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tag:blogger.com,1999:blog-17848613546847475742017-08-18T05:09:26.811-07:00Post-Doc Ergo Propter HocAlexhttp://firstname.lastname@example.com:blogger.com,1999:blog-1784861354684747574.post-19497521480388115872017-02-23T20:07:00.000-08:002017-02-24T04:49:40.943-08:00Comfortable Relativistic Space Travel: New paper on arXivI have written a new paper which has just come out on arxiv.org. It has the somewhat long title of "Relativistic minimization with constraints: A smooth trip to Alpha Centauri" and can be found here. My co-author on this paper was Riccardo Antonelli, who also writes a physics blog, called Hologrammata. It discusses and derives the most comfortable way to plan a relativistic journey, which may become relevant in several thousand years when we can actually go fast enough for this to matter. In this post, I will explain what this paper is about and what inspired me to write it.

Artist's impression.

Every month, the American Journal of Physics puts out a new edition with about twenty articles. These are usually a delight to read and can be read by anyone with a basic physics background. Some of the articles are about physics education, but most of the research articles involve applications of established physics to problems nobody had quite thought of before. (This is where my tunnel-through-the-Earth paper is published).

In the October 2016 edition, I read an article called "The Least Uncomfortable Journey from A to B" (possible free version here) by three Swedish and possibly French researchers, Anderson, Desaix, and Nyqvist. In it, they ask the following question: on a linear trip taking time T from point A to point B separated by distance D, what is the trajectory (instantaneous velocity as a function of time or distance) that minimizes the time-integrated squared acceleration? If you can't mentally convert that sentence into LaTeX, the "discomfort" here is the total acceleration that the passenger feels throughout the trip, pushing them into their seat on the first half of the trip, and thrusting them out of their seat on the second half.

The solution, which they find through Lagrangian-style variational calculus, turns out to be a trajectory with constant jerk, where the acceleration decreases linearly over the entire trip, such that the velocity is a quadratic function of time. A lot of the paper talks about various analytical approximation methods, and also computes an alternate "least uncomfortable" trajectory where the squared jerk is minimized instead of the squared acceleration.

The interesting thing about the solution is that there is only one. The relative velocity (v/vmax) as a function of the relative position (x/D) or time (t/T) does not actually depend on the distance or time. No matter how far or fast you want to go, there is only one set least-uncomfortable way to get there. However, there is a problem with this. If you want to calculate how to plan a train ride that covers 500 miles in an hour, the solution is fine. But if you want to plan a trip that goes from Earth to Jupiter in an hour, then it starts running into some problems. Although it wasn't stated in the paper, the solution only holds when D/T is much less than the speed of light. I was curious about finding a relativistic solution to this problem, and if I get curious enough about a problem that is actually possible to solve, I can be quite persistent.

In the relativistic version of this problem, when you consider the integral over time of the squared acceleration, it matters in which reference frame the time and acceleration are being measured. Because we care about the comfort of travellers and not of stationary observers, we consider the proper acceleration and not the coordinate acceleration. This problem can be formulated either in terms of "lab" time or proper time on the ship, there are benefits to both. My first attempt at this essentially involved following Anderson et al.'s derivation except with a lot of extra Lorentz factors, which is generally not a good way to go about things, and I stalled pretty quickly.

Then one day, I was talking to my internet friend Riccardo, who had just finished his master's thesis in string theory and holography. I figured a lot of the relativistic analytical mechanics formalisms might still be fresh in his mind, so I mentioned the problem to him. He too found the problem interesting, and came to the realization that if the problem was formulated in terms of the rapidity (the hyperbolic tangent of velocity relative to c) since its derivative is proper acceleration, it could be expressed as a much neater problem than my "add a bunch of Lorentz factors" approach.

The way to derive the solution, Riccardo discovered, is to treat it as a Lagrangian function of the rapidity rather than the position (such that you write down L(r, $\dot{r}$, ) instead of L(x, $\dot{x}$) and apply the Euler-Lagrange to higher-order derivatives of position than normal). Even though we were unable to derive a closed-form solution, it turns out, the rapidity of the least uncomfortable solution evolves like a particle in a hyperbolic sine potential, and I was able to generate solutions using the Runge-Kutta algorithm, which has been serving me well since I learned how to use it in 2008.

As I said, there is only one classical solution, it is universal for all distances and times. However, when relativity it taken into account, the solution depends on how close to the speed of light it gets: there is now a single free parameter that characterizes the solution, and we can express it in terms of its maximum velocity, rapidity, or Lorentz factor. Our solution recovers the classical solution in the low-velocity limit (which is good, otherwise we'd be obviously wrong), but as the maximum speed increases, a greater fraction of the time is spent close to the maximum (which makes sense, as the path is some form of "accelerate close to light speed, decelerate down from close to light speed" and as you approach light speed, your velocity changes less and less)

A figure from the paper showing relative velocity as a function of relative time. The classical solution is the inner parabola with the car next to it, the solutions get faster as you go out, the fastest has a maximum Lorentz factor of 100 and is marked by a rocket. While making this figure I had a minor legal quandary about the copyright status of unicode emojis.

Now, we had solved the problem that I was inspired to solve, the relativistic version of Anderson et al.'s least uncomfortable journey. However, this whole thing may be moot for space travel: you don't necessarily want to keep acceleration to a minimum, instead you might want to contintually accelerate with Earth-like gravity for the first half of the trip, then reverse thrust and decelerate with Earth-like gravity for the second half, so that the entire ride (besides the switcheroo in the middle) feels like Earth. We calculated a trip like this to Alpha Centauri, which would take six years relative to Earth and 3.6 proper years on-board due to time dilation, reaching 95% light speed. With our solution, covering that distance in the same proper time would only reach 90% the speed of light, and might be more appropriate for sending mechanically sensitive electronic probes or self-replicating machines than it would be for sending people.

Anyway, we wrote up the paper and now it's online for the world's reading pleasure. It was fun seeing this paper go from a persistent idea to a finished product in a few weeks time, and dusting off some of the long-forgotten special relativity tools. It was a nice little collaboration with Riccardo, where I came up with the problem and he came up with the solution. This is the first random physics idea I've had in while that has actually come to fruition, and I hope there are many more to come.Alexhttp://firstname.lastname@example.com:blogger.com,1999:blog-1784861354684747574.post-22200107122381030812017-02-03T07:02:00.002-08:002017-02-03T07:02:32.960-08:00New article on the early history of scienceI wrote an article on the first science experiments, over at PhysicsForums. Check it out!Alexhttp://firstname.lastname@example.com:blogger.com,1999:blog-1784861354684747574.post-38632628810055501692017-01-14T08:12:00.000-08:002017-01-14T08:27:50.691-08:00Empirical Searches for Nominative DeterminismNominative determinism is the idea that a person's name influences their life or career. The Seven Dwarfs are an example, and humorous lists have been written about people with names that suit their vocation. The world's fastest man is named Bolt, there was a Russian hurdler named Stepanova, a meteorologist named Julie Freeze, and eugenics pioneer Eugen Fischer. Considerable attention is given to a urology paper by Weedon and Splatt.

To my knowledge, the idea is not taken too seriously. However, I was recently curious about the extent to which people have looked for evidence of nominative determinism. I went on Google Scholar and found some results which I'm sharing with you now.

Most analyses of nominative determinism focus on medical professions, looking at whether specialists tend to have names associated with their speciality.

Examples of nominative determinism in medical research, with statistics of N=1. Ref.

Although this study was not entirely serious, it had a number of methodological flaws as it failed to consider the prominence of each surname in the medical fields relative to its prominence in the population at large.

One recent study was by four members of a family surnamed Limb, all doctors (two of whom named C. Limb are avid climbers). They surveyed a directory of over 300,000 medical doctors grouped by speciality, and went through the list and independently noted which were suited to medicine in general and to their speciality. The lists were then merged by consensus. They found that 1 in 149 doctors had medically appropriate surnames, and 1 in 486 had names relevant to their field. They noted:

"Specialties that had the largest proportion of names specifically relevant to that specialty were those in which the English language has provided a wide range of alternative terms for the same anatomical parts (or functions thereof). Specifically, these were genitourinary medicine (eg Hardwick, Kinghorn, Woodcock, Bell) and urology (eg Burns, Cox, Dick, Koch, Cox, Balluch, Ball, Waterfall). Some terms for bodily functions have been included in the frequency counts but are not mentioned because they are colloquial terms that may lower the tone of this publication."
Another study looked at nominative determinism in patients rather than doctors, trying to see if surname influences health. Based in Ireland, the authors looked through the Dublin phone book to estimate the percentage of people surnamed Brady (0.36%), and then tried to ascertain what fraction of them had been diagnosed with bradychardia (slow heart rate). The only data they had available was for pacemaker implants, and found that 8 out of 999 pacemakers had gone to people named Brady. Despite the small numbers, this was statistically significant (p=0.03) relative to Brady's prominence in the Dublin telephone directory. The authors don't speculate on the causes of this, but neglected to discuss the idea that doctors may be subtly biased towards diagnosing Bradys with bradycardia.

"This increased bradyphenomenon in Bradys could be attributable to increased levels of bradykinin"

In the analysis of particle physics experiments there is something called the "look elsewhere effect" where any significant effect has to be considered in light of all the other effects that are being looked for (this xkcd gives an example of the folly), which reduces the overall significance of the interesting part. I suspect that if they had looked at what other surnames were over-represented in bradycardia diagnosis, the significance of Brady would be diminished. I think this proviso should be applied to any search for nominative determinism: look at the boring names as well as the silly ones.

There are a few other studies out there, mostly tongue-in-cheek. These do go a step beyond compiling humourous lists, but not too much further.Alexhttp://firstname.lastname@example.com:blogger.com,1999:blog-1784861354684747574.post-70007013050637524302016-12-04T10:06:00.002-08:002016-12-04T12:00:50.413-08:00Strange Moments in Science: The Battle for the Planck LengthMany people use Wikipedia as a go-to source of information on a topic. People know that anyone can edit Wikipedia at any time, but generally trust that it is largely accurate and trustworthy because of the large number of vigilant volunteers and the policy of requiring sources for facts. A few years ago, I noticed that the article on the Planck length had some very detailed yet weird and incorrect information that was causing a lot of confusion among physics enthusiasts, and I decided to look into it.

This image taken from the Wikipedia commons and based on an xkcd.

I have written about the Planck length and its physical meaning on PhysicsForums. Besides being a "natural" unit, I stressed that the main significance of the Planck length is that it is approximately the order of magnitude at which quantum gravity becomes relevant, but that the idea that it makes up some sort of "space pixel" is a common misconception. There may be additional significance to it, but that wouldn't be part of established physics.

