Gibbs’ theory.

There is also a dynamical systems interpretation of entropy, where entropy serves as a Lyapunov function for a dynamical system. This is the viewpoint taken here:

• Wassim M. Haddad, A Dynamical Systems Theory of Thermodynamics, Princeton U. Press.

In each of these viewpoints, however, the maximization of entropy is not global, but rather constrained. The dynamical system only maximizes entropy along its orbit, and the statistical mechanical system maximizes entropy with respect to constraints on the probability distribution.

We can think of thermostatics as a ‘common refinement’ of both of these points of view. We are agnostic as to the mechanism by which constrained maximization of entropy takes place and we are simply interested in investigating its consequences. We expect that a careful formalization of either system should end up deriving something similar to our thermostatic theory in the limit.

See all four parts of this series:

Part 1: thermostatic systems and convex sets.

Part 2: composing thermostatic systems.

Part 3: operads and their algebras.

Part 4: the operad for composing thermostatic systems.

Submission to arXiv

4 February, 2022

guest post by Phillip Helbig

Monthly Notices of the Royal Astronomical Society is one of the oldest and most prestigious journals in the fields of astronomy, astrophysics, and cosmology. My latest MNRAS paper was not allowed to appear in the astro-ph category at the arXiv (, the main avenue of distribution for scientific articles in many fields) because it was reclassified to a category which is inappropriate for several reasons. This is definitely not due to some technical error, misunderstanding, or oversight. It took more than three months for me to even be told why it had been reclassified, and that only after a well known cosmologist threatened the Scientific Director of arXiv that he would complain to the arXiv sponsors if things weren’t cleared up. Also, there is evidence that the reason I was given is not the real one.

Although I would like my paper to appear in astro-ph, this in not about just my paper. Rather, it is about the question whether the community wants arXiv to decide which papers, and hence which people, are allowed to be part of that community, as opposed to peer review by respected journals such as MNRAS. Below, after some general background on arXiv, I mention some policies which are probably not as well known as they should be, before briefly describing my own odyssey.

Like it or not, many if not most astronomers rely on arXiv at least for learning about new papers; some rely on it exclusively, despite the facts that not everything is on arXiv, that that which is there is not always in the definitive version, and that even if the definitive version is there, then that might not be clear. The last two (and, in some cases, the first as well) can be due to lazy authors or to restrictions imposed by journals as to what version, such as the ‘author’s accepted manuscript’, is allowed to appear; more-definitive versions hence either don’t appear or if so then that fact is not advertised. At the same time, publication in a respected journal is generally recognized as a mark of quality. In fact, the main reason that the quality of papers at arXiv is so high is that most of them will eventually appear in respected journals. So essentially journals are for separating the wheat from the chaff while arXiv has become the main method of distribution, because no subscription is required and because a majority of articles can be found at one website with a reasonably useful interface (the former is crucial for those without access to a subscription to every journal they might want to access and the latter saves large amounts of time). There is thus a problem if standards of acceptance between journals and arXiv differ.

The main reason, at least for me, to have my papers on arXiv is visibility. All else being equal, papers on arXiv are almost certainly read more, and probably cited more, than those which are not. (In a field in which a large fraction are on arXiv, the reason can’t be that only the better papers are put on arXiv. Also, at least a few years after the paper has appeared, having it on arXiv before it has appeared in the journal probably won’t substantially increase the number of times it is read and/or cited due to the only slightly increased time during which it has been available; the increased citation rate is due to the higher visibility from being on arXiv.) The ‘stamp of approval’ comes from the journal. It is easy to distribute open-access versions of the paper, although implementing a robust long-term storage strategy is not. Finding them is more difficult; that would be easiest via arXiv, but author-supplied links at the corresponding ADS& abstract web page are good enough.

People often look for open-access versions of papers via links on such web pages, especially if they want to make sure that they find the official version, not whatever version might be on arXiv; arXiv itself is not an option for papers which are not on arXiv; of course, ADS can be and is used completely independently of arXiv. Lack of visibility at arXiv is a serious disadvantage to an author and such decisions should be made only in extreme cases. (Also, having the paper at arXiv but in the wrong category can be worse than not having it there at all.)

arXiv is under no obligation to allow even a paper which has been accepted by a leading journal in the field to appear in the appropriate category (e.g., astro-ph for astronomy / astrophysics / cosmology), or even to appear at all. There are also some other things which are documented but not as well known as they should be, some things which are at best poorly documented, and inconsistent and/or incomplete recommendations. I think that it is important to alert the community to those in order to counter the impression held by many that everything worth reading is on arXiv and/or if something is not on arXiv then it must be a matter of the author excluding himself from the community, rather than being excluded by arXiv (references intentionally not included to avoid public shaming). (Of course, most who claim that all papers in their field worth reading are on arXiv are not in a position to make that claim, because they don’t read any papers which are not on arXiv.) I suspect that at least some of those things are known by many, but also that there is a fear of criticizing arXiv in public for fear of getting banned, which is the modern-day equivalent of excommunication.

According to the submission agreement, “[t]he Submitter waives…[a]ny claims against arXiv…based upon actions…including…decisions to include the Work in, or exclude the Work from, the repository…the classification or characterization of the Work.” “arXiv reserves the right to reject or reclassify any submission.” In other words, the idea that any serious paper (‘serious’ being defined here as having appeared in a respected journal) can (assuming, of course, that the journal allows it) be uploaded to arXiv is wrong. Also, arXiv reserves the right to reclassify the article, e.g. a paper submitted to astro-ph can be reclassified to gen-ph. Moreover, after such a reclassification, the author is not allowed to withdraw the paper (Steinn Sigurdsson*, personal communication; Eleonora Presani@, personal communication), although that is technically possible (by first ‘unsubmitting’ it then ‘deleting’ it).

Of course, journals also decide which papers they accept and reject. However, the comparison of arXiv with journals is not appropriate, for several reasons: arXiv does not peer-review submissions and claims to do only a minimal amount of moderation. Also, journals offer something between acceptance and rejection, namely the possibility of revision, coupled with the opportunity to discuss the degree of revision, or even reasons for rejection, with the referee(s) and/or editor(s). Of course, revision of an article accepted by a journal doesn’t make sense, but the fact that it is not offered is another piece of evidence that interaction with arXiv shouldn’t be compared to interaction with a journal. Moreover, if an article is rejected by a journal, it is not automatically submitted to another journal, much less without any possibility for the author to choose to withdraw it completely, hence the claim that the various arXiv categories are comparable to various journals with different standards (Eleonora Presani, personal communication) is dubious at best. In addition, there is usually more than one journal of comparable reputation in a given field, so the author has the chance of getting an independent evaluation. In that case, competition between journals is good. In the case of arXiv, however, a monopoly is actually good, as long as it works, because one of the main advantages of arXiv is that there is only one place one needs to look in order to find most papers. This is the main point of my criticism: arXiv’s unique relevance to the community means that excluding a paper from its intended category should be done only under extreme circumstances. arXiv has become one of the most important resources for the astronomical community but that community has essentially no control over arXiv. Great power should be accompanied by great responsibility. Quis custodiet ipsos custodes?

