Sunday, 30 October 2016

Media appearance: Discover Magazine

An article about my work on falling through the Earth was published in this month's edition of Discover Magazine. The article, based on some phone interviews I did with them, can be found here. Unfortunately you will need a subscription to read it, and because they were nice enough to write this article I'm not going to blatantly infringe their copyright and post a scanned version here. Enjoy, if you can.

Saturday, 22 October 2016

Visit to the Alcator C-Mod Fusion Reactor at MIT

I recently became aware that MIT had an experimental fusion reactor on campus, so I decided to check it out. After a few back-and-forth emails, I ended up joining a tour that was being given to the Tuft's energy club (yes, Tufts) on Friday afternoon.

MIT's tokamak research began as an offshoot of its magnet research lab (where MRI was developed), that eventually became so large as to spawn a new center and envelop its host. Alcator is an elision of Alto Campo Toro (high field torus), and this is the third iteration of it. It was recently in the news because they had achieved a fusion reaction at a comparatively high pressure, immediately before it was shut off because the US government diverted all fusion research funds to ITER.

The tour began in the lecture room with an overview of the why's and what's of fusion. In discussing the when, the speaker showed the improvement towards the break-even point in fusion reactors over the past several decades, but said that because the tokamaks had gotten so much bigger, they timescale of the experiments had slowed down considerably and now they have to wait several decades for ITER to be ready before their next big learning experience. I talked more about the timeline to fusion here; it's a bit more pessimistic.

 "Why use fusion to give plants energy to make sugar and then bury them for millions of years until they form fuel and burn the fuel to make electricity, when we can just use fusion to make electricity?"

Another amusing tidbit the speaker mentioned is that more money has been spent in recent years making movies about fusion (e.g. The Dark Knight Rises, Spiderman 2, etc) than on fusion research itself. He was also overjoyed when someone asked why tokamaks were shaped like donuts and not spheres and he got to cite the hairy ball theorem.

After the speech we went across the street to the facility. There was a big control room where a few people were watching a video and looking at data from the last experiment. There was also a neat demonstration where a plasma was established by running a high voltage across a mostly-evacuated tube, and then a magnetic field was activated that visibly pinched the plasma.

We then entered the room with the tokamak. Most of the room, however, was full of electrical equipment. One of the neatest things I learned was about the way the thing is actually powered. It requires very powerful short lived surges of electricity to power the magnetic coils, on the order of 200 megawatts. To achieve this, they integrate power from the grid and store it in a massive rotating flywheel. The infrastructure for transforming the grid electricity is actually much bigger than the tokamak and its other supporting infrastructure. I was informed that the flywheel is bigger than the tokamak and its shielding, but I wasn't allowed to see it. It is apparently 75 tons and spins up to 1800 RPM, storing 2 gigajoules of kinetic energy.

 The flywheel at the Joint European Torus, bigger than Alcator's.

On the wall outside the reactor room was the floor plate from a previous version. It was large.

The tokamak itself is big enough for a person to crouch in, and is surrounded by the wires that generate the magnetic field, and a lot of shielding. The whole thing is about the size of four elephants. It is painted light blue on the outside. On top there is a tank of liquid nitrogen that they use to cool down the copper wires after each heating pulse (which lasts about two seconds). People took photos and asked our tour guide some questions.

 Here I am with the reactor (light blue tank thing behind me). You can see the tank of liquid nitrogen on top. The black pipe behind me is the power supply cable for the magnets. Not sure what the cheese grater thingy is.

Afterwards in the lobby there were a few artefacts we could play with, including some of the copper wires used for the field and the superconductors that are replacing them, and some tungsten shielding plates that were visibly damaged from years of hot plasma abuse.

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Overall, I didn't learn much that I didn't already know about fusion (except the flywheel!) but I'd beat myself up if I knew I worked down the street from a fusion reactor and never went to see it. I appreciate the Plasma Science and Fusion Center guys for putting on the tour, and the Tufts club for (perhaps unknowingly?) letting me crash their event.  Also, one of the safety signs had a typo.

