More on bugs in Mathematica

In comments to my previous post, Rytis Jursenas pointed at an entire article in the Notices of AMS discussing several bugs in Mathematica. Since such problems are apparently quite common, I feel it deserves a separate post.

Here it comes:

"The Misfortunes of a Trio of Mathematicians Using Computer Algebra Systems. Can We Trust in Them?" by Antonio J. Durán, Mario Pérez, and Juan L. Varona

(just in case, here is the preprint)

"...We have been using Mathematica as a tool in our mathematical research. All our computations with Mathematica have been symbolic, involving only integers (large integers, about 10 thousand digits long) and polynomials (with degree 60 at most), so no numerical rounding or instability can arise in them, and we completely trusted the results generated by Mathematica. However, we have obtained completely erroneous results."

"...Software bugs should not prevent us from continuing this mutually beneficial 

relationship in the future. However, for the time being, when dealing with a problem whose answer cannot be easily verified without a computer, it is highly advisable to perform the computations with at least two computer algebra systems."

Well, I didn't expect such a conclusion to be drawn in 2014.

Take care,

Misha

A bug in Mathematica - 6 years later

In general, I'm a big fan of Wolfram Mathematica and their customer support. For instance, it happened that after my question "Do you guys have/plan to implement the X method to solve ordinary differential equations?" the support team would code and send me an implementation within a few days (although I obviously didn't ask for it).

Also, back to the day, when I found bugs in Mathematica routines I'd describe them in a blog post, and the support team would usually contact me very fast (sometimes within hours) and assist with solving the issue.

However, sometimes the bugs remain. One of the examples was a bug in evaluation of Clebsch-Gordan coefficients I've bumped into six years ago.

Today I mentioned it to a colleague ("...can you imagine how careful we had to be doing angular momentum algebra in grad school!"), and decided to check it, just for fun.

Here is my output from today:

Yep, after 6 years and 10 updates to Mathematica, it's still there...

Take care (especially using the Clebsch-Gordan routines),

Misha

Standing on the shoulders of giants

All of you have heard Newton's famous quote:

"If I have seen further, it is by standing on the shoulders of giants" *

It turns out that not only doesn't this quote originate from Newton, in fact, it was a very common saying at the time (such that Newton wouldn't even think there were people who hadn't heard it before).

That's how the "How to fly a horse" book describes it:

"Newton’s line was, in fact, close to a cliché at the time he wrote it. 

... Newton got it from George Herbert, who in 1651 wrote, "A dwarf on a giant’s shoulders sees farther of the two."

... Herbert got it from Robert Burton, who in 1621 wrote, "A dwarf standing on the shoulders of a giant may see farther than a giant himself."

... Burton got it from a Spanish theologian, Diego de Estella, also known as Didacus Stella, who probably got it from John of Salisbury, 1159: "We are like dwarfs on the shoulders of giants, so that we can see more than they, and things at a greater distance, not by virtue of any sharpness of sight on our part, or any physical distinction, but because we are carried high and raised up by their giant size."

... John of Salisbury got it from Bernard of Chartres, 1130: "We are like dwarfs standing upon the shoulders of giants, and so able to see more and see farther than the ancients." 

We do not know from whom Bernard of Chartres got it." 

Take care,

Misha


* Some of you might have heard Murray Gell-Mann's interpretation:

"If I have seen further than others, it is because I am surrounded by dwarfs."

Superfluidity and Bose-Einstein Condensation

A few times I heard people from the outside of the atomic physics community wondering why was the discovery of Bose-Einstein Condensation (BEC) in alkali gases so special, since the existence of BEC in superfluid helium was considered to be an accepted fact.

Of course, very soon after a BEC of ultracold atoms was created, the implication of the employed technology (as well as of several related developments) became crystal clear. By now, the field of ultracold gases grew into one of the mainstream areas of physics; it already allowed to use our knowledge about atoms and molecules to understand solid state physics, photonics, and even chemistry better.

However, a direct experimental observation of the BEC, as a novel state of matter, was of crucial importance.

Almost immediately after the discovery of superfluidity by Kapitza and Allen and Misener,* Fritz London suggested that this phenomenon was closely related to Bose-Einstein condensation. László Tisza, who was together with London in Paris (no pun intended) at the time, got excited and quickly elaborated on these ideas. Tisza developed the basis for what is known as the "two-fluid model." He conjectured that one can understand the superfluid phase of helium as a mixture of two components. The first, superfluid component, represents a Bose condensate of the atoms occupying the same single-particle quantum state. This results in a macroscopic coherence allowing a flux without friction or viscosity. The second, normal component, whose fraction depends on temperature, behaves as a regular viscous fluid.

Three years later, Lev Landau derived his version of the two-fluid model, based on the quantization of classical hydrodynamics equations. His theory was phenomenological and didn't require the particles to obey Bose statistics. Moreover, he started the paper by bluntly opposing the ideas of Tisza:

...Tisza’s well-known attempt to consider helium II as a degenerate Bose gas cannot be accepted as satisfactory – even putting aside the fact that liquid helium is not an ideal gas, nothing could prevent the atoms in the normal state from colliding with the excited atoms; i.e., when moving through the liquid they would experience friction and there would be no superfluidity at all.