It is possible to appeal a decision. However, the appeals process is not well documented, in part because astro-ph is sometimes seen as a top-level category, sometimes as one of the physics categories. As part of the appeals process, “[e]xtreme cases may be addressed to the appropriate advisory committee chair only”. The value of a successful appeal is questionable, because most rely on the abstract lists for recent papers in a particularly category, either sent via email or available at the arXiv website. As far as I know, a paper reclassified after a successful appeal would not appear in the ‘recent’ list for that category. The main problem with such an appeal, though, is that arXiv is policing itself.

For various reasons, in recent years so-called arXiv-overlay journals have sprung up. There is even one for astrophysics, The Open Journal of Astrophysics, and I have published a review paper there. The basic idea is that there is a robust distribution structure already in place, namely arXiv, so the job of the journal is essentially only to provide refereeing. Such journals usually assume that all potential authors could post their paper to arXiv before submitting it to the journal, but obviously that is not the case. (Some even use the arXiv category as a filter to determine whether the paper could even be considered to be appropriate for the journal.) It is sometimes possible, though usually not widely advertised, to submit to the journal first and submit the paper to arXiv only after acceptance, which is what I did (like many, I prefer to put papers on arXiv only after acceptance). That paper had no problems at arXiv, but based on the reasons I’m presenting here, arXiv-overlay journals are no longer an option for me. (I have long suggested not only that should the possibility to submit to the journal before submitting to arXiv be more widely advertised, but also that the journal should have some sort of agreement with arXiv that any paper accepted by the journal automatically qualifies for the corresponding category at arXiv (after all, the purpose of a journal is publication); alas, the Open Journal of Astrophysics does not plan to pursue that at all: “OJA has no power to compel arXiv to accept submissions, nor would we want to. We see arXiv as the most important resource in astrophysics….”.) Despite the longevity and robustness of some traditional journals, the scientific publishing landscape is changing rapidly. That is a topic for another discussion, but part of it involves arXiv-overlay journals, and wrong assumptions about arXiv mean that a substantial part of the new system is built on shaky foundations.

Those who are interested in high-quality, free-for-readers-and-authors, well organized, open-access journals should check out Is there any valid reason to submit anywhere else? Their astronomy journals are just getting underway; please consider supporting them.

I learned about some of the things discussed above the hard way when my latest MNRAS paper was reclassified from astro-ph to gen-ph (general physics). Of course, I appealed the decision quickly, after discussing the matter with a few colleagues, some of whom assumed that it must have been some sort of technical glitch. It took more than three months before I was told a reason for the classification (after having escalated up to the highest levels of arXiv)§, and more than four before the appeals process finally ended. That paper is not on arXiv, and I don’t intend to post anything else to arXiv before the procedure becomes fairer, more transparent, and more accountable (if it ever does). I had escalated as highly as possible within arXiv before I asked Cornell University (which hosts arXiv) to investigate possible academic misconduct, which led to an email from Eleonora Presani. Her stance is essentially the same as that of Licia Verde#: my accusations themselves don’t seem to have been investigated and authors just have to live with the fact that arXiv can reclassify papers at will and even prevent authors from withdrawing them completely before announcement if they disagree with the reclassification. Unfortunately, Cornell takes the point of view that although Cornell maintains and sustains arXiv, it is not the university’s role to interfere in the moderation or appeal process.

There is evidence that I wasn’t told the real reason why my paper was reclassified$, and no-one with whom I have discussed the matter thinks that arXiv was right to reclassify my paper. (That doesn’t mean that they necessarily have a high opinion of my paper, but those are two separate issues. One colleague stated (though not in reference to my paper) that even the occasional papers which appear in respected journals obviously by mistake should appear on arXiv; that would put pressure on journals to be more careful and also benefit those wishing to critically discuss or refute them.) However, I will discuss that and other aspects (hopefully) unique to my case elsewhere (perhaps in the comments if there is interest), and here concentrate on problems which the astronomical community should recognize and try to correct.

I certainly regard reclassifying a paper which has appeared in MNRAS to a category other than astro-ph, giving reasons for the reclassification only after threat from a famous colleague, and then giving me a completely different reason, to be an extreme case. Thus, I did contact the chair of the physics advisory committee, Robert Seiringer; that he is the appropriate person was also confirmed by Licia Verde. Nevertheless, his response was that he could not investigate disputes involving individual submissions, which was also Verde’s reply to my complaint. Hence, not only is there disagreement between arXiv’s documented appeals procedure and how those involved actually behave, there seems to be no system of checks and balances within arXiv, not to mention the problem that the community, despite relying on arXiv, in practice has no way to arbitrate disputes with it; it is judge, jury, and executioner.

All who believe that my paper should be on arXiv in the astro-ph.CO category if I so desire are encouraged to contact the Scientific Director, the Executive Director, the Chair of the Scientific Advisory Board, and the Chair of the Physics Advisory Committee and complain. It is not necessary to think that my paper is great. It is enough if one thinks that it is not so bad that it should be banned from astro-ph, or even if one can point to worse papers which are in astro-ph. (Of course, if one agrees that my paper should appear in astro-ph.CO, the reason why arXiv has not (yet?) let it appear are irrelevant.)

Of course, my bad experience with arXiv is not the main point. The main point is that arXiv can, and does, make decisions which experts in the field (see third footnote; Tegmark wasn’t the only expert consulted by me) cannot understand at all. Due to fear of the consequences of criticizing arXiv, most of those probably go unnoticed. While arXiv does need the possibility to reject or reclassify some papers, that needs to be done transparently and fairly. However, in view of its value to the community, there should be some simple rules, such as a ‘white list’ of journals so that papers accepted by them automatically qualify for the corresponding category at arXiv. Fortunately, my own livelihood does not depend on submitting to arXiv (in either sense of the word). Imagine the consequences of a young scientist who, after a year or so of work, gets their first paper accepted by a serious journal, only to have it rejected by arXiv or reclassified into a category where no colleague, potential employer, and so on will see it. Not only that, but the decision is made by someone (or some thing; arXiv is now moving to classification based on machine learning, but that was not relevant to the reclassification of my paper (Licia Verde, personal communication)) via an untransparent algorithm and no reason is given. Any appeal is within arXiv itself and essentially consists of some people asking others if they are guilty and accepting the expected answer. Such behaviour should be an embarrassment to the scientific community.