The Wikipedia page on the Planck length made the claim that a photon with a Planck length wavelength would collapse into a black hole. This is physically problematic (what does it even mean for a photon to collapse into a black hole?) in part because it violates Lorentz symmetry, because you could just observe the photon in a different reference frame and it would have a longer wavelength. Would the black hole "uncollapse" if you moved away from it? Is it only a black hole in some privileged reference frame. There was a large "proof" of this purported fact based on the Schwarzschild metric and the gravitational self-energy of a photon (which doesn't really make sense either because photons don't have a rest frame...nor are they uniform spheres). If you're using general relativity to contradict special relativity, you have done something wrong.

I thought this was strange, so I wanted to look at the source, which was a paper with a Russian title that in English meant "Comprehending the Universe." I did a search for it, and all that came up was the same Wikipedia page. The author of the referenced paper suspiciously had the same name as the person who had made the majority of the recent edits to the Wikipedia page. The reference is published by Lambert Academic Publishing, which is widely known to be a scammy vanity press that will publish your manuscript and then sell you a paper copy. They have recently been emailing me asking if I want them to publish my thesis. I was suspicious and perturbed, so I made a comment on the Wikipedia talk page mentioning that the reference couldn't be found. The author then posted a link to a page where it could be purchased for $60.

So basically, the situation was that there was this incorrect information being passed off on Wikipedia as "proven" physical truth, based on an article published in Russian in a vanity press and added to Wikipedia by the author. This was Bad, and it was misleading people, and I wanted to fix it. It can be quite difficult to actually change things on Wikipedia, because of the large number of policies that have to be followed and protectively zealous editors who will revert a lot of the changes they see. I figured if I just started removing the relevant material, it would quickly be undone by this guy who wanted his stuff on Wikipedia. So, I went to a friend who I knew to be a pretty serious Wikipedian, who goes by the nickname Oreo Priest. I explained the situation to him, and he logged in to back me up. He pointed out on the talk page the various policies that were being violated (original research, reliable source, conflict of interest, etc) and gave me carte blanche to remove the offending material.

So, I went ahead and started removing it, and when the author protested, my friend and some other Wikipedians backed me up by citing official policies. The author's final argument was that his work was presented at the "5th International Conference on Gravitation and Astrophysics of Asian-Pacific Countries" in Moscow in 2001, and was therefore legit. Then, the final smackdown was laid:

The fact that you also presented your work at a conference doesn't change any of the above points, sorry.

After that, the pollution stopped, although I haven't checked the guy's profile lately to see what he's up to. A large chunk of the article was removed, and hasn't been added back. However, there hasn't been a lot of improvement to the article since, so the "Theoretical Significance" section is a bit disjointed and sparse. It, and some of the other Planck unit articles, could use the attention of some quantum gravity researchers. Still, it's much better than the page on topological insulators. That page is terrible.
Alexhttp://firstname.lastname@example.com:blogger.com,1999:blog-1784861354684747574.post-22802014263805470122016-11-29T04:58:00.002-08:002016-11-29T04:58:39.614-08:00The Bremen Physics CatapultIn the town of Bremen in Northwest Germany there is a facility where researchers can launch their experimental apparatus by catapult. It is at a research facility called ZARM, the Center for Applied Space Technology and Microgravity (the acronym makes more sense in German).

The tower in all its glory (from the ZARM website).

The main goal of this facility is to provide microgravity environments for physics experiments, on a scale more accessible than orbital space launch or parabolic aircraft. Besides answering the kinds of questions that keep us up at night, like how cats behave in zero gravity, microgravity is a useful method to control the effective gravitational field in an experiment. As Albert Einstein reminds us, true freefall is indistinguishable from inertial rest, and remaining static in a gravitational field is indistinguishable from uniform acceleration. If you want your experiment to take place in a truly inertial reference frame, dropping it down a tower is a simple way to achieve this.

The entire experiment is built into a capsule like this, and then launched.

Initially, it was simply a drop facility: an experiment built into a modular capsule could be raised by an elevator and dropped from the top of the tower, falling 120 meters and experiencing four seconds of freefall. In 2004 the facility was modified to include a catapult launch with an initial speed of 168 km/h, giving it twice the freefall time but making the facility at least eight times as cool.

A gif of a lanch and landing, pillaged from this youtube video.

Based on a few videos of the facility, it appears that immediately after launch, a crash pad full of protective foam slides into place, protecting the experiment as it lands. Some of the videos are quite well done, check them out if you're interested.

Launch and landing.

I first became aware of this experiment when I read a paper (or possibly a news article) about launching a Bose-Einstein condensate in the catapult. In quantum mechanics class, we learn that a system whose wavefunction that is initially a Dirac delta will expand isotropically through time-evolution, becoming more spread out. Bose-Einstein condensates, functionally made of cold atoms in an electromagnetic trap, serve as a good experimental system for wavefunction dynamics because they are much larger than an individual particle while still behaving like one. However, none of those quantum mechanics homework problems included an anisotropic acceleration vector like we have on Earth's surface, and breaking the symmetry like that causes the condensate to expand differently than it would in true freefall. To measure this predicted isotropic expansion, the research group launched their experimental from a catapult. Because these Bose-Einstein condensates are so sensitive to acceleration, the group was proposing they could be used for accelerometers in spacecraft navigation systems. We will see if that actually happens.

The Bose-Einstein Projectile, from the Humboldt University group.

Not every experiment is as exotic, sometimes it's used to characterize apparati that will eventually be launched on satellites. Some experiments are testing the expansion of explosions in microgravity rather than atomic condensates, both for science's sake and also to gauge safety protocols aboard spacecraft. There are also some biological experiments, for example studying how fish orient themselves and turn in microgravity (I guess they couldn't get the approval to use cats), and others to test how plants perceive gravity (I feel like "how to plants know to grow up" can become a pretty complicated question. The projects are listed on the ZARM website, some in English and some in German.

My experiments typically take place in a little pocket of fluid on a piece of glass on top of a microscope on an optical table in a stationary lab. Compared to launching an experiment from a catapult, that seems rather mundane.Alexhttp://email@example.com:blogger.com,1999:blog-1784861354684747574.post-75076099702131572312016-11-03T20:46:00.001-07:002016-11-03T20:46:45.941-07:00DNA, topological ropelength, and minimal lattice knotsThis post is about some diversions into knot theory, a branch of topology, that I took while contemplating my experiments. My job involves performing and analyzing experiments where I stretch out DNA molecules with knots in them. We study DNA because it serves as a good model polymer that's analogous to the much smaller polymer molecules that plastics and other modern materials are made of, and we study knots because we want to understand how entanglement affects polymer dynamics. There are also some genetic sequencing technologies for which we might want to add or remove knots, but right now I'm mainly just trying to figure out how polymer knots work. The theoretical side of the project (largely carried out by my colleague Vivek and Liang) involves running simulations of polymer chains with different kinds of knots, or simulating the formation of knots, and thinking about that is where I start to dip into knot theory.

A DNA molecule with two knots in it, from one of my experiments. The molecule is about 50 microns when stretched.

Mathematically, knots are only truly defined in loops, as a knot in an open string can just be slid off the end. Knots are loosely categorized by the minimum number of times the loop crosses over itself in a diagram (see below). The simplest knot, the trefoil, when drawn must cross itself a minimum of three times. For a given crossing number, there can be multiple kinds of knots. The two simplest knots, with three and four crossings, only have one type, then there are two for five, three for six, seven for seven, then it starts to explode: 165 for ten crossings, 9988 for thirteen, and over a million for sixteen. Beyond this, it is unknown how many kinds of knots there are (although there are constraints). Each type of knot has certain parameters that are unique to the knot, called invariants, and calculating these invariants is how knots can be distinguished. When dealing with knots in real ropes and polymers and DNA, we have to remember that it doesn't count as a knot unless the ends are closed. If the ends are very far from the interesting knotty part we can just pretend this is the case without too much issue; in simulations of tightly bunched knots however there are various algorithms called "minimally interfering closure" that make knots in open strings topologically kosher.

Knots with up to seven crossings.

The typical model of a polymer is that of a self-avoiding walk, which I've talked about in the past. When the walk is forced to have closed ends, it represents a topologically circular molecule, and can be thought of as a self avoiding polygon (even though polygons are typically imagined in two dimensions). These self-avoiding polygons can take the form of different types of knots, and in fact it can be shown that as a self-avoiding walk becomes longer and longer, the probability of having a knot in it approaches 100%. A number of computational papers have shown that knotted ring polymers have a size (quantified by the radius of gyration, the standard deviation of the location of each monomer in the chain) which is smaller than an unknotted chain of the same length. This makes intuitive sense, there are topological constraints which prevent the chain from adopting a widespread conformation.

The chains in these simulations tend to be quite "loose," and I was curious about the scaling of tight or "ideal" knots, which are knots that have the minimum possible length-to-diameter ratio. Each type of knot has a parameter called the ropelength, which is the minimum length required to make a knot of unit diameter. The ropelength can be a difficult quantity to calculate, and one group has written a few papers just slightly hammering down the ropelength of the simplest knot, bringing it from 16.38 in 2001 to 16.37 in 2014.

These knots are "ideal" meaning they have the smallest possible length-diameter ratio.

The ropelength comes up in my research in part because of a theory from Grosberg and Rabin, who derived the free energy of a knot in a polymer. The free energy arises from the fact that the length of polymer in the knot is confined, reducing entropy, and bent, increasing energy. They argued that the energy of a given knot type is minimized at a certain length, and such a knotted conformation constitutes a metastable local minimum in the free energy (compared to the unknotted global minimum) that implies that an open knotted polymer wouldn't spontaneously unknot itself, but instead would unknot through a slower diffusion process when the knot reaches one end. One of the terms in the Grosberg-Rabin is the excess knot contour, which is the length of polymer in the knot minus the ropelength, and thus it is necessary to know the ropelength to calculate the metastable knot energy. Another paper argued that metastable knots don't exist, and that is one of the things I'm trying to figure out with my knotted DNA experiments.

The most topologically accurate DNA knot experiment comes from Robert Bao, Heun Jin Lee, and Stephen Quake in 2003, where knots of known topology were tied in individual DNA molecules using optical tweezers. I get the impression that Bao developed the expertise to do this, wrote his paper, and made the sanity-promoting decision never to do it again. In their paper, they measured the amount of DNA in different kinds of knots and the diffusion of the knots along the molecule, and used this to ascertain how much intramolecular friction was occurring within the knot. Grosberg and Rabin claimed to have postdicted Bao et al.'s findings with their model. In my experiments, unfortunately, the knots are much more complicated and I don't know what kind they are.