As a matter of fact, it took several decades to unify the ideas of Landau with the ones of London and Tisza. In the end of the 1950's and beginning of the 1960's, several hard-core many-body calculations allowed to theoretically prove that superfluidity is indeed accompanied by Bose-Einstein condensation of helium atoms.**

However, from the experimental side, establishing the existence of the BEC state in superfluid helium turned out to be extremely challenging. Namely, it was possible to obtain only indirect evidence that about 10% of the atoms form a condensate, based on high-energy neutron scattering and spectra of atoms evaporated from the helium surface.

Thus, it was the observation of a BEC in ultracold gases and later experiments on their superfluidity which allowed to establish a connection between the two concepts beyond all possible doubt.

Take care,

Misha

* While these two papers appeared back-to-back in Nature, only Kapitza was awarded a Nobel Prize for this discovery (and only 40 years later!). Quite unfortunately, even nowadays the contribution of Allen and Misener remains widely disregarded. There are several great articles discussing this peculiar story; there are even gossips that Kapitza refused to accept the prize together with Allen which made the Nobel committee postpone the decision for decades.

** Since the phenomenological theory of Landau happened to successfully reproduce the experimental data, the contributions of London and Tisza were not acknowledged as widely as they should have been. Among other things, it seems that Fritz London was the first person ever to recognize the effects of quantum mechanics at the macroscopic scale, and think about the emergence of quantum phenomena in many-particle systems. Phil Anderson wrote a nice essay about London's forgotten contributions, which resonates with his own "More is different"  very well.

"Ambulance chasing" in particle physics

The progress in particle physics crucially depends on a few large experiments, releasing new data to the entire community of theorists several times a year.

Such a format of theory-experiment interaction leads to what is called the "ambulance chasing" phenomenon. Namely, every time after an announcement of a preliminary experimental result (i.e. one with less than 6 sigma deviation), the arxiv.org preprint server explodes from the amount of theory submissions explaining it. Widely-known examples would be the detection of superluminal neutrinos five years ago, or the recent ATLAS observation of a two-photon peak at 750 GeV.

Quite often, it turns out that the result happened to be within an experimental error bar and is not confirmed by future measurements. However, whether right or wrong, the theory papers receive a fair amount of citations - the more the earlier the preprint appeared.

The race for priority is so intense that dozens of theory preprints are submitted within hours (!) after the official announcement, which means that they were written in advance based on some insider information from the experimentalists.

In any case, in a recent preprint Mihailo Backović (Catholic University of Louvain, Belgium) developed a theory describing the ambulance chasing phenomenon both qualitatively and quantitatively.

The analysis is based on the assumptions that the number of papers on a particular topic can be described using Poisson statistics, and that the interest in the topic as well as the number of available ideas decrease in time (he considers power-law and exponential decays). This resulted in a two-parameter model, which provided a perfect fit to 9 cases of ambulance chasing, considered by the author.

Thus, even if all these theory papers were wrong in explaining the measurements, they at least can be used to study universality in complex systems. :-)

Take care,

Misha

Surprisingly elegant expressions for fundamental constants and other applied numerology

A few years ago, when I was still at Harvard, Ariel Amir knocked on my door. Somehow, he bumped into my old blog post on the (probably) shortest physics article ever, and was wondering whether such a numerological coincidence is truly random or not.

I've seen this paper by Friedrich Lenz for the first time in the Fall of 2006, visiting the Fritz Haber Institute to interview for a PhD position (I cannot believe it was almost ten years ago). There was a talk by an experimentalist who was aiming to measure the change in the proton-to-electron mass ratio using a molecular fountain (I think that was someone from the Rick Bethlem's, but I'm not absolutely sure). There, the "shortest paper ever" was shown as a joke to spice up the introduction.

Since then I was somewhat puzzled whether Lenz actually meant what he wrote, or was basically trolling the scientific community.

Thus, Ariel and I chatted and decided that the only way to answer this was to do an actual calculation. It turned out that Tadashi Tokieda was on sabbatical at the Radcliffe Institute – basically around the corner – and we decided to discuss the idea with him first.

To prove the point, we considered all possible combinations of the 0-9 digits and standard mathematical constants (such as pi or e) and calculated the probability that a combination of three of them would reproduce a 5-digit number of the form xxxx.x.

The probability turned out to be as low as 1.2%!

In other words, the Lenz observation was indeed quite intriguing, in the sense that such an elegant expression for the proton-to-electron mass ratio might have signaled for some underlying physical theory. While, as far as I know, this theory has not ever been revealed (and later measurements actually deviate from 6pi^5), paying attention to such surprises might pay off and deepen our understanding of physics.

Tadashi, Ariel, and I wrote a little popular piece about such surprising coincidences, which will appear in the June issue of the American Mathematical Monthly.

I hope you will enjoy reading it!

Take care,

Misha

Blogging While Untenured and Other Extreme Sports

Hello there,

In the good old times I used to be a blogger. Moreover, this blog still appears on the first page of the Google output if one searches for my name.

Long story short, I decided to follow the classic piece by Christine Hurt and Tung Yin (or did I misunderstand their advice?) and give blogging another try after several years of silence.

As a faculty, I do not really have more time as I used to have being a postdoc, when blogging became a luxury that I could not afford. I feel, however, that now I might have more stories to tell, and those will be lost if I don't put them out somewhere.

Not much has changed concerning the things that excite me - it's still mostly funny stories from science and its history, although we'll see how it goes (and how long I will last as a blogger this time :-)

Take care,

Misha