I think that some action on the part of the community would be in order even if my paper were the only one affected. However, the problem is much larger. Many colleagues have told me that they disagree with the reclassification of my paper, but are afraid to say so publicly for fear of getting banned from arXiv themselves. Also, I have been told that I am far from the first person to make such complaints about arXiv. Since I have started discussing this with colleagues, a few other similar cases have been mentioned to me. Considering that many of those affected probably don’t mention it at all out of a false sense of shame, the number of people affected is probably larger than many might at first guess. (I am not on FaceBook, but I understand that a similar problem was recently discussed within a FaceBook group for professional astronomers.)

A new development is that arXiv, by its own admission, doesn’t have the necessary means to do its job properly, and that I am not the only one complaining about it:

• Daniel Garisto, reaches a milestone and a reckoning, Scientific American, 10 January 2022.

A red herring is that the American Astronomical Society has made all of its journals (which are some of the major journals in cosmology/astrophysics/astronomy) open-access. That probably won’t diminish the importance of arXiv—and hence the importance of making sure that it is run responsibly—for several reasons. First, an attraction of arXiv is that it is a one-stop shop with a reasonable interface, and by following it one can keep of with much of the literature in one’s field (though of course not all papers are posted to arXiv, but if it is run responsibly then there should be no reason for them not to be, except if the journal forbids posting (some version of) the paper to arXiv). Even if all papers were open-access, that would mean following websites, or RSS feeds, of several or even dozens of websites, not nearly as convenient as the abstract listings at arXiv. Second, the AAS journals have rather expensive publication fees, which are becoming increasingly hard to justify, especially in the case of online-only publications. (Note that there are journals with no publication fees which actually encourage the author to post something equivalent to the final version on arXiv with no embargo period; MNRAS is an example.) Third, items which would otherwise have limited circulation, such as theses and conference proceedings, can (in principle) be on arXiv.

I’m all for giving arXiv more support, but first my paper needs to be rehabilitated by being allowed into astro-ph, and the policies should be changed, and publicly communicated, so that such problems do not happen in the future (neither to me nor anyone else); I could then post my backlog. The evidence is that the goof is so large that a public apology is called for. The minimum which needs to be done:

  1. When a paper is reclassified, authors should be informed (now, there is not even an automatic email; that makes sense because arXiv thinks that it needs to reclassify some papers against the will of the submitter) and given a chance to approve the reclassification, delete the submission entirely, suggest another reclassification, or appeal. Until the matter is resolved, the submission should stay in the ‘hold’ status with no action required to keep it there (now, one has to unsubmit and resubmit it to keep it from going away).
  2. When a paper is reclassified, the submitter must be given concrete reasons.

  3. The appeals process needs to be overseen with some authority outside of arXiv which has the power to overrule arXiv’s decisions, otherwise it is more or less a farce. It seems to me that some committee in the corresponding professional organization would be a good choice, e.g. the International Astronomical Union for papers on cosmology / astrophysics / astronomy. There can be an internal appeals process, but the final authority of arXiv’s decisions should not reside with arXiv if arXiv is to provide a meaningful service to the community.

  4. Papers from the major journals should be essentially white-listed. If a paper is really so bad that it is obvious that it somehow slipped in by mistake, arXiv should request the journal to formally withdraw it. If the journal does so, then arXiv shouldn’t accept it either. If not, then it should go onto arXiv. (It should go on even if it is bad, to put pressure on journals to uphold quality and so that it can be discussed and rebutted).

  5. arXiv needs to publicly apologize for reclassifying papers for reasons other than quality or content (e.g. my case), and invite those papers to be resubmitted after the other points above have been implemented.

  6. The points above should make (re)submissions by wrong authors viable, but perhaps some sort of special protection is needed for whistle-blowers such as myself.

  7. I was going to call for the resignation of Seiringer, Verde, and Presani, but it seems that they have all no longer in the posts they were when interacting with me. The main guilty person, though, Sigurdsson, is still Scientific Director. How anyone can be aware of my story (which can be backed up with evidence, in court if necessary) and still think that Sigurdsson should have anything at all to do with arXiv is beyond me. Also, although they have chosen (probably with good reason) to remain nameless, if arXiv were not drastically wrong on this point, the distinguished colleagues who put in a lot of time and effort trying to get arXiv to reverse its decision would not have done so. I am extremely grateful to them for their courage.

Of course, a boycott will not put pressure on arXiv. (It would actually remove pressure if people who are critical of arXiv stop using it.) If really famous people publicly announce that they will stop posting to arXiv until the points I raise have been cleared up, that might lead to something.

It is not clear how large the problem is, in part because not everyone feels able to complain. I don’t think that my case is a one-off, or even part of a small minority, because otherwise arXiv would not have invested so much time and effort to prevent one more abstract from appearing in astro-ph. I have given them several opportunities to revert their decision and hence cut their losses, but never even received a reply to such requests. Thus, the problem is probably substantial, and hence should be of interest to the entire community.

Information based on the web pages pointed to by the URLs in the reference list reflects the state of those pages on 28 August 2020; that based on the technical behaviour of the arXiv interface reflects my experiences between 20 April and 25 July 2020. References to ‘arXiv’ reflect my experience with the astro-ph category.

I would be interested in hearing anything relevant to this topic by email (my address is easy enough to find). Please indicate the degree of confidentiality you wish.

Please point as many people as possible, by all means at your disposal, to this post and related discussion. I am probably taking a big risk by going public, but if I do so, I want it to have the maximum effect. I see the lack of accountability of arXiv as a serious problem in modern academia.


* Steinn Sigurdsson is the Scientific Director of arXiv.

@ Eleonora Presani was the first Executive Director of arXiv, the post having been created only in 2020, while arXiv itself was created in 1991. She used to work for Elsevier. On 21 December 2021, it was announced that she would step down. According to the same announcement, Steinn Sigurdsson is still Scientific Director. Robert Seiringer is no longer Chair of the physics committee. I don’t see a new Executive Director listed on the arXiv Leadership Team web page.

§ Even that happened only after noted cosmologist Max Tegmark had threatened to complain to arXiv’s sponsors if my paper wasn’t taken out of limbo. Before, I had received only an extremely brief reply from Sigurdsson, and that only after a colleague who has known him for a long time discussed my complaints with him. Tegmark not only agrees that arXiv is overstepping its bounds by essentially overriding the refereeing process of a respected journal, but also that there is no reason that my paper should not be allowed to appear in astro-ph. He was also kind enough and brave enough to give me permission to quote from his emails to me. These do contain quotations of emails he received from arXiv. Ethically, I think that trying to correct the tremendous harm done to me and others because of wrong reclassification overrides any concerns about quoting without permission (which of course would not be given), especially since such quotations make my case much stronger than merely paraphrasing what others have told me or even just my own suspicions; this is a typical whistle-blower situation.