In my experiments, I'm interested in figuring out what kind of knots I'm looking at, and one way to do that is to estimate the amount of DNA in the knot and backstrapolating to figure out the number of essential crossings. To aid in this endeavor, I looked up the known ropelength of knots up to 11 crossings, which was not super-easy nor super-difficult to find. I don't think anyone has analyzed this specific relationship before, but not too surprisingly, with more crossings you tend to have a longer knot, although there is overlap between knots with adjacent or superadjacent crossing number. This relationship fits well with either a linear or square-root function (the best-fit power is 0.8...not sure that's meaningful but I love power-law fits). Once you get past 11 crossings there are problems because A. nobody has bothered calculating this for 12-crossing knots (of which there are 2176) and B. the mean isn't a useful parameter anymore because there are two populations: alternating and non-alternating knots, which have their own behaviours. Complex knots often have more in common with other knots of a certain class (e.g. twist, torus knots) than with the same crossing number.

More complex knots require more rope to tie.

My idea was to use this trend to figure out or constrain the topology in my experiments. For example, if my knot has 2 microns of DNA in it, I can figure out (hypothetically) that it must have at most 30 crossings based on extrapolating the ropelength curve or Bao's data, or something like that. I still haven't solved the topology-constraining problem. However, looking into the ropelength scaling got me interested in other aspects of maximally tight knots, even though maximally tight knots don't really occur in my experiments.

In addition to the lengths of ideal knots, I was interested in their size in general, which for a polymer is typically quantified by the radius of gyration, and a lot of polymer physics looks at the scaling of the radius of gyration with length. It's established that loose floppy knots grow more slowly than loose floppy rings, but what about tight knots? It turns out, the radii of gyration for ideal knots hasn't been calculated (at least not to my knowledge), so to examine this I'd have to generate the knots based on tables of 100 Fourier coefficients and then discretize and measure them. Instead I opted to look at a less-ideal system, minimal knots on cubic lattices. These were tabulated by Andrew Rechnitzer for a paper on the topic, and exist for knots up to 10 crossings (for which there are 165). These are useful for doing simulations of knotted polymers because they allow the initialization of complex knots, and the lack of tabulation for more complex knots makes them difficult to systematically simulate.

A minimal 10-crossing knot on a cubic lattice.

These are the knots whose coordinates on a cubic lattice have the least number of points, but they are still longer than ideal knots. How much longer? The length of each minimal lattice knot (which I call the cubelength) is roughly 40% greater than the equivalent ideal knot, decreasing slightly as a function of crossing number. This makes handwaving sense, because you have an extra factor of square root of two if you go across a diagonal instead of taking two steps. We can see all the different knots (from 3 to 10 crossings) in the graph below; notice there is overlap between the largest 7-crossing knots and the smallest 10-crossing knots. I had a hypothesis that these tight knotted lattice polygons would be more compact in their scaling (the radius of gyration would increase with a weaker power of length) than regular compact lattice walks, and went to examine it.

Every minimal lattice knot up to 10 crossings (except one of the 10-knots...I only ended up with 164 for some reason).

The size of a random walk grows with the square root or 0.5 power of its length. The size of a self-avoiding walk grows faster, with roughly the 0.6 power, for reasons I explain here. A compact globule (the form a polymer takes when it collapses) which grows by wrapping around itself, will grow with a volume proportional to its length, so its radius is expected to grow with the 0.33 power of its length. What do we see when we look at lattice knots? The ensemble of compact lattice knots grows at the 0.33+-0.02 power, entirely consistent with a compact globule.

Left: Average izes of minimal cubic lattice knots as a function of the crossing number. Right, same as a function of their length. I intuitively guessed the function Rg=(L/6)$^{1/3}$ and it fit the data really well.

Nothing too exciting there, but another question one could ask is whether minimal knots grow faster or slower than random self-attracted walks of comparable length, as polymers scale differently as they grow, before they reach the asymptotic limit. To figure this out, I generated self-avoiding random walks on a cubic lattice up to length 60, biasing the steps they take (with a Metropolis method) to have a strong preference for being next to non-adjacent occupied sites. Even with very strong self-attraction (well past the theta-point in polymer language), the scaling was 0.36, and not 0.33, but looking at a graph there is not a noticeable difference between the scaling of knots and tight lattice walks.

So, what's the moral of this story? A tangent from my research into DNA knots lead me to look into some of the scaling properties of ideal and minimal cubic lattice knots. I didn't find anything too profound: more complex ideal knots are bigger, and minimal cubic knots grow like the compact globules they are. It was a fun little investigation that I don't think anyone had bothered doing before. I didn't find anything interesting enough to publish, so I thought I'd share it here.Alexhttp://firstname.lastname@example.com:blogger.com,1999:blog-1784861354684747574.post-47218112777657527682016-10-30T14:17:00.002-07:002016-10-30T14:17:42.518-07:00Media appearance: Discover MagazineAn 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.Alexhttp://email@example.com:blogger.com,1999:blog-1784861354684747574.post-22024329364330985612016-10-22T17:16:00.001-07:002016-10-22T17:19:42.111-07:00Visit to the Alcator C-Mod Fusion Reactor at MITI 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.

W

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.

Alexhttp://email@example.com:blogger.com,1999:blog-1784861354684747574.post-49289029424723566342016-10-10T08:39:00.000-07:002016-10-10T13:03:03.390-07:00The Nested Logarithm ConstantsDepending 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.
Alexhttp://firstname.lastname@example.com:blogger.com,1999:blog-1784861354684747574.post-88969355903460273232016-08-27T09:16:00.000-07:002016-08-27T09:16:22.251-07:00What's the deal with fusion power?
"Fusion is the energy of the future...and always will be."
I recently attended a seminar at the MIT Plasma Science and Fusion Center from a director of the ITER project about the prospects of a commercial fusion reactor. I decided to write a post about why fusion always seems to be 50 years away and what needs to happen to make it a reality. What it comes down to, is that it requires a huge initial investment that nobody wants to pay for.

The SimCity 2000 transit advisor understands the challenges facing the fusion plants in his city.

The most-studied system for fusion is the tokamak (from the Russian for toroidal chamber with magnetic coils), which is a big donut-shaped chamber with a plasma inside, with a strong applied magnetic field that keeps the plasma confined in thin ring-shaped region in the middle of the torus. The plasma is heated up by driving it with electric fields and inducing Joule heating. If the plasma gets hot enough, it can begin to fuse hydrogen into helium, which produces a lot of excess energy in the form of neutrons, gamma rays, and overall heat. The heat is used to boil water to run a turbine. While the sun typically fuses four protons into helium (the p-p chain), the most accessible reaction is deuterium-tritium fusion, which has a lower energy barrier. Tritium is extremely rare naturally, but it can be produced by surrounding a nuclear reactor or the tokamak itself with lithium, which becomes activated by the neutron flux and then decays into tritium.

The inside of the Joint European Torus tokamak, with a view of the plasma.

Fusion research began after the Second World War, not only towards the development of hydrogen bombs but also for power generation, with plasma systems getting bigger and hotter over the next few decades. Due to the Cold War, a lot of this research was classified.

The sad graph

In 1976, fusion researchers in the US wrote a report about future prospects for fusion power, suggesting that with appropriate funding fusion power would be realized within 15 to 30 years. They also predicted that with current levels of funding, they would not be able to achieve their goals in the foreseeable future. The actual funding reality since 1976 has been even bleaker than that. So it's not just scientists always saying that it's 20 years away and not making any progress, it's the lack of investment that prevented those 20 years from counting down. Their prediction was correct.

Prediction of fusion progress from 1978 based on funding, compared to the actual historic funding for fusion in the US. Even though this graph looks like it was drawn in Microsoft Paint, I have looked up the source material and this graph represents it accurately.

The reality is that the science and technology required for fusion power is ready to go, there are just a number of engineering and economic challenges that need to be overcome. Not engineering challenges like "how do you make a magnet that big" but more like "we need a gigantic building with complicated plumbing that won't expose its workers to radiation." We know how to build a giant magnet donut and fill it with plasma, it just needs to be made big enough to generate power, and big is expensive.

The Future

The next step in this direction is ITER (International Thermonuclear Experimental Reactor), a giant tokamak being built in France. Its main goal, from what I've read, is to generate ten times more power than it requires to operate (although this still won't make it viable as a power plant). It has gone massively overbudget and is now expected to run a tab of 20 gigaeuros. I think, moreso than any plasma science objectives, this will shed a lot of light on what is required to engineer a facility of this magnitude.

ITER, dream and reality.

At the fusion seminar I attended, a director of ITER was talking about his ideas for a commercially viable fusion reactor, which would have to be much bigger than ITER. One of the reasons tokamaks are so expensive is that they have to be really big. The fusion reactor would require about 500 megawatts just to maintain the fusion reaction: beyond that, power can be sold to the grid. However, if it's just selling an extra 50 megawatts after that 500, the electricity would have to be extremely expensive to cover the costs of the plant. It is estimated that a fusion plant would have to generate at least 2.5 gigawatts of electricity (slightly more than Hoover Dam) in order to sell the electricity at a cost comparable to current power sources. Thus, the minimum sensible infrastructure investment is that which is needed to make a plant that big. Not all of this goes into the generator itself, a lot of it goes into the building that houses it, as well as the plumbing necessary to extract tritium such that the reactor can keep making its own fuel.

The price tag that he quoted was 30 billion dollars, plus maintenance costs and morgtage payments over the next sixty years totalling over 100 billion, but it would produce enough power that that electricity could be sold at grid prices. A debate arose in the seminar about the economics of choosing the ideal initial size, about which costs scaled super-linearly with size and which scaled sub-linearly. It was pointed out that the first fission reactor was not sufficiently large to produce electricity that could be sold at a reasonable price, but the fact that it demonstrated that the technology was viable lead to investments in bigger nuclear plants. After the first viable fusion reactor is built, it won't be as difficult to build the next one.

The speaker claimed that what was required for this to actually exist was a rapid increase in fossil-fuel prices that would be driven by scarcity and increased energy usage in China and India. He cited $200/barrel as roughly the price at which a 30 billion dollar fusion plant would seem like a not-crazy investment. However, I've heard this story before; the rise in prices in the last decade made it viable to extract oil from fracking and from the Canadian tar sands and didn't give us a renewable energy revolution. Someone in the audience mentioned that General Electric was now developing its coal power technology, even dirtier than oil, to placate the rising energy demand in China.

The National Ignition Facility

I'm mainly discussed magnetic confinement fusion, but I'll also mention the National Ignition Facility (NIF) that was built for fusion research and then hastily re-purposed. The idea was to fire an extremely powerful laser at a small deuterium-tritium pellet such that it rapidly compressed and heated up until it was so hot and dense that fusion ignited. To this end, they built the world's most powerful laser array that was basically in inverted Death Star, with all the lasers focused on a little target at the center. They gradually ramped up the power, occasionally publishing papers about the behaviour of shock waves and the radiation emitted from these tests, and just as they were on the verge of getting powerful enough for fusion, that aspect of the project was halted and the facility was turned into a materials characterization facility and a way to test whether the fuel in nuclear bombs still works, without detonating the bombs themselves. I really wish they'd keep trying for fusion.