Monday, 10 October 2016

The Nested Logarithm Constants

Depending on your philosophical interpretation of mathematics, I have either discovered or invented a new number: the Nested Logarithm Constant.

$\alpha$=log(1+log(2+log(3+log(4+log(5+....)))))=0.820359862208789788....

As is natural, the logarithms are base-e. I first calculated this number while reading about the nested radical constant, and I wanted to see if something similar existed for logarithms...and it did! While I cannot be certain that others have not examined this number previously, googling the first few digits only yields random lists of digits. I have entered its decimal expansion into the Online Encyclopedia of Integer Sequences, the closest thing that exists to a Wiki of mildly interesting numbers.

It converges quite rapidly with respect to the final digit, being effectively exact after about 5. The plot below shows the convergence with respect to the final digit, and the fractional deviation from the asymptotic value, with an exponential decay shown for comparison.

While it seems obvious empirically that this converges, it is not proven [by me, see update below]. As far as I can tell, a proof of the convergence of the nested radical constant is also lacking. I would suspect that the nested logarithm constant is both irrational and transcendental, and there is a short proof that the logs of integers are irrational, but I'm not aware of a proof that nested logarithms are irrational.

There are a few extensions one can examine. Changing the base of the logarithm changes the value of the final number, decreasing with increasing base towards zero, and towards infinity with decreasing base.

Another extension is to include an exponent on the integers, such that the number becomes:

$\alpha_{N}=\log(1+\log(2^{N}+\log(3^{N}+...)))$

When N is large, 2$^N$ will be much larger than log(3$^N$+...), so we can take the N out of the exponent and see that $\alpha_{N}$ approaches log(1+N·log(2))  log(N)+log(log(2)). The difference between $\alpha_{N}$ and its large-N approximation decreases inversely with N, with a prefactor empirically* very close to the square root of two. With this reciprocal correction, I can almost write down a closed form expression for $\alpha_{N}$, except it starts to diverge close to N=1, and I don't know if it's a coincidence or not (where does the square root come from?). I would be interested in exploring this further to try to find a closed form expression for the 0.82...constant.

 The nested logarithm constants as a function of the exponent, and the deviation of two approximations.
There is one more related number of interest. I can calculate log(1+2·log(1+3·log(1+4·log...)))) which converges to roughly 1.663.... I have not investigated this in as much detail, but it is empirically very close to the Somos Quadratic Recurrence Constant, which can be calculated as √(1√(2√(3√(4...)))) and converges to 1.662. Unless I'm lead to believe otherwise I will assume that this is a coincidence.

This investigation started with me playing around with logarithms and seeing what came out, and lead to a few findings that I think are kind of interesting. Maybe when I have the time and energy I'll investigate this further and be able to determine if the nested logarithm constant has an exact value. There is a journal called Experimental Mathematics, which is basically what this is, so perhaps I can send it there.

UPDATE: A commenter on reddit, SpeakKindly, has put forward a simple proof of convergence by establishing an upper bound using induction. I will repeat it here for the readers:

First, consider the related term log(a+log(a+1+log(a+2+log...(log(a+k)))) for some k and a>1. SpeakKindly proved this is less than a for all k. The base case, for k=0, we have no nested terms and just have log(a), which is less than a. Then we assume this is true for k-1 nestings, so we have log(a+log(a+1+log(a+2+log...(log(a+k)))) < log(a+(a+1))=log(2a+1), which is always less than a. The nested logarithm constant is the log(1+(the above term for a=2)), which is less than log(1+2), thus the nested logarithm constant is less than log(3).

Another user, babeltoothe, used a different argument using the integral definition of the logarithm, where the upper limit of each integral was another integral.

*Best power-law fit is 1.004√2 for the prefactor and -0.998 for the exponent.