$ The only reason which I was given is the alleged lack of “substantiveness” of the paper. Max Tegmark, on the other hand, wasn’t told that, but was told that my case is “complicated” and that “[t]he reason for this [arXiv not automatically accepting a paper accepted by a journal] is partly the SCOAP3 agreement, which arXiv is not party to but still put certain obligations on us, and partly because we can not privilege any one journal or publisher for legal reasons. We get sued.” (Max Tegmark, personal communication.) I certainly don’t think that arXiv should automatically accept a paper just because it has been accepted by any journal, but do think that rejecting or reclassifying a paper which has been accepted by a respected journal should be done only under extreme circumstances, via a transparent and fair process, and for reasons which can be explained. Also, no one I have talked to has any idea how SCOAP3 could be relevant to my paper. Apart from Max Tegmark, several other colleagues (all full professors of cosmology / astrophysics / astronomy at major research universities) tried to intervene with arXiv (which did not want even discuss the matter with a low-life such as myself). That none of them want their names mentioned publicly is a problem in itself: the people whom arXiv is supposed to serve do not feel free to offer constructive criticism in public. Between the lines (or even in them, if one is allowed to see them), it seems that, in my case, the reclassification was not due to the contents or quality of my paper, but rather indicates another, possibly even more serious, problem: arXiv appears to be afraid of getting sued by crackpots. Apparently they abuse the gen-ph category (which is a mix of papers about general physics, papers which at first or even second or third glance obviously belong in another category and have nothing obviously wrong with them, and genuine crackpot stuff) by reclassifying some real papers to it and also letting through a few crackpot papers, thus avoiding the accusation of white-listing the major journals (which shouldn’t be a problem) and the crackpots can be appeased by having their papers in the same category as some major-journal papers. Of course this is not a policy which arXiv has published, but when several people get the same message behind the scenes, it is as certain as it needs to be to make my case. Although I believe that the concept still would have been deeply flawed, I offered to leave the paper in gen-ph but get have it cross-listed to astro-ph, but that suggestion was rejected by arXiv. Of course, if their goal is to appease the crackpots but at the same time keep them out of the major categories, that strategy wouldn’t work, because they would then have to cross-list crackpot papers or make a distinction, which is what they are trying to avoid (or rather they want to have a few alibi papers with no distinction).

# Licia Verde was Chair of the arXiv Scientific Advisory Committee. The Chair is now Ralph Wijers, who is also chair of the Physics Advisory Committee. I did contact him, but he sees no reason to investigate my case, as it happened before his posts as Chairman.

& The SAO/NASA Astrophysics Data system is the most important bibliographic database in astronomy/astrophysics/cosmology, operated by the Smithsonian Astrophysical Observatory (part of the Harvard/Smithsonian Center for Astrophysics, which also includes the Harvard College Observatory) under a grant from the National Aeronautics and Space Administration.

Hardy, Ramanujan and Taxi No. 1729

30 January, 2022

In his book Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, G. H. Hardy tells this famous story:

He could remember the idiosyncracies of numbers in an almost uncanny way. It was Littlewood who said every positive integer was one of Ramanujan’s personal friends. I remember once going to see him when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”


10^3 + 9^3 = 1000 + 729 = 1729 = 1728 + 1 = 12^3 + 1^3

But there’s more to this story than meets the eye.

First, it’s funny how this story becomes more dramatic with each retelling. In the foreword to Hardy’s book A Mathematician’s Apology, his friend C. P. Snow tells it thus:

Hardy used to visit him, as he lay dying in hospital at Putney. It was on one of those visits that there happened the incident of the taxicab number. Hardy had gone out to Putney by taxi, as usual his chosen method of conveyance. He went into the room where Ramanujan was lying. Hardy, always inept about introducing a conversation, said, probably without a greeting, and certainly as his first remark: “I thought the number of my taxicab was 1729. It seemed to me rather a dull number.” To which Ramanujan replied: “No, Hardy! No, Hardy! It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways.”

Here Hardy becomes “inept” and makes his comment “probably without a greeting, and certainly as his first remark”. Perhaps the ribbing of a friend who knew Hardy’s ways?

I think I’ve seen later versions where Hardy “burst into the room”.

But it’s common for legends to be embroidered with the passage of time. Here’s something more interesting. In Ono and Trebat-Leder’s paper The 1729 K3 surface, they write:

While this anecdote might give one the impression that Ramanujan came up with this amazing property of 1729 on the spot, he actually had written it down before even coming to England.

In fact they point out that Ramanujan wrote it down more than once!

Before he went to England, Ramanujan mainly published by posting puzzles to the questions section of the Journal of the Indian Mathematical Society. In 1913, in Question 441, he challenged the reader to prove a formula expressing a specific sort of perfect cube as a sum of three perfect cubes. If you keep simplifying this formula to see why it works, you eventually get

12^3 = (-1)^3 + 10^3 + 9^3

In Ramanujan’s Notebooks, Part III, Bruce Berndt explains that Ramanujan developed a method for finding solutions of Euler’s diophantine equation

a^3 + b^3 = c^3 + d^3

in his “second notebook”. This is one of three notebooks Ramanujan left behind after his death—and the results in this one were written down before he first went to England. In Item 20(iii) he describes his method and lists many example solutions, the simplest being

1^3 + 12^3 = 9^3 + 10^3

In 1915 Ramanujan posed another puzzle about writing a sixth power as a sum of three cubes, Question 661. And he posed a puzzle about writing $1$ as a sum of three cubes, Question 681.

Finally, four or five years later, Ramanujan revisited the equation a^3 + b^3 = c^3  + d^3 in his so-called Lost Notebook. This was actually a pile of 138 loose unnumbered pages written by Ramanujan in the last two years of his life, 1919 and 1920. George Andrews found them in a box in Trinity College, Cambridge much later, in 1976.

Now the pages have been numbered, published and intensively studied: George Andrews and Bruce Berndt have written five books about them! Here is page 341 of Ramanujan’s Lost Notebook, where he came up with a method for finding an infinite family of integer solutions to the equation a^3 + b^3 = c^3  + d^3:

As you can see, one example is

9^3 + 10^3 = 12^3 + 1

In Section 8.5 of George Andrews and Bruce Berndt’s book
Ramanujan’s Lost Notebook: Part IV, they discuss Ramanujan’s method, calling it “truly remarkable”.

In short, Ramanujan was well aware of the special properties of the number 1729 before Hardy mentioned it. And something prompted Ramanujan to study the equation a^3 + b^3 = c^3  + d^3 again near the end of his life, and find a new way to solve it.

Could it have been the taxicab incident??? Or did Hardy talk about the taxi after Ramanujan had just thought about the number 1729 yet again? In the latter case, it’s hardly a surprise that Ramanujan remembered it.

Thinking about this story, I’ve started wondering about what really happened here. First of all, as James Dolan pointed out to me, you don’t need to be a genius to notice that

1000 + 729 = 1728 + 1

Was Hardy, the great number theorist, so blind to the properties of numbers that he didn’t notice either of these ways of writing 1729 as a sum of two cubes? Base ten makes them very easy to spot if you know your cubes, and I’m sure Hardy knew 9^3 = 729 and 12^3 = 1728.