NIF from the outside and inside.

ITER and NIF aren't the only extant fusion projects, there is also a really cool looking one in Germany called the Wendelstein 7-x and the American Z-Machine as well as smaller facilities around the world, and I hope the future of fusion is brighter than what was laid out at that seminar.Alexhttp://email@example.com:blogger.com,1999:blog-1784861354684747574.post-83010864048324278822016-08-21T12:16:00.002-07:002016-08-21T12:32:40.482-07:00The duplication of BioMed Central.This post is about something strange that happened in 2014 and my attempts to deal with it.

Towards the end of my master's, I wasn't sure what to do next, so I picked up a part time consulting job, basically looking into public health statistics and compiling the trends and developing models to forecast them. I ended up doing a Ph.D. after my master's but I kept doing this job on the side. The first project I worked on involved the epidemiology of tuberculosis in Quebec, which is a fairly serious problem up North. It lead to two papers, one looking at the statistics and one forecasting them into the future. Working on these papers was a good learning experience but I would say that they are not as good as my physics papers, which I am more proud of.

The second paper was published in BioMed Central (BMC) Public Health. BMC is a massive journal network owned by a massive publishing company, Springer, and there are over 100 BMC journals, which I believe are all pay-to-publish open access. Pay-to-publish always seems kind of sketchy but there was definitely a peer-review process.

One day, I was compulsively checking my citations on Google Scholar and noticed that there was a new citation to my first TB paper, bringing the total up to two. I clicked to see who cited me, and found that the two citations were 1. my second TB paper on BMC Public Health, and 2. also my second TB paper, on the domain science-project.ru (don't bother going there, it's just an ad site now). My paper was randomly hosted on this Russian website with no explanation, so I decided to look into it. I found that the entire BMC database, consisting of hundreds of journals and tens of thousands of papers, was completely duplicated on this Russian server, with no explanation. I don't understand why someone would do this, and the BMC papers are free to read anyway. In addition to duplicating my paper, they also had links to rough pre-publication versions of the paper, including MS Word track-changes notes, that were still hosted on the BMC server.

I decided to contact the editor-in-chief of BMC, who was nominally in the UK but actually in Cyprus. I mentioned the serious issue that their whole website was duplicated on a Russian website, and the less serious issue that my rough drafts were still being linked to. The editor responded to me saying:

Please note that authors of articles published in BMC Public Health are the copyright holders of their articles and have granted to any third party, in advance and in perpetuity, the right to use, reproduce or disseminate the article, in its entirety or in part, in any format or medium, provided that no substantive errors are introduced in the process, proper attribution of authorship and correct citation details are given, and that the bibliographic details are not changed. If the article is reproduced or disseminated in part, this must be clearly and unequivocally indicated. Please see the BioMed Central copyright and license agreement for further details.

Furthermore, as stated in our 'About this journal' page (http://www.biomedcentral.com/<wbr></wbr>bmcpublichealth/about#<wbr></wbr>publication) the pre-publication history including all submitted versions, reviewers' reports and authors' responses are linked to from the published article. We are unable to remove the pre-publication history of the manuscript following its publication in the journal.

 In other words, she completely ignored the fact that I was trying to tell her about HER ENTIRE JOURNAL NETWORK being pirated, and instead decided to focus on a minor editorial policy. That was honestly a bit disturbing, how little she cared. I sent another email telling her that she ignored the important part of my email and asking if it bothered her that her entire journal database was copied on another website.

Meanwhile, I looked for a better way to get into contact with somebody in charge, because the editor of the journal didn't seem to care. I managed to find the secure whistleblower ombuds page for Springer, which is basically for people who want to report scientific fraud and related misconduct. I left a message explaining the situation, and my concern as to how casually the editor seemed to be taking it. A few days later I got two messages, an email from the editor apologizing for the misunderstand, and a secure message in the ombuds inbox from the legal team:

we have looked into this and the Legal Department will approach the contact that it given on the (fake) site. Please rest assured that Springer takes piracy very serious. In order to protect our authors´ rights and interests, Springer proactively screens websites for illegal download links of Springer eBooks and subsequently requires hosts of such download sites to remove and delete the files or links in question. This necessary action has become increasingly important with the growing number of eBooks within the Springer eBook collection.

Eventually, the fake version of BMC on science-project.ru disappeared and my citation count went back down to 1, and the issue appeared to be resolved. I am still left wondering, however, why someone would bother duplicating a massive network of free papers to their own server, and the whole incident with the editor left me a little wary of BMC.Alexhttp://firstname.lastname@example.com:blogger.com,1999:blog-1784861354684747574.post-21052170611968876032016-07-31T10:01:00.002-07:002016-10-17T17:52:48.336-07:00DNA Waves: New paper in Physical Review E.In June I submitted my latest paper to Physical Review E, and today it was published. I also uploaded a preprint to arxiv.org, and that free version can be found here. It is (probably) the last paper from my PhD work, which I wrote on-and-off in my spare time over the last year. Here, I'll briefly summarize what the paper is about.

A DNA molecule snaking its way through an array of cavities (which are separated by about a micron). The paper is about how "waves" appear to propagate along the DNA molecule.

As I've mentioned in some  other articles, my Ph.D. work was about the physics of DNA molecules trapped in cavities connected by a narrow slit. I was studying this both to better understand polymers in geometries using DNA as a model system, and to possibly develop genetic sequencing technology. The first paper I published on this was about diffusion through these cavities, and most of my Ph.D. I spent working on measuring the entropy loss involved in confining DNA. Towards the end I started working on a paper looking at how long it takes DNA to fluctuate from one cavity to another, which we described in terms of modes of a coupled harmonic oscillator system. That was published last summer.

From my two-pit fluctuations paper.

One day I did an experiment using much longer DNA than usual, and I noticed something cool: when you look at videos of the molecules, it looks almost like there are waves propagating back and forth along the molecule. I decided to investigate that, and that's what the paper was about.

Do you see the waves?

Looking at these movies you can sort of convince yourself that there are these transient waves, and a good way to look at how these things propagate over time is through a kymograph, which averages out one spatial dimension so you can see how the profile in the other dimension evolves over time. Doing this, you can see diagonal streaks of brightness, which are the propagating waves. You can see both positive waves, where excess DNA propagates between the pits, and negative waves, where paucity propagates, which are analogous to electron holes in a semiconductor. I also thought I could see evidence of waves reflecting off the end of the molecules, although that didn't make it into the final paper.

A wave of brightness propagating through a molecule.

The main way I analyzed the data was through correlation functions. A correlation function basically measures the probability that a deviation from the mean of one thing leads to a deviation from the mean in another thing. In my two-pit fluctuation paper, I was looking at the cross-correlation between the intensity in one pit and the other, and since DNA is just going from one pit to the other, if the intensity pit 1 deviates upward from the mean, it's very likely that pit 2 has deviated downward, so they are anti-correlated. At short time-scales this anti-correlation is strong, while at long time-scales it decays towards an uncorrelated noise floor.

This is from an actual presentation I gave at group meeting.

With two pits this is simple, but with a larger number it gets complicated. With three pits you have one-two, two-three, one-three, plus the autocorrelation functions one-one, two-two, and three-three. In general for N pits you have N(N-1)/2+N unique correlation functions, which provide a lot of information. My biggest molecule was in 15 pits, which would give 120 correlation functions. We decided to focus on correlations between neighboring pits: 1-2, 2-3, 3-4 etc. This would allow us to look at the process of DNA leaving one pit and going to the next one, then from that one to the next one, etc.

Basically what we observed was that any given point in time, neighboring pits were anti-correlated (as expected, because if there's more DNA in one pit it's less likely to be in another), and then that grew to a positive correlation at some later time (this is the propagation of the "wave": excess DNA in one pit at one time is more likely to be found in a neighboring pit at a short time later), and then a long-time decay (everything averages out due to random thermal motion).

Cross-correlation functions between different pit intensities. At zero time (A), they are anti-correlated because DNA in one pit is less likely to be in another. At a short time later (B), they are positively correlated, because the excess DNA at A is likely to have reached B at this time. At long times (C), random fluctuations bring correlation down to zero. This is the meat of the analysis.

This pattern was repeatably observable for all these large-N systems, which gave us a lot of data on how these waves propagate through confined DNA. We could also see something similar looking at next-nearest neighbors, and even next-next-nearest neighbors. The hard part was understanding all this data and what it was telling us about the underlying physics that lead to these waves. Presumably, such an explanation would allow us to predict what these correlation functions look like.

When the DNA molecule is at equilibrium, there is a certain length in each cavity (the ideal length balances its own self-repulsion and the entropy loss from the slits), and a certain tension in the strands linking each cavity. If, due to a thermal fluctuation, one cavity has an excess of DNA, the whole system gains some energy that is harmonic with respect to the excess length of DNA, and this excess is diminished as DNA is transferred to adjacent cavities through propagating changes in tension in the linking strands.

Because of the harmonic energy cost and my previous work mapping the two-pit system onto harmonic oscillators, the way I initially thought of the waves was in terms of a chain of harmonic oscillators, where a disturbance in one propagates down the chain as a phonon. It is a bit tricky to map this phenomenon onto a polymer in solution, because it is overdamped and effectively massless, so there is no momentum that is conserved. I spent a while trying to figure out the theoretical correlation function for an overdamped harmonic chain in thermal reservoir, and writing Monte Carlo simulations thereof, but that only got me so far.

One dimensional random hopping on an array. This model turned out to describe our system very well.

A simpler model turned out to work better: if you just imagine a bunch of Brownian random walkers on a one-dimensional lattice, each with some random probability of hopping in either direction at some rate, you can show that this system gives rise to collective motion which is exactly solvable and evolves in time in a way that looks like our observed correlation functions. In our system, we can treat the DNA as randomly fluctuating in either direction, but fluctuations are more likely in a direction that reduces excess DNA in a cavity, and that is essentially what the multi-hopping model is describing. A lot of the paper involves matching the predictions of the model to our observed correlation functions. It turned out to be different than I had initially envisioned it; I was originally thinking of it in terms of sound waves propagating through the molecule as tension perturbations.

I like this paper because it started out as an investigation of a neat phenomenon I happened to observe, and lead to something more systematic that we eventually understood in terms of some fairly fundamental statistical physics. I'm glad the reviewers liked it too!