Second of all, how often do number theorists come out and say that a number is uninteresting? Except in that joke about the “least uninteresting number”, I don’t think I’ve heard it happen.

My wife Lisa suggested an interesting possibility that would resolve all these puzzles:

Hardy either knew of Ramanujan’s work on this problem or noticed himself that 1729 had a special property. He wanted to cheer up his dear friend Ramanujan, who was lying deathly ill in the hospital. So he played the fool by walking in and saying that 1729 was “rather dull”.

I have no real evidence for this, and I’m not claiming it’s true. But I like how it flips the meaning of the story. And it’s not impossible. Hardy was, after all, a bit of a prankster: each time he sailed across the Atlantic he sent out a postcard saying he had proved the Riemann Hypothesis, just in case he drowned.

We could try to see if there really was a London taxi with number 1729 at that time. It would be delicious to discover that it was merely an invention of Hardy’s. But I don’t know if records of London taxi-cab numbers from around 1919 still exist.

Maybe I’ll let C. P. Snow have the last word. After telling his version of the incident with Hardy, Ramanujan and the taxicab, he writes:

This is the exchange as Hardy recorded it. It must be substantially accurate. He was the most honest of men; and further no one could possibly have invented it.

The Kepler Problem (Part 4)

27 January, 2022

The Kepler problem is the study of a particle moving in an attractive inverse square force. In classical mechanics, this problem shows up when you study the motion of a planet around the Sun in the Solar System. In quantum mechanics, it shows up when you study the motion of an electron around a proton in a hydrogen atom.

In Part 2 we saw that the classical Kepler problem has, besides energy and the three components of angular momentum, three more conserved quantities: the components of the eccentricity vector!

This was discovered long ago, in 1710, by the physicist Jakob Hermann. But thanks to Noether, we now know that in classical mechanics, conserved quantities come from symmetries. In the Kepler problem, conservation of energy comes from time translation symmetry, while conservation of the angular momentum comes from rotation symmetry. Which symmetries give conservation of the eccentricity vector?

As we shall see, these symmetries are rotations in 4-dimensional space. These include the obvious rotations in 3-dimensional space which give angular momentum. The other 4-dimensional rotations act in a much less obvious way, and give the eccentricity vector.

In fact, we’ll see that the Kepler problem can be rephrased in terms of a free particle moving around on a sphere in 4-dimensional space. This is a nice explanation of the 4-dimensional rotation symmetry.

After that we’ll see a second way to rephrase the Kepler problem: in terms of a massless, relativistic free particle moving at the speed of light on a sphere in 4-dimensional space. Our first formulation will not involve relativity. This second will.

All this is very nice. You can read some fun explanations of the first formulation here:

• Greg Egan, The ellipse and the atom.

• John Baez, Planets in the fourth dimension.

But how could you guess this 4-dimensional rotation symmetry if you didn’t know about it already? One systematic approach uses Poisson brackets. I won’t explain these, just dive in and use them!

Remember, the particle in the Kepler problem has various observables, which are all ultimately functions of its position and momentum:

• position: \vec q

• momentum: \vec p

• energy: H = \tfrac{1}{2} p^2 - \tfrac{1}{q}

• angular momentum: \vec L = \vec q \times \vec p

• the eccentricity vector: \vec e = \vec p \times \vec L - \tfrac{\vec q}{q}

I’ll use conventions where the Poisson brackets of the components of position q_k and momentum p_\ell are taken to be

\{q_k,p_\ell\} = \delta_{jk}

From this, using the rules for Poisson brackets, we can calculate the Poisson brackets of everything else. For starters:

\{H, L_k\} = \{H,e_h\} = 0

These equations are utterly unsurprising, since they are equivalent to saying that angular momentum \vec L and the eccentricity vector \vec e are conserved. More interestingly, we have

\begin{array}{ccl} \{L_k, L_\ell\} &=& \epsilon_{jk\ell} L_\ell  \\ \{e_k, L_\ell\} &=& \epsilon_{jk\ell} e_\ell \\ \{e_k, e_\ell \} &=& -2H \epsilon_{jk\ell} L_\ell \end{array}

where all the indices go from 1 to 3, I’m summing over repeated indices even if they’re both subscripts, and \epsilon_{jk\ell} are the Levi–Civita symbols.

Now, the factor of -2H above is annoying. But on the region of phase space where H < 0—that is, the space of bound states, where the particle carries out an elliptical orbit—we can define a new vector to deal with this annoyance:

\displaystyle{    \vec M = \frac{\vec e}{\sqrt{-2H}} }

Now we easily get

\begin{array}{ccl} \{L_k, L_\ell\} &=& \epsilon_{jk\ell} L_\ell  \\ \{L_j, M_k\} &=& \epsilon_{jk\ell} M_\ell \\ \{M_j, M_k \} &=& \epsilon_{jk\ell} M_\ell \end{array}

This is nicer, but we can simplify it even more if we introduce some new vectors that are linear combinations of \vec L and \vec M, namely half their sum and half their difference:

\vec A = \tfrac{1}{2} (\vec L + \vec M), \qquad \vec B = \tfrac{1}{2}(\vec L - \vec M)

Then we get

\begin{array}{ccl} \{ A_j, A_k\} &=&  \epsilon_{jk\ell} A_\ell \\ \{ B_j, B_k\} &=&  \epsilon_{jk\ell} B_\ell  \\ \{ A_j, B_k\} &=& 0 \end{array}

So, the observables A_j and B_k contain the same information as the angular momentum and eccentricity vectors, but now they commute with each other!

What does this mean?

Well, when you’re first learning math the Levi–Civita symbols \epsilon_{jk\ell} may seem like just a way to summarize the funny rules for cross products in 3-dimensional space. But as you proceed, you ultimately learn that \mathbb{R}^3 with its cross product is the Lie algebra of the Lie group \mathrm{SO}(3) of rotations in 3-dimensional space. From this viewpoint, the Levi–Civita symbols are nothing but the structure constants for the Lie algebra \mathfrak{so}(3): that is, a way of describing the bracket operation in this Lie algebra in terms of basis vectors.

So, what we’ve got here are two commuting copies of \mathfrak{so}(3), one having the A_j as a basis and the other having the B_k as a basis, both with the Poisson bracket as their Lie bracket.

A better way to say the same thing is that we’ve got a single 6-dimensional Lie algebra

\mathfrak{so}(3) \oplus \mathfrak{so}(3)

having both the A_j and B_k as basis. But then comes the miracle:

\mathfrak{so}(3) \oplus \mathfrak{so}(3) \cong \mathfrak{so}(4)

The easiest way to see this is to realize that S^3, the unit sphere in 4 dimensions, is itself a Lie group with Lie algebra isomorphic to \mathfrak{so}(3). Namely, it’s the unit quaternions!—or if you prefer, the Lie group \mathrm{SU}(2). Like any Lie group it acts on itself via left and right translations, which commute. But these are actually ways of rotating S^3. So, you get a map of Lie algebras from \mathfrak{so}(3) \oplus \mathfrak{so}(3) to \mathfrak{so}(4), and you can check that this is an isomorphism.

So in this approach, the 4th dimension pops out of the fact that the Kepler problem has conserved quantities that give two commuting copies of \mathfrak{so}(3). By Noether’s theorem, it follows that conservation of angular momentum and the eccentricity vector must come from a hidden symmetry: symmetry under some group whose Lie algebra is \mathfrak{so}(4).

And indeed, it turns out that the group \mathrm{SO}(4) acts on the bound states of the Kepler problem in a way that commutes with time evolution!

But how can we understand this fact?

Historically, it seems that the first explanation was found in the quantum-mechanical context. In 1926, even before Schrödinger came up with his famous equation, Pauli used conservation of angular momentum and the eccentricity to determine the spectrum of hydrogen. But I believe he was using what we now call Lie algebra methods, not bringing in the group \mathrm{SO}(4).

In 1935, Vladimir Fock, famous for the ‘Fock space’ in quantum field theory, explained this 4-dimensional rotation symmetry by setting up an equivalence between hydrogen atom bound states and functions on the 3-sphere! In the following year, Valentine Bargmann, later famous for being Einstein’s assistant, connected Pauli and Fock’s work using group representation theory.

All this is quantum mechanics. It seems the first global discussion of this symmetry in the classical context was given by Bacry, Ruegg, and Souriau in 1966, leading to important work by Souriau and Moser in the early 1970s. Since then, much more has been done. You can learn about a lot of it from these two books, which are my constant companions these days:

• Victor Guillemin and Shlomo Sternberg, Variations on a Theme by Kepler, Providence, R.I., American Mathematical Society, 1990.

• Bruno Cordani, The Kepler Problem: Group Theoretical Aspects, Regularization and Quantization, with Application to the Study of Pertubation, Birkhäuser, Boston, 2002.

But let me try to summarize a bit of this material.

One way to understand the \mathrm{SO}(4) symmetry for bound states of the Kepler problem is the result of Hamilton that I explained last time: for a particle moving around an elliptical orbit in the Kepler problem, its momentum moves round and round in a circle.

I’ll call these circles Hamilton’s circles. Hamilton’s circles are not arbitrary circles in \mathbb{R}^3. Using the inverse of stereographic projection, we can map \mathbb{R}^3 to the unit 3-sphere:

\begin{array}{rccl} f \colon &\mathbb{R}^3 &\to & S^3    \subset \mathbb{R}^4  \\  \\ & \vec p    &\mapsto & \displaystyle{\left(\frac{p^2 - 1}{p^2 +1}, \frac{2 \vec p}{p^2 + 1}\right).} \end{array}

This map sends Hamilton’s circles in \mathbb{R}^3 to great circles in S^3. Furthermore, this construction gives all the great circles in S^3 except those that go through the north and south poles, (\pm 1, 0,0,0). These missing great circles correspond to periodic orbits in the Kepler problem where a particle starts with momentum zero, falls straight to the origin, and bounces back the way it came. If we include these degenerate orbits, every great circle on the unit 3-sphere is the path traced out by the momentum in some solution of the Kepler problem.

Let me reemphasize: in this picture, points of S^3 correspond not to positions but to momenta in the Kepler problem. As time passes, these points move along great circles in S^3... but not at constant speed.

How is their dynamics related to geodesic motion on the 3-sphere?
We can understand this as follows. In Part 2 we saw that

L^2 + M^2 =  - \frac{1}{2H}

and using the fact that \vec L \cdot \vec M = 0, an easy calculation gives

H \; = \; -\frac{1}{8A^2} \; = \; -\frac{1}{8B^2}

In the 3-sphere picture, the observables A_j become functions on the cotangent bundle T^\ast S^3. These functions are just the components of momentum for a particle on S^3, defined using a standard basis of right-invariant vector fields on S^3 \cong \mathrm{SU}(2). Similarly, the observables B_j are the components of momentum using a standard basis of left-invariant vector fields. It follows that

K = 8A^2 = 8B^2

is the Hamiltonian for a nonrelativistic free particle on S^3 with an appropriately chosen mass. Such a particle moves around a great circle on S^3 at constant speed. Since the Kepler Hamiltonian H is a function of K, particles governed by this Hamiltonian move along the same trajectories—but typically not at constant speed!

Both K and the Kepler Hamiltonian H = -1/K are well-defined smooth functions on the symplectic manifold that Souriau dubbed the Kepler manifold:

T^+ S^3 = \{ (x,p) : \; x \in S^3, \, p \in T_x S^3, \, p \ne 0 \}

This is the cotangent bundle of the 3-sphere with the zero cotangent vectors removed, so that H = -1/K is well-defined.

All this is great. But even better, there’s yet another picture of what’s going on, which brings relativity into the game!

We can also think of T^+ S^3 as a space of null geodesics in the Einstein universe: the manifold \mathbb{R} \times S^3 with the Lorentzian metric

dt^2 - ds^2

where dt^2 is the usual Riemannian metric on the real line (‘time’) and ds^2 is the usual metric on the unit sphere (‘space’). In this picture x \in S^3 describes the geodesic’s position at time zero, while the null cotangent vector p + \|p\| dt describes its 4-momentum at time zero. Beware: in this picture two geodesics count as distinct if we rescale p by any positive factor other than 1. But this is good: physically, it reflects the fact that in relativity, massless particles can have different 4-momentum even if they trace out the same path in spacetime.

In short, the Kepler manifold T^+ S^3 also serves as the classical phase space for a free massless spin-0 particle in the Einstein universe!

And here’s the cool part: the Hamiltonian for such a particle is

\sqrt{K} = \sqrt{-1/H}

So it’s a function of both the Hamiltonians we’ve seen before. Thus, time evolution given by this Hamiltonian carries particles around great circles on the 3-sphere… at constant speed, but at a different speed than the nonrelativistic free particle described by the Hamiltonian K.

In future episodes, I want to quantize this whole story. We’ll get some interesting outlooks on the quantum mechanics of the hydrogen atom.

Learning Computer Science With Categories

26 January, 2022

The first book in Bob Coecke’s series on applied category theory is out, and the pdf is free—legally, even!—until 8 February 2022. Grab a copy now:

• Noson Yanofsky, Theoretical Computer Science for the Working Category Theorist, Cambridge U. Press, 2022.

There are already books on category theory for theoretical computer scientists. Why the reverse? Yanofsky explains:

There’s just one catch: you need to know category theory.

But it’s worth learning category theory, because it’s like a magic key to many subjects. It helps you learn more, faster.

Categories in Chemistry, Computing, and Social Networks

26 January, 2022

This article does two things:

• John Baez, Simon Cho, Daniel Cicala, Nina Otter, and Valeria de Paiva, Applied category theory in chemistry, computing, and social networks, Notices of the American Mathematical Society, February 2022.