Update: this post was modified after the paper was published.Alexhttp://email@example.com:blogger.com,1999:blog-1784861354684747574.post-69373513652533392272016-07-28T20:51:00.000-07:002016-07-28T20:51:19.187-07:00Do radioactive things glow?In a lot of depictions of radioactive materials, there is a green glow that is emitted. This is so common the green glow itself is associated with radioactivity. Does this green glow exist, and if so, where does it come from?

This animated plutonium rod should not be confused with the inanimate carbon rod, which does not glow.

Radioactive materials such as uranium and plutonium do not, by themselves, glow. Pure uranium looks like a boring grey metal, and plutonium is slightly shinier. The glow associated with radioactivity originates from materials containing radioactive isotopes, but is due to electronic rather than nuclear transitions, similar to how certain materials glow under a blacklight. These materials can glow without an external power supply, because the atomic transitions can be excited by the radioactive decay. This is known as radioluminescence.

Radioluminescent materials were more common in the early 20th century, before it was understood how incredibly bad for you extended radiation exposure is. Clock faces often had dials painted with radium, so that they could be read in the dark. The luminescence was actually produced by zinc sulphide, which was activated to an excited state by the radium decay. Radioluminescent watches can still be purchased, but they use the safer tritium as an isotope.

Another common product was uranium glass, which was marketed as vaseline glass, apparently because it was the same colour as petroleum jelly was at the time. Once again, it is not the uranium producing the light, but the transitions excited by its decay.

Uranium glass glowing green.. The blacklight in the background is the main source of illumination.

Perhaps the most common radioactive green movie trope is plutonium, but plutonium is not used in radioluminescent compounds. It can appear to glow red, but that is due to a chemical reaction with oxygen (aka burning) rather than anything associated with radioactivity.

All of these heavy-element radioluminescent materials rely on alpha-decaying isotopes. Alpha particles tend to move slowly (compared to light) and are comparatively safe; alpha radiation can essentially be blocked by clothes, but if you inhale a piece of alpha-emitting dust you can be in serious trouble. The main scenario where radioactivity does produce a glow is not from alpha decay but beta decay*, typically in nuclear reactors. Nuclear reactors are often kept underwater, for cooling and for neutron shielding, and when emitted beta particles (electrons) exceed the speed of visible light in water, they emit Cerenkov radiation, which manifests itself as a blue glow (This is analogous to a sonic boom when something exceeds the speed of sound).

Cerenkov radiation from a nuclear reactor.

So just to summarize, radioactive materials do not emit light just because they are radioactive, but essentially act as a power source to make other things glow. Except nuclear reactors, those actually glow.

*Electrons are 1/7000th the mass of alpha particles, making them much faster at the same energy.Alexhttp://email@example.com:blogger.com,1999:blog-1784861354684747574.post-17275386063255135262016-07-18T07:44:00.005-07:002016-07-18T07:44:55.308-07:00"What's the application?": Genetic sequencing and polymer physicsLong time no post. I wrote an article on PhysicsForums trying to explain a few of the applications of my field of research, and how they relate to some of the physics problems. Check it out!

https://www.physicsforums.com/insights/whats-application-polymer-physics-genetic-sequencing/Alexhttp://firstname.lastname@example.com:blogger.com,1999:blog-1784861354684747574.post-42319456990771057582016-06-01T19:44:00.000-07:002016-06-01T19:44:46.688-07:00The Fall of the Twin Prime GapThe twin prime conjecture states that there are an infinite number of primes separated by a gap of two, e.g. 17 and 19 or 29 and 31. In May 2013, a relatively unknown mathematician named Yitang Zhang proved that there are infinitely primes separated by a gap of some number smaller than 70,000,000. This number is much bigger than the postulated two, but much smaller than infinity, which is where the number theory community was stuck at before. This finding heralded in a rush of work attempting to lower this bound, causing it to drop rapidly over the next several months, finally coming to a rest at 246 less than a year later, where it has sat. This post is about cataloguing and quantifying this drop.

Timeline of the twin prime gap, on log-linear and log-log axes.

Most of this work was carried out by the Polymath Project, which is a group of mathematicians that collaborate on unsolved math problems. The results have been catalogued here, which is where I got the data for those graphs. The Polymath Project can be really interesting to read through, because you can see chunks of the problem getting solved in real time on an online discussion forum, like reading the world's most high-brow youtube comments. The retrospective on the project  is a nice read even for non-mathematicians such as myself. Midway through this project, an independent mathematician (working in Montreal!) named James Maynard developed another method to shrink this gap, which was eventually worked into the project.

I will not say too much about the techniques that were used to prove these bounds; there are people who are more qualified to do so who have already done so, such as Terry Tao, who was part of the project. From what I gather, the method involving generating a large list of numbers and then "sieving" them until the solution converges*. Two assumptions can be made to make the bound easier to shrink: the Elliott-Halberstam conjecture, and Deligne's Theorem. While their validity in this situation is unproven, assuming the Elliott-Halberstam can reduce the bound to 6. I have tried to take only data that does not rely on these conjectures (the proven bound, if you will), and have ignored data marked as conditional in the table.

Ok, so let's look at the falling bound. There are clearly three phases: the "let's get the ball rolling" phase, the wild-mathematical-rollercoaster phase, and the "honeymoon is over" phase. The first two weeks were just people starting to realize what could be done (Zhang kind of caught the mathematical community by surprise). In fact, the first post-Zhang paper was called "A poor man's improvement on Zhang's result" and subsequently there was a blog post called "I just can't resist: there are infinitely many pairs of primes at most 59470640 apart." Then Terry Tao proposed the Polymath Project, and that's when it really started dropping. If you look through their website, the progress involves people posting tables of their iterated sieving data and discussing it; sometimes it works and sometimes it doesn't.

In my line of work, data is described by two types of functions: power laws and exponential curves, both of which look like straight lines if you have the right number of logarithmic axes (or the power is 1 or the x-variable covers a small range). Exponentials crop up more when something is varying with time (e.g. the length of the DNA molecule I'm looking at), while power laws come up when looking at how one quantity varies with another (e.g. how stretched a molecule is vs how confined it is). A big part of my thesis was demonstrating how we shouldn't just blindly look at power law fits when analyzing data, and that is exactly what I'm going to do now.

 When I fit the steep part of the bound dropping curve to a power law, the exponent is -9.6 with respect to time. That's pretty crazy; it falls 1000 times as much in a fortnight compared to a week. If instead I fit it to exponential decay, I find that the characteristic time for the gap to fall by a factor of e is about three days, although this doesn't fit the data as well as the power law. I have also ignored the slowdown that occurred around day 30 (although, within a month of Zhang's news, they had already dropped by bound by a factor of 300).

The results sort of petered out in the thousands, and then they were re-invigorated by Maynard's discoveries, which is what got them down to 246. If I do the same type of analysis for the slow phase, there is an inverse-square decay of the gap with respect to time, or a characteristic exponential decay of about 72 days. Can this teach us anything about the nature of collaborative number theory? Probably not.

The gap data with unjustified exponential (left) and power law (right) fits to the fast and slow portions of the drop.

I was watching this gap fall as it was happening, and I was curious as to why it stopped at the seemingly arbitrary 246. A googling came up with an excellent reddit comment explaining it from a user named BoundingBadger. It was pretty neat seeing an unsolved problem get hammered down by professional and sometimes famous mathematicians in real time and openly discussing their results, and I'm glad I caught that neat little bit of history. I was curious as to how the bound would end up behaving over time, and now I know!

*That is a really bad explanation but it's the limit of my understanding right now.Alexhttp://firstname.lastname@example.com:blogger.com,1999:blog-1784861354684747574.post-31801209925202050662016-05-24T09:08:00.004-07:002016-05-24T11:33:52.496-07:00Perturbative Champions: Cohen and Hansen take it to next-to-next-to-next-to-next-to-next-to-next-to-next-to-next-to leading order.In 1999, in a preprint on arXiv.org, Thomas D. Cohen and James M. Hansen, physicists at the University of Maryland, claimed the following:

If one insists on an accuracy of ∼ 20%, one estimates contributions at their nominal order and Λ is taken to be 300 MeV, then one has to work to order (Q/Λ)$^7$ , this corresponds to next-to-next-to-next-to-next-to-next-to-next-to-next-to-next-to leading order.
This octet is, to my knowledge, the largest string of next-to's ever to appear in scientific literature. As far as I can tell, the runner up is a recent paper by Eltern et al. in Physical Review C, with a paltry five next-to's, although this may be the champion of peer-reviewed literature. Below this, four next-to's is fairly common; it has its own notation, N4LO. I found a reference to N6LO in the literature, but it becomes hard to google these.

What does this actually mean, and why does it sound so silly? A powerful tool in physics is perturbation theory, where we approximate the solution to a problem with a power series, and then compute first the simplest terms, then terms of increasing complexity until we have a solution that is close enough to the exact solution that it's useful to us. There will be the zeroth-order solution that tells us the order of magnitude, the first-order solution with its basic dependences on system parameters, then second and higher order solutions for non-linear effects. In many cases, certain powers will be zero, so the "leading order" term might be the first non-zero term besides zeroth-order, although not necessarily first order. For example, the leading order term of something that is perturbed about equilibrium is second-order (which is one of the reasons why treating things as harmonic oscillators is so useful). Feynman diagrams are essentially a way to express perturbation theory in a graphic form, first drawing interactions with no loops, then one loop, then two loops, etc.

Anyway, it may have been the case that all of the even and odd powers of the perturbation series in Cohen and Hansen's paper were zero, they couldn't just say "eighth order," but thought they had to expand all those next-to's. Each term makes the approximation more accurate, and for their 20% desired accuracy, they needed to compute all these Feynman diagrams really far past leading order, but unfortunately, as they claim, "it is implausible that such calculations will ever prove tractable."

Another, similar situation involves describing sites on a lattice. For example, if you're sitting at a site on a square lattice (with lattice constant x), you have four "nearest neighbors" at distance x from you, four "next-nearest neighbors" across the diagonals at distance $\sqrt{2}$x, "next-next-nearest neighbors at distance 2x, etc. This paper uses the term "next next next next nearest neighbors" which is designates 4NN.

There was perhaps a better way to visualize this.

These record nextings are fairly insignificant but perhaps mildly interesting.Alexhttp://email@example.com:blogger.com,1999:blog-1784861354684747574.post-77332510796038864602016-05-17T11:41:00.000-07:002016-05-17T11:41:08.388-07:00Measuring Newton's Constant with a Space-Borne Gravity TrainI recently came across a paper which was published in Classical and Quantum Gravity, a respected journal, after it had initially appeared on arXiv. It proposes a space mission consisting of a metal sphere with a cylindrical hole, floating through space as a smaller reflective object oscillates back and forth along the hole, pulled by the gravitational field of the sphere. The position of the the smaller object can be monitored by another space probe, and the period of oscillation can be used to measure Newton's gravitational constant, big G. I like this idea, it draws upon some of my recent work and I place it firmly in the "just crazy enough to work" paper category, which are my favourite papers to read.

Diagram of the proposed experiment, from the arXiv version of the paper.

The proposal is motivated by some recent analysis (by some of the current proposal's authors) of independent measurements of the gravitational constant, which showed that even though they are measuring the constant with smaller and smaller uncertainty, the different measurements are not in precise agreement of each other, sometimes deviating by 40 times the standard error on the measurements. The analysis makes the bolder claim that the difference between the measurements has the same periodicity as Earth's length-of-day variations, which are due to large-scale seismic effects. They conclude that there are systematic effects caused by the fact that all these experiments (which typically involve monitoring a rotating pendulum near large masses) take place on the Earth, and desire a way to measure this constant away from the Earth. The National Science Foundation has put out a call for proposals for more accurately measuring big G; I recommend reading its three paragraphs if you're wondering why anyone would bother caring about this.

The various non-agreeing measurements of G over the years, from Anderson et al. Pay more attention to the red than the black. The black is the "bold proposal" I mentioned.

To make this measurement, the authors, lead by independent researcher Michael Feldman, suggest sending a miniature gravity train into space. A gravity train, something I have written about in great detail, consists of a spherical mass (often taken to be the Earth, but not here) with a tunnel through it (often through its center), with a smaller object falling through the hole. It builds up speed due to gravitational attraction to the sphere, passes the halfway point, and then starts decelerating, coming to a rest on the other side. Inside the Earth, it would take 38 minutes to fall through this tunnel. Feldman and friends propose a small metal sphere, roughly 10 cm in diameter and 1.3 kg in mass, that would take about two hours for the small object to fall through.

How could G be measured from such a device? For a uniform density sphere, it can be shown that the period of a gravity train is $T=2\pi\sqrt{\frac{R^3}{MG}}$. If the mass and radius are known, and the time is measured, G can be extracted. In the proposed setup, the position of the small object will be monitored by a laser aimed at the tunnel from another nearby space probe, and from these periodic measurements of position, the time can be measured, and sent back to Earth by an antenna on the probe. The paper consists of more detailed derivations of the G-T measurement, unique to the proposed design (which consists of two layers of different materials).

Zoomed out schematic of the probe and sphere, from the arXiv version of the paper.

The authors are concerned with the precision of such a device, and which systematic errors contribute to the overall uncertainty in G. These include the metallurgy of the sphere and hole (uncertainties in R, M, and the uniformity thereof), the initial placement of the small reflecting object in the hole (which must be extremely gentle), the ideal place to position sphere with respect to the host probe so that the probe is close enough to block the sun but not so close that its own gravity affects the experiment, the radiation pressure from the probe laser on the device, deformations of the sphere due to the tidal influence of the sun, possible charging of the tunnel's interior due to cosmic rays, and more. They even calculate the change in the period if general relativity is taken into account, which is something I was curious about for my gravity tunnel research, but didn't have the tools to solve. The hypothetical uncertainty analysis was probably the most fun part of the paper to read.

They estimate that, given advances in metallurgy and aerospace deftness, they can get the precision of the G measurement down to an optimistic 63 parts per billion. The current record for Earth-based G measurements is 13,000 parts per billion. This would be a huge improvement if it actually worked, and would eliminate some of the systematic issues with measuring the strength of gravity in Earth's gravitational field.

The question, of course, is whether this thing will actually exist, and whether the budgetary will exists to make it so. The authors suggest that the experiment would not be the main mission payload of a space launch, but rather would piggyback on a larger, more important probe headed out of the solar system.

I was interested in this paper because I like crazy yet scientifically rigorous ideas, and it draws upon a system that is close to my scientific heart [disclaimer: the paper cites mine]. It was a pleasure to read about all the potential effects that could skew the time measurement, and how they planned to deal with them. I hope the available metallurgy, metrology, and money becomes sufficient to launch this thing into space.

Alexhttp://email@example.com:blogger.com,1999:blog-1784861354684747574.post-3356196740368845662016-05-16T09:11:00.000-07:002016-05-16T09:11:41.566-07:00PhysicsForums: The Interaction of Sound and LightMy last post about sound propagating through light inspired me to write about the more conventional interactions between sound and light. Read about it on physicsforums.com.

https://www.physicsforums.com/insights/interaction-sound-light/Alexhttp://email@example.com:blogger.com,1999:blog-1784861354684747574.post-27276201260172282312016-05-09T19:33:00.003-07:002016-05-09T19:33:56.820-07:00The Speed of Sound in LightThis post will discuss something that I think is interesting, and wanted to think more about: the propagation of sound through light. One can show that the speed of sound in light is 57% the speed of light. I will talk about what the hell that actually means, and where that number comes from.

Sound in light?

Sound in light? Huh?

The first thing one must wrap their head around is: what does it mean for sound to propagate through light? I devised a simple thought experiment to help me understand that.
Imagine two mirrors with springs on their backs, facing each other with light propagating between them. You can either imagine this as photons bouncing back and forth, or a standing electromagnetic wave. The mirrors are in mechanical equilibrium: the radiation pressure from the light reflecting off them is balanced by compression of the springs holding the mirrors in place.

Two mirrors with a standing electromagnetic wave between them. Springs balance the radiation pressure, but if one mirror is moved, the induced Doppler shift will increase the radiation pressure on the opposite mirror, causing that spring to compress. This is my first foray into hand-drawn diagrams...sorry if it's terrible.

Now imagine you boop one of the mirrors, pushing it towards the other one. As light reflects off this mirror, it will be blue-shifted from the perspective of the other mirror. This blue-shifting will increase the magnitude of the radiation pressure, pushing the mirror back and compressing the spring a bit more. In a sense, pushing the first mirror caused an acoustic signal to propagate through the light to the other. The mirrors are both now in simple harmonic motion, coupled by the transfer of radiation overpressure and underpressure through blueshifting and redshifting as the mirrors move.

This scenario can be extended further, to an array of two-sided mirrors in an optical cavity, springs only on the mirrors on the end. Booping one end mirror would cause this blueshift-induced radiation overpressure to propagate down the line of mirrors. As the last mirror compresses the spring and rebounds, the wave will reverse direction.

The way I am picturing this scenario, the signal propagates through the light at the speed of light: as soon as you blueshift the first photon, it's already heading towards the other mirror at the speed of light.

The speed of sound in a photon gas

The scenario above envisioned light in mechanical equilibrium with mirrors. But what if it is also in thermal equilibrium? Now in this thought experiment, we have a box with mirrors on the inside, that is really hot (students of physics may recognize this as the canonical example of a blackbody). The mirrors are so hot that they start emitting blackbody radiation, which is reflected (or absorbed and re-emitted, depending on how you think about it) by the other mirrors in the box. Now what happens if you push on one side of the box?

Left. A really crappy depiction I made of a photon gas: a mirrored box with really hot walls, with radiation bouncing around in the box. Right. The best online depiction I found on google images, from photonics.com.

The way I am picturing the situation, basically the same thing happens but not deterministically: the blackbody radiation from the mirror you push on gets blueshifted, increasing the radiation pressure felt on the opposite side. However, the radiation is emitted isotropically from the surface, and some of it goes off on an angle and reaches the side walls, and the radiation pressure excess from the boop gradually propagates towards the other side. So instead of just one c-retarded propagation from side to the other, there is a distribution in the arrival time of the mechanical information. I suppose the same thing happens when we push on one side of a box full of gas, and because there are so many gas molecules, the sound propagation is effectively deterministic.

At this point I have exhausted my ability to picture what would happen, and will examine a more rigorous argument. The speed of sound in a photon gas can be derived from its thermodynamic equation of state and associated equipartition theorem, which is basically similar to the ideal gas law, except it is derived by assuming that all the particles are going so fast that they are effectively massless (in that the lion's share of their energy is kinetic). The main difference between the regular and relativistic results is that for slow particles like atoms in a gas, the equipartition tells us that the average kinetic energy is 3/2 kT per atom, whereas in the relativistic case is is 3kT. This is basically because you're writing down the energy as pc instead of 1/2mv$^{2}$=$\frac{p^2}{2m}$, so there's no factor of two. There are some more involved derivations to get the number of photons and calculate the pressure based on the temperature, which essentially involve integrating the Planck spectrum over all frequencies (similar to the derivation of the Stefan-Boltzmann law). This is not necessary here.

To find the speed of sound in a gas we want the ratio of pressure to density:

$v^{2}=\frac{P}{\rho}$.

And from the photon gas equation of state we have a relationship between pressure and internal energy.

U=3PV

With a bit of relativity we can relate the regular density to the energy density by dividing it by $c^2$:

$\rho=\frac{1}{c^2}\frac{U}{V}=\frac{3P}{c^2}$

Plugging this back into the equation or the velocity, we have:

$v^{2}=P\frac{c^2}{3P}=\frac{c^2}{3}$.

Thus, we find that the speed of sound through a photon gas is one-over-root-three, or 57%, the speed of light. Notice that the factor of three comes from the number of dimensions used in the equipartition theorem. In one dimension, which my initial mirror scenario took place in, the speed of sound in light is the same as the speed of light itself.

Photon-Photon Interactions

In the above discussions, the "sound waves" are essentially just signals of increased radiation pressure that are transmitted through Doppler shifts. However, at high-enough energy densities, you can start to have the light interacting with itself, via electron-positron pairs that are produced from the vacuum. This self-interaction of light causes a slight slowdown in the speed. It is derived from quantum electrodynamics by Partovi here, and he derives a correction to the speed that depends on the fourth power of the ratio of the temperature of the blackbody to the temperature-equivalent of the electron mass, 5.9 gigakelvin (511 keV divided by Boltzmann's constant). The correction to the velocity is of order 10$^-5$ even at gigakelvin temperatures:

This equation copied from Partovi's paper.

With this in mind, I can picture poking one side of a hot mirror box and expecting a small delay before the other side of the hot mirror box gets jolted.

The Relativistic Sound Speed in Astrophysics

This idea is neat, but is it important? This photon gas sound speed comes up in a few other places. The main one is baryon acoustic oscillations, sound waves through the very early universe that dictated the large-scale structure of the cosmos as the universe cooled down, still present in the periodic distribution of galaxy clusters. The very early universe was very hot, so there was a high energy photon gas just from thermal equilibrium, and all the massive particles zipzapping around were also highly relativistic. Thus, the speed associated with these baryon acoustic oscillations is that relativistic 0.57c.

A telescope attempting to measure Baryon Acoustic Oscillations, from BNL.gov

Another question that arises is whether this is the absolute maximum for sound speed propagation. As I demonstrated in my thought experiment, one-dimensional coherent systems can exceed it, but what about in thermodynamic systems? A paper in Physical Review Letters last year argues that certain models of neutron star interiors may predict sound speeds that exceed this limit. There is a lot that is unknown about neutron star interiors, and many models put constraints on the maximum and minimum sizes of neutron stars. These models then get selectively ruled out by newer neutron star discoveries. Bedaque and Steiner argue that given the available models and certain observations of particularly small neutron stars, the speed of sound inside some neutron stars must exceed 0.57c. Unfortunately I don't really understand enough of their analysis to say anything for insightful. Alternately, if this 0.57c is the true upper limit, it will rule out a lot of these models.

To summarize, I think that the idea of a speed of sound in light is a cool idea (this is how most of my blog posts end). I make it make sense to me using a thought experiment involving radiation pressure and Doppler shifting. For a more complete understanding, we go to photon gas thermodynamics and QED. There is one point where my research experience touches upon this concept: my undergraduate thesis in general relativity, the first independent physics research I conducted, involved finding fluid mass distributions that solve Einstein's equations. These mass distributions then had to satisfy certain conditions to make them physically realistic: mass couldn't be negative, density had to always decrease with radius, I wanted to avoid singularities, etc, but I also required that the speed of sound not exceed the speed of light. This extended the previous work of my supervisor Kayll Lake, who showed that many existing solutions to Einstein's equations violated these conditions. I wonder if anything would change if we demanded that v<0.57 c instead of v<c.

Alexhttp://firstname.lastname@example.com:blogger.com,1999:blog-1784861354684747574.post-74229114303832507012016-04-30T08:39:00.002-07:002016-05-01T06:50:16.014-07:00Strange moments in science: The theft of the EBEX telescopeIn early 2013, a balloon-born telescope, EBEX, was launched from Antarctica and circled the South Pole over the course of two weeks, attempting to observe the Cosmic Microwave Background and its polarization. A number of things went wrong with this mission, but it almost didn't happen because the truck containing the telescope was stolen from a motel in Texas.

EBEX before and during launch. Images from the Ebex in Flight blog via space.com and NewScientist.

The EBEX (E and B EXperiment) telescope was assembled in Minnesota with input from an international team of collaborators. One group was at McGill where I was doing my PhD, and I played on a hockey team with a few people involved in the mission, including my friend Kevin who was the one of the main McGill guys on the collaboration. The telescope had a large number of bolometers (measuring incident radiation through a change in temperature) which, when floated above the lower atmosphere with a balloon would attempt to measure the polarization of the cosmic microwave background and look for B modes as a signature either of cosmic dust or primordial gravitational waves, something that was later somewhat controversially discovered by the BICEP2 telescope, also in Antarctica.

In May 2012, it was on its way from Minnesota to NASA's balloon centre in Palestine, Texas, being transported in the back of a truck. However, the truck failed to arrive, so the trucking company looked into it. According to its GPS, it was last seen at a truck stop Hutchins, Texas, and the trucking company attempted to contact the driver, but could not reach him. Several days later, he was found sleeping in the cab of his truck, but the trailer and the 8 million dollar cargo were still missing. The driver claimed the truck was stolen from the motel he was staying at. Eventually the trailer was found at a carwash in Dallas, with the telescope still inside and undamaged.

Artist's rendition.

According to a 2012 article, the driver had been fired, and the head of the trucking company commented "Why didn't he decide to do this on a load of potato chips?"

After it was recovered, the telescope was configured to the balloon and shipped to Antarctica, where it was launched. A number of things went wrong with EBEX: one of the orientation controllers overheated in the sun, causing the telescope to lose control of where it was pointing (I recall Kevin calling it "a dance party in space"), meaning it only imaged a donut-shaped swath of sky. It eventually landed out on the ice, and without sufficient time and resources to launch an immediate recovery the scientists had to wait until the next season to go access it. Not much has been heard from the collaboration since, although there may be plans to build and launch a new and improved version, hopefully one that will maintain orientation. To read more about EBEX and its launch and recovery (including the truck disappearance), I recommend this blog.

The flight of EBEX, from the EBEX in Flight blog.

As an experimental scientist I'm accustomed to things going wrong, but I can't imagine showing up to work one day to find my microscope missing.

Alexhttp://email@example.com:blogger.com,1999:blog-1784861354684747574.post-85878707825990290532016-04-20T17:30:00.002-07:002016-04-25T17:29:07.109-07:00Visit to the Roger Babson Anti-Gravity monument at TuftsIn the early 20th century lived an eccentric businessman named Roger Babson. He was a sort of Donald Trump of his time, writing books, founding a college, and running for president. Instead of hating Muslims and Mexicans, what Babson really hated was gravity, and declared it to be mankind's foremost enemy.

He founded an organization devoted to defeating gravity's grip on humanity, part of which involved organizing the yearly Gravity Research Foundation essay contest, which has been won by the likes of Stephen Hawking and entered by the likes of me. He left a series of monuments around New England dedicated to his crusade. The closest one to me is at Tufts (yes, Tufts), and I visited it this evening. There used to be a yearly tradition of digging up the monument and carrying it to see if gravity had been defeated yet, but that was discontinued.

I plan to write a longer article about this history of his contest and his foundation, and for now you can find other articles about it online. I wrote an article about the history of the contest on PhysicsForums!

The text of the monument reads:

"This monument has been erected by the Gravity Research Foundation, Roger W. Babson founder. It is to remind students of the blessings forthcoming when a semi-insulator is discovered in order to harness gravity as a free power and reduce airplane accidents. 1961"

Alexhttp://email@example.com:blogger.com,1999:blog-1784861354684747574.post-72565825768110107652016-04-16T17:29:00.001-07:002016-04-16T17:56:00.977-07:00Tiglath-Pileser's Hunting OmissionThis post is on a different topic than usual post-doc ergo propter hoc content.

In the discussions or conclusions of academic papers, you will often find a description of work that the author had done but chose not to include in the paper. The reason is often because it would make the paper too long without adding much interest, or is a bit too tangential to keep the paper on topic. These are slightly different than the "future work will involve..." statements that often come at the end of papers. Including these references to omitted work can inform the reader that due-diligence was performed by the investigator, and also makes the author sound smarter and more impressive.

I was reading an account of the conquests of Tiglath-Pileser I, king of Assyria around 1100 BC. These accounts were inscribed on clay tablets in cuneiform script in the Akkadian language, and were found in the ruins of Assur, in what is now Iraq.

This is actually the equally ferocious Tiglath-Pileser III, who has a better picture than Tiglath-Pileser I.

Assyria was a dominant military power in the Middle East in the late Bronze Age and early Iron Age, before being supplanted by the Babylonians. They are known for their use of chariots in battle, their ferocious descriptions of their conquests, and the scattering of the ten "lost tribes" of Israel after conquering the Northern Kingdom. Tiglath-Pileser I ruled at the end of the Bronze Age, before a period of tumult involving the invasion of the Sea Peoples and the collapse of the Mediterranean Bronze age. Tiglath-Pileser I went on a rampage Eastwards, conquering literally dozens of kingdoms in several pitched battles, looting the region and defiling the temples of the defeated. He writes, in typical Assyrian fashion:

The city of Khunutsa, their stronghold, I overthrew like a heap of stubble. With their mighty troops in the city and on the hills I fought fiercely. I defeated them; their fighting men in the middle of the forests I scattered like chaff. I cut off their heads as if they were carrion; their carcasses filled the valleys and (covered) the heights of the mountains. I captured this city; their gods, their wealth, and their valuables I carried off, and burnt with fire. Three of their great castles, which were built of brick, and the entire city I destroyed and overthrew, and converted into heaps and mounds, and upon the site I laid down large stones; and I made tablets of copper, and I wrote on them an account of the countries which I had taken by the help of my Lord Ashur, and about the taking of this city, and the building of its castle; and upon it I built a house of brick, and I set up within it these copper tablets.

There are many such descriptions. However, after he finished his descriptions of all the armies he defeated and cities he destroyed, he left the following note:

I have omitted many hunting expeditions which were not connected with my warlike achievements. In pursuing after the game I traversed the easy tracts in my chariots, and the difficult tracts on foot. I demolished the wild animals throughout my territories.
In mentioning these omitted hunting achievements, he saves the reader the time that would have gone into taking in all this out-of-scope extraneous information, but uses the stated omission to bolster his credibility as a martial conqueror. I believe that this is one of the first published omissions to other relevant work, setting the stage for scholarly literature for the next three millennia.

This may not be the first reference to a deliberate omission; I haven't thoroughly scoured the Egyptian and Akkadian records.

Alexhttp://firstname.lastname@example.com:blogger.com,1999:blog-1784861354684747574.post-59151371682100872872016-04-10T09:55:00.001-07:002016-04-10T15:10:19.840-07:00Physics Shower Thoughts Part II: Cosmic Radiation Pressure and Relativistic SpacecraftThis is the second instalment of "Physics Shower Thoughts," an exploration of an idea that I found interesting that ultimately may not have monumental consequences in the grand scheme of our understanding of the universe. In this case I was inspired by a question on reddit that was answered by another physics blogger, VeryLittle of Quarks and Coffee. The question was about the limits of acceleration given special relativity, and my shower thought was about the role of blue-shifted radiation pressure in limiting that acceleration.

The setup is thus: an object (let's call it a spaceship) under the influence of a constant force in its reference frame will accelerate. The acceleration will seem uniform but as it gets very fast relative to its initial rest frame, its velocity will plateau asymptotically towards light speed. In our universe, however, there is a cosmic microwave background that will be blueshifted in front of us as we move in one direction, and redshifted behind us. The radiation pressure from this background will become stronger and counter the force that our spaceship is producing. This may limit the ability of our spaceship to accelerate. So, to what extent does this matter?

It's a good picture, ok.

To figure this out, we need to work out two things: the blue-shifted blackbody radiation pressure, and the relativistic acceleration under special relativity. The shifting of the blackbody spectrum can be considered with varying degrees of complexity, including a Lorentz transformation of the entire spectrum and a mixing of polarizations. However, I am interested in a simple solution and an order-of-magnitude estimate, so I will be using a simpler method: I will consider the Stefan-Boltzmann law given a relativistic Doppler shift of the peak frequency of the blackbody distribution.

Incident electromagnetic power can be converted to a radiation force by dividing by the speed of light. The Stefan-Boltzmann law gives us areal power density, so we can multiply by the cross-sectional area as well to from power density to pressure to force:

$F_{r}=\frac{A}{c}\sigma T^4$

The Stefan-Boltzmann constant $\sigma$ is a product of several powers of Boltzmann and Planck's constants as well as the speed of light, but its value in SI units is easy to remember: 5-6-7-8, or 5.67x10$^-8$ Watts per meter squared per kelvin fourthed.

According to Wien's displacement law, the temperature is proportional to the peak frequency of the distribution, which I'll call f. In our universe at this time, f is about 160 GHz. The proportionality constant is based on numerically minimizing the Planck spectrum, and in terms of frequency Wien's law can be simply expressed:

$T=\alpha f$

The proportionality constant for frequency is actually very close to two times the constants of Pythagoras* and Boltzmann divided by that of Planck, or in SI units about 5.9x10$^{10}$ Hertz per kelvin.

Moving toward a source, the frequency experiences a Doppler shift and is transformed into f', and if the motion is fast enough we should take into account the full relativistic Doppler shift:

$f'(v)=f\sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}$

The radiation hitting us from behind is redshifted, and I will deem it small enough to ignore. Plugging this all together, the transformed cosmic radiative force on our spaceship is:

Now we consider the relativistic acceleration of this spacecraft. The way to incorporate special relativity into Newton's law of motion is to remember that the change in momentum is the product of force and time. Momentum is the product of mass, velocity, and the Lorentz factor. If we just have a constant force then we can find the acceleration as a function of time:

If you plot this vs time for some values of F and m you will find that it increases and asymptotically approaches light speed. Somewhat coincidentally, if you set F/m to be Earth's gravity, it will take about a year to get close to light speed.

Accelerating under Earth-gravity-equivalent for three years. To convert seconds to years, remember that pi seconds is a nanocentury.

If we then incorporate our radiation pressure into the force side of the equation, it gets a bit more complicated.

This does not have a closed form expression for the velocity, but if I solve it numerically I find.... that it increases and asymptotically approaches light speed. For the values I use, it is essentially indistinguishable from the radiation-free solution. However, if I calculate values for long enough times with high enough numerical precision, the radiation solution does converge on a sliiiiiightly subluminal value (99 point a bunch of 9's percent light speed). So, what's going on?

This is going on.

Basically when the spaceship is going fast enough, the radiation force becomes strong enough to counter the accelerating force, and the net force is zero. This velocity is easier to calculate, we just solve the transformed radiation pressure equal to constant F, and solve for v. To simplify notation, I have introduced the variable pressure P=F/A:

Just the general form of this expression is $(1+x-2\sqrt{x})/(1-x)$, which is monotone decreasing, passing zero when x=1. In our scenario, that corresponds to $f^{4}/P=\alpha^{4}c/\sigma$, which I guess is the frequency and pressure required to have zero acceleration from the rest frame.

So, let's plug in some realistic values for this critical velocity and see what happens. Using the space shuttle as a framework for a spaceship, it has a mass of two million kg, a thrust of 12.5 meganewtons, a cross-sectional area of roughly 200 square meters. This puts its maximum velocity at 99.99999992% the speed of light, given constant thrust.

Now there are many other factors I didn't take into account. I could use a more rigorous transformation of the blackbody radiation. The length-contraction of the spaceship might change the incidence of the radiation, compounding the effect (it's kind of funny that shape becomes important again, not for aerodynamics but for electrodynamics). The mass could decreases as fuel is ejected from the rocket (maintain fuel supply for this journey could perhaps be attained with a Bussard ramjet, but that's beyond the scope of this article), which increases the acceleration. And it would take so long to accelerate to these high speeds that the cosmic background radiation could become more significantly redshifted due to the intervening cosmic expansion.

As VeryLittle pointed out in his reddit post, such a cosmic speed limit already exists for protons: the GZK limit is the speed at which cosmic ray protons see the CMB blueshifted to the point that it interacts with the now-gamma rays. This corresponds to an energy of 8 joules, but the speed is close enough to light that the number of 9's in the percentage doesn't really matter (whereas for the space shuttle it's only 7 decimal 9's). I guess protons are more electroaerodynamic than spaceships.

So, to summarize, the blueshifting of the cosmic microwave background may have an extremely tiny effect on the limiting speed of relativistic spacecraft. However, if I were a relativistic spacecraft engineering, I wouldn't worry about it.

*also known as the square root of two.Alexhttp://email@example.com:blogger.com,1999:blog-1784861354684747574.post-51213969802361737862016-03-31T06:42:00.001-07:002016-07-20T12:49:23.679-07:00Where might Planet Nine be hiding?On January 20, 2016, the inferred presence of a large planet far beyond Neptune was announced. The finding was a joint collaboration between Mike Brown, who is known for discovering the Kuiper belt objects that helped nix Pluto's planet status, and Konstantin Batgyin who is an expert in orbital simulations. The finding was based on alignments in the orbits of several trans-Neptunian objects, whose alignment can be explained by the presence of a distant planet about ten times the mass of Earth and 30 times the distance of Neptune. It can also be the result of rare events occurring due to random chance. In March, Brown and Batygin released another paper (initially on ArXiv) trying to figure out where this planet can be found, if it does indeed exist. The paper has since been published  in Astrophysical Journal Letters, a respected journal for short communications about astronomy and astrophysics.

As in many aspects of physics research, even in the absence of positive data one can learn things by asking the right questions about negative results. In this case the question is "Given that we have not yet detected this planet, where could it be?" rather than repeatedly asking "Is it there? No? Ok how about there?" at various points in the sky.

There are a number of telescopes that scan the sky looking for various things (well, that's basically what telescopes do...I'll refer to these as survey telescopes to distinguish them from telescopes like Hubble that look at specific targets when directed). There is one called WISE that is looking for distance planets and nearby brown dwarves (distant from the solar system's perspective and nearby from the galaxy's). So they consider where this planet could be and not be detected by WISE. The same goes for other survey telescopes: one called CRTS, one called Pan-STARRS, one called the Dark Energy Survey. Another group of researchers, in the time since the January announcement, did an analysis of Saturn's orbit given hypothetical perturbations from this planet, and used the fact that Saturn and the asteroids have not been very perturbed to rule out a few more regions.

What remains is a band of sky where Brown and Batygin's analysis shows this planet must be in order to explain the orbital alignments of the trans-Neptunian objects, and various regions of that band are excluded from these various non-detections.

The band of sky where this planet is expected, and all the ruled out regions of that band, leaving only the black areas. This is an image from their paper which I have annotated. I respect Brown and Batygin as scientists but I disapprove of their use of .jpg for a paper figure! And yet here I go using yellow text on white background.

 So, all that remains in the realm of possibility is a medium chunk of sky in the Northern hemisphere, and a tiny one in the Southern hemisphere. Interestingly, Brown gave his first public lecture on the topic at MIT, which I attended. He showed a similar figure, except there was a much larger region in the Southern hemisphere. Since then, it has been ruled out by the Saturn analysis.

Where is this big region on the actual sky? I have approximately traced it out on a sky map, which is shifted 180 degrees compared to the image above. In the Northern hemisphere, the big region encompasses Orion, which is one of the easiest constellations to find!

Very approximate locations. If you think you can do a better job of mapping the graph regions onto the sky map, please show me and I will credit you. Update (July 20 2016): An anonymous commenter has made a better version, with the yellow showing the search regions.

This tells us how to focus our telescope searches to best find this hypothetical planet. In the Northern hemisphere, the best telescope for the job is the Japanese Subaru telescope (named after the Japanese word for the Pleiades, not the car company whose name has the same origin). This may take up to five years, unless they can get more dedicated telescope time, or refine their search with more accurate computations or the discovery of more aligned trans-Neptunian objects.

So, by looking in this region, it could find this new planet, or it could rule out its hypothetical existence.
Alexhttp://email@example.com:blogger.com,1999:blog-1784861354684747574.post-11824794027924960692016-03-19T17:46:00.001-07:002016-03-19T17:46:39.818-07:00The Robophysics session at the APS March Meeting was delightfulI recently returned from the March Meeting of the American Physical Society (APS) in Baltimore. It is the largest annual physics conference, with about 10,000 attendees and 50 simultaneous lecture sessions laden with technical comic sans. Everyone gets 10 minutes to speak and 2 minutes for questions, and then are affirmatively ushered off to stay on schedule. My talk was on Tuesday. There are about 10 research groups in my field and all of them were there, so it was a very relevant session. On Friday afternoon I had seen everything I needed to see and was pretty exhausted, so decided to go to a session called Robophysics: Physics Meets Robotics. I am glad I did.

This lively fellow made an appearance.

There were about 30 talks spread over two sessions, of which I saw about ten. They generally followed the following format.

1. Here is a video of an animal. Look at how smoothly it moves.
2. Here is our best attempt to make a robot to do that. Look at how terrible it did.
3. Here are several slides of variational calculus to guide us towards a better robot.
4. Here is our new robot. We had an undergrad film it for hours on end to gather data. Enjoy.

This video was played in multiple talks:

There was a variant of the format above, where they filmed animals with high-speed cameras in weird situations to see how they function, for example filming mudskippers pushing themselves up sandy hills. One researcher showed how a centipede rapidly increases its vibration frequency and amplitude when poked, and attempted to make a robot centipede that mimicked this.

The main motivation for the research was to improve the way robots interact with their environment, which currently has to be hard-coded (e.g. for manufacturing robots) and is not at all adaptive, and his holding back the expansion of robotic utility. Much like in nanotechnology, researchers are trying to move forward by drawing inspiration from nature.

I will summarise a few of them. I look a few potato-quality pictures which I might try to supplement by looking up the papers.

One was titled "Is a snake a wave or a particle" and was about the angle that snakes scatter when they slither between two posts. They built a snake robot, had an undergrad gather some data on single-snake diffraction, and found that the snake scattering angle distribution has discrete peaks. Someone in the audience asked how a snake could interact with itself like that, which prompted a brief discussion of the history of quantum mechanics. There were quite a few snake-related talk, including one about dropping flying snakes from the top of a room full of high-speed cameras.

Another was about the propagation of vibrations through a spider web. They showed a high speed video of a spider sensing and attacking a moth in its web, then built a large-scale mechanical spider web full of accelerometers with varying tension to study how the perturbation of the prey reaches the spider. After the talk, somebody asked the question:

"In 30 years when we live in a post-apocalyptic robotic hellscape, and I got caught in a human-sized robotic spider web trap, do you have any recommendations as to how I can try to vibrate the web to avoid detection the longest?"
 Another talk was inspired by a scene from The Simpsons, where Homer kicks a table in order to vibrate a bowl of dip closer to himself so he could dip his chip without getting up. They managed to invent a mechanism to vibrate a surface to create arbitrary flow fields along the surface and move objects around in precisely controlled ways.

He embiggened the session with his cromulent research.

I only made it to the second session, so I missed the keynote invited lecture in the first. Overall it was a nice coda to week of intensive science exposure.

One of the labs that presented a few talks has a website about this with many videos, which I suggest perusing.

Alexhttp://firstname.lastname@example.com