It urges you — or your friends, or students — to apply for our free summer school in applied category theory run by the American Mathematical Society. It’s also a quick intro to some key ideas in applied category theory!

Applications are due Tuesday 2022 February 15 at 11:59 Eastern Time — go here for details. If you get in, you’ll get an all-expenses-paid trip to a conference center in upstate New York for a week in the summer. There will be a pool, bocci, lakes with canoes, woods to hike around in, campfires at night… and also whiteboards, meeting rooms, and coffee available 24 hours a day.

You can work with me on categories in chemistry, Nina on categories in the study of social networks, or Valeria on categories applied to concepts from computer science, like lenses.

There are also other programs to choose from. Read this, and click for more details:

The Kepler Problem (Part 3)

23 January, 2022

The Kepler problem studies a particle moving in an inverse square force, like a planet orbiting the Sun. Last time I talked about an extra conserved quantity associated to this problem, which keeps elliptical orbits from precessing or changing shape. This extra conserved quantity is sometimes called the Laplace–Runge–Lenz vector, but since it was first discovered by none of these people, I prefer to call it the ‘eccentricity vector’

In 1847, Hamilton noticed a fascinating consequence of this extra conservation law. For a particle moving in an inverse square force, its momentum moves along a circle!

Greg Egan has given a beautiful geometrical argument for this fact:

• Greg Egan, The ellipse and the atom.

I will not try to outdo him; instead I’ll follow a more dry, calculational approach. One reason is that I’m trying to amass a little arsenal of formulas connected to the Kepler problem.

Let’s dive in. Remember from last time: we’re studying a particle whose position \vec q obeys

\ddot{\vec q} = - \frac{\vec q}{q^3}

Its momentum is

\vec p = m \dot{\vec q}

Its momentum is not conserved. Its conserved quantities are energy:

H = \frac{1}{2} p^2 - \frac{1}{q}

the angular momentum vector:

\vec L = \vec q \times \vec p

and the eccentricity vector:

\vec e = \vec p \times \vec L - \frac{\vec q}{q}

Now for the cool part: we can show that

\displaystyle{ \left( \vec p - \frac{\vec L \times \vec e}{L^2} \right)^2 = \frac{1}{L^2} }

Thus, the momentum \vec p stays on a circle of radius 1/L centered at the point (\vec L \times \vec e)/L^2. And since \vec L and \vec e are conserved, this circle doesn’t change! Let’s call it Hamilton’s circle.

Now let’s actually do the calculations needed to show that the momentum stays on Hamilton’s circle. Since

\vec e = \vec p \times \vec L - \frac{\vec q}{q}

we have

\frac{\vec q}{q} = \vec p \times \vec L - \vec e

Taking the dot product of this vector with itself, which is 1, we get

\begin{array}{ccl}  1 &=& \frac{\vec q}{q} \cdot \frac{\vec q}{q}  \\ \\  &=& (\vec p \times \vec L - \vec e) \cdot (\vec p \times \vec L - \vec e) \\ \\  &=& (\vec p \times \vec L)^2 - 2 \vec e \cdot (\vec p \times \vec L) + e^2  \end{array}

Now, notice that \vec p and \vec L are orthogonal since \vec L = \vec q \times \vec p. Thus

(\vec p \times \vec L)^2 = p^2 L^2

I actually used this fact and explained it in more detail last time. Substituting this in, we get

1 = p^2 L^2 - 2 \vec e \cdot (\vec p \times \vec L) + e^2

Similarly, \vec e and \vec L are orthogonal! After all,

\vec e = \vec p \times \vec L - \frac{\vec q}{q}

The first term is orthogonal to \vec L since it’s the cross product of \vec L and some other vector. And the second term is orthogonal to \vec L since \vec L is the cross product of \vec q and some other vector! So, we have

(\vec L \times \vec e)^2 = L^2 e^2

and thus

\displaystyle { e^2 = \frac{(\vec L \times \vec e)^2}{L^2} }

Substituting this in, we get

\displaystyle { 1 = p^2 L^2 - 2 \vec e \cdot (\vec p \times \vec L) + \frac{(\vec L \times \vec e)^2}{L^2} }

Using the cyclic property of the scalar triple product, we can rewrite this as

\displaystyle { 1 = p^2 L^2 - 2 \vec p \cdot (\vec L \times \vec e) + \frac{(\vec L \times \vec e)^2}{L^2} }

This is nicer because it involves \vec L \times \vec e in two places. If we divide both sides by L^2 we get

\displaystyle { \frac{1}{L^2} = p^2 - \frac{2}{L^2} \; \vec p \cdot (\vec L \times \vec e) + \frac{(\vec L \times \vec e)^2}{L^4} }

And now for the final flourish! The right hand is the dot product of a vector with itself:

\displaystyle { \frac{1}{L^2} = \left(\vec p -  \frac{\vec L \times \vec e}{L^2}\right)^2 }

This is the equation for Hamilton’s circle!

Now, beware: the momentum \vec p doesn’t usually move at a constant rate along Hamilton’s circle, since that would force the particle’s orbit to itself be circular.

But on the bright side, the momentum moves along Hamilton’s circle regardless of whether the particle’s orbit is elliptical, parabolic or hyperbolic. And we can easily distinguish the three cases using Hamilton’s circle!

After all, the center of Hamilton’s circle is the point (\vec L \times \vec e)/L^2, and

(\vec L \times \vec e)^2 = L^2 e^2

so the distance of this center from the origin is

\displaystyle{ \sqrt{\frac{(\vec L \times \vec e)^2}{L^4}} = \sqrt{\frac{L^2 e^2}{L^4}} = \frac{e}{L} }

On the other hand, the radius of Hamilton’s circle is 1/L. So his circle encloses the origin, goes through the origin or does not enclose the origin depending on whether e < 1, e = 1 or e > 1. But we saw last time that these three cases correspond to elliptical, parabolic and hyperbolic orbits!


• If e < 1 the particle’s orbit is an ellipse and the origin lies inside Hamilton’s circle. The momentum goes round and round Hamilton’s circle as time passes.

• If e = 1 the particle’s orbit is a parabola and the origin lies exactly on Hamilton’s circle. The particle’s momentum approaches zero as time approaches \pm \infty, so its momentum goes around Hamilton’s circle exactly once as time passes.

• If e > 1 the particle’s orbit is a hyperbola and the origin lies outside Hamilton’s circle. The particle’s momentum approaches distinct nonzero values as time approaches \pm \infty, so its momentum goes around just a portion of Hamilton’s circle.

By the way, in general the curve traced out by the momentum vector of a particle is called a hodograph. So you can learn more about Hamilton’s circle with the help of that buzzword.

The Periodic Table

21 January, 2022


I like many kinds of periodic table, but hate this one. See the problem?

Element 57 is drawn right next to element 72, replacing the element that should be there: element 71. So lutetium, element 71, is being denied its rightful place as a transition metal and is classified as a rare earth. Meanwhile lanthanum, element 57, which really is a rare earth, is drawn separately from all the rest! This is especially ironic because those rare earths are called ‘lanthanoids’ or ‘lanthanides’.

Similarly, element 89 is next to element 104, instead of the element that should be there: element 103. So lawrencium, element 103, is also being denied its rightful place as a transition metal. Meanwhile actinium, element 89, is banished from the row of ‘actinoids’, or ‘actinides’ — even though it gave them their name in the first place. How cruel!

Here Wikipedia does it right. Element 71 is a transition metal — not element 57. Similarly element 103 is a transition metal, not element 89.

This stuff is not just an arbitrary convention. Transition metals are chemically different from lanthanides and actinides. You can’t just stick them wherever you want.

In simple terms, as we move across the transition metals, they fill 1, 2, 3, … , 10 of their outermost d orbitals with electrons. Similarly as we move across the lanthanides or actinides, they fill 1, 2, 3, … 14 of their outermost f orbitals with electrons. I wrote about this here a while ago:

• John Baez, The Madelung rules, Azimuth, December 8, 2021.

There are some exceptions to the Madelung rules, but the bad periodic tables are not motivated by those exceptions. The Wikipedia periodic table accurately reflects the chemistry. The Encyclopedia Brittanica table completely ruins the story by arbitrarily sticking lanthanum and actinium in amongst the transition metals instead of the elements that should be there: lutetium and lawrencium. I see no good reason for doing this.

Here’s another common kind of periodic table that I hate. It cuts a hole into the bottom two rows of the transition metals, and moves the metals that should be there — elements 71 and 103 — into the rare earths and actinides.

This amounts to claiming that there are 15 rare earths and actinides, and just 9 transition metals in those two rows. That’s crazy: the fact that the p subshell holds 10 electrons and the d subshell holds 14 is dictated by group representation theory. Subshells hold 2, 6, 10, 14 electrons — twice odd numbers.

The periodic table is a marvelous thing: it shows how quantum
mechanics and math predict patterns in the elements. Have fun making up new designs — but if you’re going to use the old kind, use the good one!

If you don’t believe me, listen to this guy:

But unlike him, I don’t think experiments were necessary to realize that the bad periodic tables were messed up. It’s not as if they were designed based on some alternative theory about which elements are transition metals.

Interestingly, the International Union of Pure and Applied Chemistry were supposed to meet at the end of last year to settle this issue. What did they decide? If you find out, please let me know!

Rapid Variable B Subdwarf Stars

19 January, 2022

A subdwarf B star is a blue-hot star smaller than the Sun. A few of these crazy stars pulse in brightness as fast as every 90 seconds! Waves of ionizing iron pulse through their thin surface atmosphere.

What’s up with these weird stars?

Sometimes a red giant loses most of its outer hydrogen… nobody is sure why… leaving just a thin layer of hydrogen over its helium core. We get a star with at most 1/4 the diameter of the Sun, but really hot.

It’s the blue-hot heart of a red giant, stripped bare.

Iron and other metals in the star’s thin hydrogen atmosphere can lose and regain their outer electrons. When these electrons are gone, the metals are ‘ionized’ and they absorb more light. This pushes them further out. Then they cool, become less ionized, absorb less light, and fall back down. This heats them up, so they become more ionized and the cycle begins again.

This happens in standing waves, which follow spherical harmonic patterns. You may have seen spherical harmonics in chemistry, where they describe electron orbitals. The same math is being applied here to a whole star! Now it’s not the electron’s wavefunction that’s pulsing in a spherical harmonic: it’s metals in the atmosphere of a star.

When the star is rotating, spherical harmonics that would otherwise vibrate at the same frequency do so at different frequencies. So, just by looking at the pulsing of light from a distant subdwarf B star, you can learn how fast it’s rotating!

I got the gif of a pulsing star from here:

White Dwarf Research Corporation.

Pulsating white dwarf stars also oscillate in spherical harmonic patterns, and this website shows how they look.

The figure showing frequency lines is from this cool paper:

• Stephane Charpinet, Noemi Giammichele, Weikai Zong, Valérie Van Grootel, Pierre Brassard and Gilles Fontaine, Rotation in sdB stars as revealed by stellar oscillations, Open Astronomy 27 (2017), 112–119.

This paper says “a κ-mechanism triggered by an accumulation of heavy elements (in particular iron) in the stellar envelope caused by radiative levitation is driving the oscillation.”

So, what’s the κ-mechanism and radiative levitation?

The κ-mechanism causes oscillations when a layer of a star’s atmosphere gets more opaque at higher temperatures. For example, when heavy metals near the surface of the atmosphere get hot they can ionize, and thus absorb more radiation. When the layer of ions falls in it gets hotter, more opaque, blocks more escaping heat, and the star’s pressure goes up… pushing the layer out. But when the layer shoots out it gets cooler, less opaque, blocks less escaping heat, and the pressure drops again. So we can get oscillations!

Radiative levitation can drive heavy metals to the surface of a star. They absorb light, and the light literally pushes them up. This can make
these metals thousands of times more common than you’d expect near the surface.

There’s more that can happen with subdwarf B stars, and you can learn about it here:

• Wikipedia, Subdwarf B stars.

For example, they can simultaneously oscillate in two ways, at two separate rates!

The Color of Infinite Temperature

16 January, 2022


This is the color of something infinitely hot. Of course you’d instantly be fried by gamma rays of arbitrarily high frequency, but this would be its spectrum in the visible range.

This is also the color of a typical neutron star. They’re so hot they look the same.

It’s also the color of the early Universe!

This was worked out by David Madore.

As a blackbody gets hotter and hotter, its spectrum approaches the classical Rayleigh–Jeans law. That is, its true spectrum as given by the Planck law approaches the classical prediction over a larger and larger range of frequencies.

So, for an extremely hot blackbody, the spectrum of light we can actually see with our eyes is governed by the Rayleigh–Jeans law. This law says the color doesn’t depend on the temperature: only the brightness does!

And this color is shown above.

This involves human perception, not just straight physics. So David Madore needed to work out the response of the human eye to the Rayleigh–Jeans spectrum — “by integrating the spectrum against the CIE XYZ matching functions and using the definition of the sRGB color space.”

The color he got is sRGB(148,177,255). And according to the experts who sip latte all day and make up names for colors, this color is called ‘Perano’.

Here is some background material Madore wrote on colors and visual perception. It doesn’t include the whole calculation that leads to this particular color, so somebody should check it, but it should help you understand how to convert the blackbody spectrum at a particular temperature into an sRGB color:

• David Madore, Colors and colorimetry.

In the comments you can see that Thomas Mansencal redid the calculation and got a slightly different color: sRGB(154,181,255). It looks quite similar to me: