Physics Shower Thoughts Part I: When does the Boltzmann distribution solve the Schroedinger equation?

Occasionally, as physicists, we get ideas that we find interesting, explore a bit, and realize that they're dead ends. Still, working through these ideas is a worthwhile endeavour. I call these "physics shower thoughts," and I will share them with the readers of this blog.

Before I continue, I will pose two questions to the reader:

1. You have a quantum particle in a harmonic potential. What is the probability of finding the particle at some distance from the centre of the trap?

2. You have a particle undergoing Brownian diffusion in a harmonic potential. What is the probability of finding the particle at some distance from the centre of the trap?

If you answered "Gaussian!" to both questions, you are correct. For a harmonic (quadratic) potential, the Boltzmann distribution, which is e-to-the-minus-the-potential has the same functional form as the solution to the Schroedinger equation with that potential. Is this special? Is this interesting? Is this the only example of a potential where that is true?

I thought this picture was a good idea when I started making it and I can't be dissuaded now.

There are quite a few connections between quantum and statistical mechanics. The diffusion equation and the Schroedinger equation are basically the same, if you replace the thermal energy with Planck's constant and make time imaginary. A lot of the theory for my Ph.D. thesis applied insights from established Schroedinger solutions to study diffusion solutions. There is also something called the Wick Rotation that I don't really understand that maps thermal systems onto quantum systems, and John Baez and Blake Pollard have an article called "Quantropy" discussing a lot of these connections.

Plugging the Boltzmann distribution into the solution of a physical differential equation in order to solve it isn't outlandish, the Poisson-Boltzmann equation to find the distribution of ions in a fluid is an example.

In 2013 my Ph.D. research took me in the direction of analysing experimental histograms to try to measure a thermal potential, by using the Boltzmann distribution. We were trying to show that these histograms are Gaussian*, implying that the potential is quadratic, and the standard deviation of the histograms would give us the spring constant. This got me thinking about the connection between the thermal particle on a spring, and the quantum particle on a spring.

Not really finding any physical significance to this connection, I looked for a mathematical reason, which starts by substituting the Boltzmann distribution into the Schroedinger equation. Because the units are different between thermal and quantum physics, we will write down the (one -dimensional time-independent) Schroedinger equation and Boltzmann distribution in a general form:

$-\alpha \frac{d^{2}\Psi}{d\,x^{2}}+V(x)\Psi=E\Psi$

and

$\Psi=\kappa e^{-\beta\,V(x)/2}$

I've added a factor of 1/2 in the exponential, because the probability amplitude is the square of the wavefunction, so the wavefunction in this case would be the square-root of the Boltzmann distribution.

Substituting the second equation into the first, we have a modified version of the Schroedinger equation that satisfies this constraint for the potential:

$-\alpha \frac{d^{2}}{d\,x^{2}}\left[\kappa e^{-\beta\,V(x)/2}\right]+V(x)\left(\kappa e^{-\beta\,V(x)/2}\right)=E\left(\kappa e^{-\beta\,V(x)/2}\right)$

Then I can get my trusty sidekick Maple to try to solve this. There is no simple explicit solution, but I can have it generate a Taylor series solution in x**. I can impose another constraint, that the derivative of the potential is zero at the origin. The Taylor series looks something like this:

I still have a free parameter to play with, which is the value of the potential at x=0, which would have to be balanced by a normalization constant. There is a trivial solution where V(0)=E=V(x), or I can solve for V(0) to kill the higher order terms. If I try to kill the quartic term, I get:

$V(0)=\frac{E\beta-1}{\beta}$

and if I plug this in, something cool happens, which is that every higher term dies. I'm just left with a quadratic potential!

$V(x)=\frac{E\beta-1}{\beta}+\frac{x^2}{\alpha\beta^2}$

We have shown that if we demand that the Schroedinger equation be solved by the Boltzmann distribution, one of the possible potentials is quadratic. Just to complete the quantum harmonic oscillator picture, we can kill y-intercept with $\beta=1/E$, and return to the classic Schroedinger equation with $\alpha=\hbar^{2}/2m$. Demanding that the spring constant is $m\omega^{2}$ tells us that the energy eigenvalue is $\hbar\omega/2$, the ground state energy of the quantum harmonic oscillator, so everything is tied up in a neat little package.

So, that particular value of V(0) gets me a quadratic potential, which is one of the few potentials for which there is a solution to the Schroedinger equation. Can this method be used to find others? What if I try to kill an arbitrary higher order term? It turns out that this does not work as well. You can choose V(0) to eliminate any term, but only that term and not the higher ones. Only the quartic term will cause the Taylor dominoes to collapse if you knock it over.

This argument can also be used for the excited states of the harmonic oscillator, e.g. multiplying the wavefunction by x before substituting it. This doesn't add much of interest. I also wanted to see if this would work as a general method of finding solvable potentials for the Schroedinger equation, trying various functions mapping between the wavefunction and the potential. It generally doesn't work, but I did find one solvable potential with this method:

The constraint $\Psi(x)=\frac{\beta}{V(x)}$ gives $V(x)=\frac{-E}{\cos{\sqrt{\frac{2mE}{\hbar^2}}x}-1}$ as an easily solvable Schroedinger potential. I guess that's kind of cool.

Through all this we investigated a mathematical connection between the Boltzmann distribution and the Schroedinger equation, but didn't get much physical insight. I asked John Baez, author of Quantropy, whether the connection was physically significant, and he gave me a "Yes," so I hope to learn more. Overall, I explored an idea that I had, and found that it lead somewhere kind of, but not overly, interesting. I think that's worthwhile.

*Of course, basically any large sample you measure will be Gaussian...stupid central limit theorem. This project unfortunately was a dead end, and took a lot of time and effort.

**This is one of my favourite math tricks. I highly recommend.

PhysicsForums: Can We See an Atom?

I wrote an article on PhysicsForums about whether we can "see" atoms through AFM, STM, TEM, etc. Check it out!

https://www.physicsforums.com/insights/can-see-atom/

My first discovery: Pandigital Slinky Ratios

If you plug the quotient 987654321 / 123456789 into a calculator, you will find it is very close to 8. It is approximately 8.000000072900... This seems like a coincidence, but it isn't! Figuring out the pattern to these numbers was the first thing I ever "discovered," with the help of my uncle. I tried to publish my proof, and was rejected in three minutes.

The slinky is a metaphor.

My father actually showed me this trick on an old calculator when I was about six years old. I just thought he was trying to show that two big numbers make a small number; I didn't really understand what division was at the time. I spent a lot of time in highschool playing with my calculator. To pass the time I would button-mash random four-digit numbers and try to divide them by prime factors until I got to a prime number; I generally liked playing with calculators.

I do not remember the historical sequence of events that lead to my discovery, but I always curious about why this was so close to 8. On the Windows calculator, you could enter calculations in base-8, base-16, and binary as well as decimal. So, one day I tried doing the same thing in those modes. If you do the same type of calculation in hexadecimal, FEDCBA987654321/123456789ABCDEF, you get E.0000000001D46... and the remainder is smaller than in base-10 (there are nine zeros there instead of seven). In base 8, 7654321/1234567 gets you 6.00000527, only five zeros. So, a pattern emerged: the bigger the counting base, the closer the slinky ratio gets to the second biggest digit. Around 2005, in first or second year of university, I downloaded a calculator app (we used to call them programs!) that would go up to base 36 (using 0-9 and A-Z). This was the highest I could test, and the pattern continued.

 I wanted to figure out why.

Writing this blog post and searching my email for "987654321" I found a few awkward attempts to email math professors asking for help proving this. In my training as a physicist, I learned that one of the best ways to solve a math problem is to get a computer to do it, and a program like Maple or Mathematica can figure this out after like 30 seconds of typing and pressing Enter. However, there is a more interesting way to prove this relationship. The main insight came with the help of my uncle Bob, who was a mathematician before becoming a lawyer, when I asked him about the problem. The Bob Proof is more illuminating than a computer algebra output, so I'll discuss that.

In order to figure out how this trend generalizes, the first step is to write down what the operation actually is. So, for some counting base b, the numerator is:

$\sum_{n=1}^{b-1}\left(b-n\right)b^{b-n-1}$

And the denominator is:

$\sum_{n=1}^{b-1}nb^{b-n-1}$

So we divide them and observe empirically:

$\frac{\sum_{n=1}^{b-1}\left(b-n\right)b^{b-n-1}}{\sum_{n=1}^{b-1}nb^{b-n-1}} \approx b-2 + \epsilon$

And we want to know what gives*.

Bob's insight was to rewrite each expression as a sum of oneful numbers. For the denominator, the term can be expressed as 1+11+111 etc. Do this to some value of ones, and you'll get a number that looks like 123456... because each iteration increases the final digit. You can also express it as (1+(10+1)+(100+10+1)...) with the appropriate definition of 10. It is also the case that in any base**, that a oneful number times the largest digit is one less than a power (e.g. 111x9=999=1000-1).

So, we can rewrite the oneful denominator as a chain of oneful sequences:

$\sum_{n=1}^{b-1}nb^{b-n-1}=\frac{b-1}{b-1}+\frac{b^2-1}{b-1}+\frac{b^3-1}{b-1}...$

And now factor:

$\sum_{n=1}^{b-1}nb^{b-n-1}=\frac{\sum_{n=1}^{b-1}b^{n}}{b-1}-b-1$

Now, there is a trick involved that I don't quite remember. I might have just gotten a computer to figure it out, cheapening my whole example of why this proof is more insightful than just using computer algebra, but anyway it relies on the identity:

$\sum_{n=1}^{x-1}x^{x}=\frac{x^x}{x-1}-\frac{x}{x-1}$

I honestly cannot remember why this is true, and I am failing to prove it by induction. So to prevent this whole post from falling apart, proof of this particular identity will be left as an exercise for the reader. Anyway, using this identity in the previous expression, we get a sigmaless term for the denominator:

$\sum_{n=1}^{b-1}nb^{b-n-1}=\frac{b^{b}-1}{(b-1)^2}-1$

There is a similar procedure for the numerator. Instead of 1+11+111...we add terms of equal magnitude, starting with 111111...11111 and going to 1000...0000, replacing a 1 with a zero in each term. So in base-6, we have 10000+11000+11100+11110+11111=54321. Keep adding these, and if you don't exceed the number of terms required for your base, you'll get your descending slinky.

To simplify the numerator with this method, the logic is similar for the denominator. In short:

$\sum_{n=1}^{b-1}\left(b-n\right)b^{b-n-1}=\sum_{n=1}^{b-1}\frac{\left(b^{b-n}-1\right)b^{n-1}}{b-1}=b^{b-1}-\frac{b^{b}-b}{b\left(b-1\right)^{2}}$

So now we have expressions for the numerator, the descending slinky number ending in ...54321, and the denominator, the ascending slinky starting with 12345...., and both are in "closed form." So, we can divide them and simplify, and we get:
$\frac{\sum_{n=1}^{b-1}\left(b-n\right)b^{b-n-1}}{\sum_{n=1}^{b-1}nb^{b-n-1}}=\left(b-2\right)+\frac{\left(b-1\right)^{3}}{b^{b}-b^{2}+b-1}$

And that is our result! Just looking at it, we see that it is (b-2) plus some stuff, and the stuff has a $b^{b}$ on the bottom and a $b^{3}$ on top, so it will be very small if b gets much bigger than 3. But there's more we can learn from this.

In base-10, the value we get starts with 8.0000000729. You may recognize those last three digits as the cube of 9.  If you take only the largest term in the denominator of the remainder, $b^b$, you will find that the first bit of the remainder is $(b-1)^{3}\cdot b^{-b}$, which is $9^3\cdot 10^{-10}$, which is what we see. This was before I knew about or understood perturbation theory, so nowadays I would call this the next-to-leading order term.

That remainder gets real small real fast.

Like I said, a computer can easily figure this out. Bob and I may have been the first people to do this, but weren't the last. In 2013 the question was asked on Math Stack Exchange, and a user WimC gave a similar answer, with an expression in terms of a sum of diminishing terms.

Now, I am much better at finding literature online, so I will do a retroactive literature review. A solution may have been found in 2004 but I didn't know at the time. In fact, this was discussed in 1987, the year of my birth, where P.R. Pujado shows that the eight closeness is not a coincidence, but the result of a pattern in a growing series. The generalized b-base form is not presented there.

In late 2008**, I was working as a post-bach researcher at Sunnybrook hospital studying bubbles, and decided to write up my findings and get them published; my first attempt to do so. I wrote a paper summarizing what I have explained here. Besides lab reports, my only piece of scientific writing was my undergraduate thesis on general relativity, and I still hadn't learned how to write like a grown-up. I sent it off the the Canadian Mathematical Bulletin. On December 8, 2008, at 1:25 PM I received an email saying that the editors were reviewing my paper, and it could take about six months. At 1:28 PM, I received another email saying that it had been rejected. Their reasoning? "Your paper is a cute fact." Well, they weren't wrong. They suggested I send it to the American Mathematical Monthly, which also rejected it, and then I sent it off by paper mail to the Journal of Recreational Mathematics, who never got back to me, and that was the end of that. It is perfectly reasonable that journals would reject this and I don't hold it against them.

In those three minutes, I was upgraded from Dr. to Prof.

This problem, the process of solving it, and my failure to get the solution published, ignited a passion in me that made me want to become a published mathematician. I generally find physics more interesting than mathematics, but there's a certain purity to coming up with and proving a new mathematical fact, and I wanted to have one of my own. I eventually did, about five years later, and this may be what indirectly set me on that path.




P.S. If someone can leave a comment explaining how to centre the LaTeX, I would appreciate.

*$\epsilon$ is math notation for something very small
**no rhyme intended

Exploding Collapsing Bubbles on PhysicsForums

I wrote an article on PhysicsForums about a paper about collapsing bubbles. It is not at all like my recent post about bubbles; these are in three dimensions.

https://www.physicsforums.com/insights/explosion-generated-collapsing-vacuum-bubbles-reach-20000-kelvin/

My Journey into the Hyperbubble

In 2012 I finished writing a pretty weird paper which I published a year later in the journal Physics of Fluids, about the behaviour of bubbles in spatial dimensions greater than three. This is the story of how I started writing it, what lead to me to finish it, and how it got published.

The journey is a metaphor.

After my undergrad, I got a job in a lab that developed ultrasound technology for treating diseases, for example, applying intense sound waves to a tumour until it heats up and coagulates and melts (I want to link a video here but they're all terrible). My job seemed unrelated to this: figure out how bubbles behave in a tube*. Bubbles are used as ultrasound contrast agents: a gas bubble in a liquid will oscillate under an external ultrasound field, getting bigger and smaller. This in turn releases more sound waves as the bubble displaces the fluid around it, which can then be detected. A lot of people ask if bubbles in the bloodstream cause heart attacks: not these ones, they are microscopic.

There are several ways to model how bubbles behave. The first attempt was by Lord Rayleigh, of scattering fame, who in 1917 wanted to understand the damage caused to propellers by the collapse of the bubbles they produce. He considered an infinite fluid with a spherical cavity in it, and calculated how long it would take for the fluid it fill up the cavity. A bubble with gas in it will not get totally filled by fluid: the pressure of the gas will increase until it can reverse the collapse and expand the bubble again, and this oscillation produces sound waves. By treating this as harmonic motion, Michael Minnaert derived the eponymous frequency of a bubble's oscillation in 1933, explaining that "musical air-bubbles" are responsible for the sound of babbling brooks.  These concepts were combined by Milton Plesset to describe gas-containing bubbles, leading to the Rayleigh-Plesset equation, a second-order ordinary differential equation describing how a bubble's radius evolves over time due to internal and external pressures. It is still used today to model bubble dynamics.

My task was to figure out how putting the bubbles in a tube (a proxy for a capillary blood vessel) modifies their behaviour, particularly their resonance frequency and response to ultrasound. My supervisor had grander plans for me, but I wasn't yet aware of them. This was my first independent delve into scientific literature. I found a few papers using fairly advanced modelling techniques to calculate bubble response in vessels, but I wanted something I could reproduce. Ultimately I found an equation that a Russian scientist had derived in 1992 but not solved, and I "solved" it numerically and showed that it gave the same results as some of the more complex models. I wrote a paper explaining all this, and couldn't get it published in most journals so it ended up in an obscure Russian publication. I learned an important lesson about academic publishing from this: just because you did something new, doesn't mean people will care.

The main project my supervisor wanted me to work on involved using this bubble knowledge to figure out how much heat is transferred to the brain during focused ultrasound. I obtained a 3D microscopy map of part of a rat brain from another group in the division, developed a way to populate the blood vessels with bubbles, simulate their oscillations based on the width of the vessels, calculate the heat transfer according to the "thermal damping" of the bubble, and use somebody else's solutions to the Heat Equation to figure out the effect on the tissue. This eventually lead to my first paper, in Physics in Medicine in Biology. If I were to tackle that again, I would probably not try to hack together the different methods the way I did. Unfortunately, after I left the person who was supposed to take over the modelling kind of dropped the ball, and the person who was supposed to test it experimentally quit, so the project kind of died.

Temperature in the mouse brain.

When I started graduate school, it took a while for my project to get going. In fact, it took over a year before we had a lab and I could really get into my experiments. I was still thinking about some of the stuff  from my bubble days, and one of the things that was bugging me was that the Rayleigh-Plesset equation was unsolvable. By unsolvable, I mean that there isn't an expression one can write down that describes a bubble's oscillation over time. In my first semester at McGill I was taking a course on string theory for some reason, and one thing that stuck with me was how they took something that was unsolvable in three dimensions, re-derived it in an arbitrary number of dimensions, and found the one dimensionality that it actually worked in (26, in this case). I figured I could try the same for bubble equations.

I could not. That was probably a ridiculous thing to think. I tried anyway though.

The derivation for all the equations I mentioned involved writing down the kinetic energy of the bubble by integrating the kinetic energy of a spherical shell of fluid at some distance from the bubble moving at some velocity. The velocity at the wall of the bubble decays according to the continuity equation, 1/r$^2$ in three dimensions; if that were not the case then more bubbles would form, or something. I rewrote this for some dimension N**, and now the surface element was proportional to r$^{N-1}$ and so was the velocity. With the help of Maple, a computer algebra software, I obtained expressions for the arbitrary dimension resonance frequency, the collapse time, and the Rayleigh-Plesset equation. The main result was that the resonance frequency and the collapse are both faster in higher dimensions.

Two dimensional image of a three dimensional calculation of a bubble's kinetic energy, based on the motion of fluid around the bubble. The velocity of the fluid is proportional to the rate of change of the bubble's radius (the wall velocity) and falls off as an inverse square.

As I said in a previous post, I am not a great mathematician. I hoped that I could use the differential equation solving function on Maple to spit out the answer. It didn't spit out the answer for three dimensions, and it didn't spit out the answer for N-dimensions. I could have it generate Taylor series solutions for certain initial conditions which I could try to generalize in terms of messy and boring matrices and that didn't really go anywhere.

Anyway, I figured since I had derived these things I might as well write a paper on them, even though I didn't have much in the way of implications. So I started doing that...I don't even remember when. Looking through my email, it appears I had a draft of sorts in July of 2010, when I was about halfway through my masters. In the process, I looked up a paper to find a source for something in my introduction. It was the paper "Bubbles" by Andrea Prosperetti, one of big names in bubble world, which is a thing. This paper has one of the best introductions of any I've read, and I recommend reading it. I saw that in the theory section of this paper, Prosperetti starts talking about the oscillations of a bubble in N dimensions, and derives the Minnaert frequency and the Rayleigh-Plesset equation. I was pretty bummed, because what I thought was a unique idea had already been done. However, Prosperetti's equation was not the same as mine, and I ran some numerical solutions and showed that my equation was internally consistent and his was not.

No disrespect to Prof. Prosperetti.

 So, I could write a paper about how his paper was slightly wrong and improve upon it, but that's not much of a paper. This, combined with the fact that by then my masters was probably picking up, I basically dropped the whole project and focused on my actual work.

Fast forward to 2012. I was reading through Physical Review E, which is basically a journal for "other" areas of physics. I saw a paper by Danail Obreschkow, about an analytical approximation for bubble collapse. I had encountered Danail's research during my hospital days, and somehow landed on his website. He was a few years older than me and in charge of a science mission to observe bubbles in a zero-gravity aircraft, which is inherently cool. He was also into extragalactic astronomy, oddly shaped dice, quantum dots, and was in a rock band. Coolest physicist ever basically. His new paper looked at an approximation to the solution of a re-arranged version of the Rayleigh-Plesset equation, matching it to numerical solutions and to their experiments. Describing how the radius evolves over time, they guessed that $r=\left(1-t^{2}\right)^{\alpha}$ and found the values of $\alpha$ that best describe the solution. If you care about the start of the collapse, $\alpha$ is 0.4, and if you care more about the end, $\alpha$ is about 0.42. Seeing this paper re-kindled my interest in the project, and I set myself to working out this version of the equation in my formalism. I figured it out (there may have been some guess-and-check involved), and found that the two versions of the exponent are most similar in three dimensions, and in higher dimensions they diverged to one being 23% greater than the other one***. So, I proved that three dimensional bubbles are special.

So I went back to my draft of the paper from 2010, added this section, and was about to call it a day and put it on arXiv, when I had the idea to look at some of the non-linear effects. What I mean by this is that when a bubble is strained it oscillates at its resonance frequency, but when the strain amplitude increases, the frequency starts to change and the oscillations cease to be sinusoidal...they start to resemble a shape called a cycloid****. So I looked at how the frequency changes with initial conditions, and that was a pretty good idea because I found something interesting. In three dimensions, a bigger strain just decreases the frequency. But in higher dimensions, first the frequency increases, then decreases. So three dimensional bubbles are once again proven to be special: it is the only dimension with a monotone strain response.

Would you look at that nonlinearity!

So, with the combinations of my derivations, pointing out that Prosperetti's equation was wrong, generalizing Obreschkow's approximation, and the investigation of nonlinear effects, I wrote up a paper and submitted it to arXiv in 2012. And there it sat, I was happy to have "gotten it out there."  By this time I was in the swing of my PhD and had a lot of actual stuff going on. A week later, Danail Obreschkow emailed me and said he liked my paper, and I thought that was pretty cool, it is literally the same as Alexander Friedmann getting an email from Albert Einstein in 1922.

Fast forward again to March 2013. I got another email from Danail Obreschkow, again saying he liked my paper. This time, he gave me some more encouragement and suggested that I try to get it published. I had sort of assumed that it wasn't realistic enough to be accepted in a physics journal, and not rigorous to be accepted in a math journal, but I took this encouragement to heart. That day I submitted it to Physics of Fluids, which has a self-explanatory scope.

Six weeks later I heard back from them. In typical peer-review fashion, Reviewer #1 was all positive and Reviewer #2 was a total jerk. His main points were that a) pointing out a typo in an old paper doesn't make a novel contribution and b) I should focus on something physically realistic. Personally I didn't think b) was a fair point given that we all agree that we only live in three dimensions, but the reviewers must be satisfied.

I was a bit lucky, that after I had submitted my paper, another paper came out analyzing Obreschkow's approximation. In his paper, Obreschkow improved upon his approximation by multiplying it by a truncated Taylor series, and claimed that even without truncation this approximation would never be fully accurate. Amore and Fernandez argued that it would be fully accurate, citing numerical evidence. I re-derived their analysis in arbitrary dimensions, and showed that not only is there numerical evidence in three dimensions, there is numerical evidence in every dimension, which makes a much stronger case than just a single numerical coincidence. In effect, I had used my higher-dimensional equation to settle a question in three dimensions, about whether this approximation can be exact or not. I also got to add some bigger equations, which I liked.

We're talking tertiary parentheticals here, man.

I rewrote the paper, removing some of the higher-dimensional simulation and focusing more on this issue, while toning down my criticism of Prosperetti. What once said "Obviously, experiments cannot be performed in higher dimensions to resolve this discrepancy, but the author finds fault in Prosperetti’s derivation" now reads "There is disagreement between the equations presented by Prosperetti1 for the unbounded case and those of this paper because a term was dropped from the second derivative of radius between his Eqs. (1) and (3)."  Eventually it was accepted and published in August of 2013, the same month that I finished by tunnel-through-the-Earth paper.

When I was younger I had made several naive attempts to publish something mathematical, and now I had finally succeeded. I remember at my old job thinking that Physics of Fluids was too mathy for me to understand, and now I was in it. I also learned an update to the lesson from four years earlier: even if nobody cares, work on it anyway.


There is one more addendum to the story. One of the reasons I started down this path was to try to solve the Rayleigh-Plesset equation. In 2015 two Russian mathematicians did manage to solve my equation in 4 and 6 dimensions, in terms of really complex functions. I emailed them to congratulate them, and one of the authors told me, almost word for word, the same thing I told Danail: that he was inspired to write his paper after seeing my paper.

Oh and if anyone cares but doesn't want to read the paper, here are the main results:

*My main research projects have been bubbles in tubes, DNA in tubes, and falling through tubes.

**I think the convention for an arbitrary dimension is actually D. I stuck with N.

***Coincidentally, this ratio of 1.23 is the square of the ratio of the uniform-density fall through the Earth and the constant-gravity fall through the Earth.

****Which is also the shape of the brachistochrone!

PhysicsForums: A Hand-Wavy Discussion of the Planck Length

I have an article on PhysicsForums about the Planck length!

https://www.physicsforums.com/insights/hand-wavy-discussion-planck-length/

A Surface Map of an Exoplanet

This is a summary of a paper from 2007. However, it is one of the neatest things I have read about, and I'd like to share.

Exoplanets are typically detected either by their gravitational influence on their star, or by the decrease in the star's luminosity as the planet occults it. In both these methods, you do not see the planet itself, you see changes in the star's light caused by the planet. Nevertheless, a lot can be learned about the planet: its size, its orbit...ok mainly those things. But with some very clever teloscopy, Heather Knutson and friends managed to make a map of the surface temperature of an exoplanet.

 The star is called HD189733. Stars with numbers instead of Greek or Arabic sounding names are generally part of a star catalogue, where a person or a group of people writes down a list of otherwise boring stars (in this case it is the 189733rd star in the Henry Draper catalogue). This star is in the constellation Cygnus, the swan, also known for hosting the first-discovered black hole. The star is 63 lightyears away (close, by stellar standards), and is about 80% the size of the sun.

Over there.

The planet was discovered in 2005, by the dimming of the light of the star. Every 50 hours, the light of the star dims by 2%, indicating that there is a planet 4.5 million km away from the star (less than a tenth the distance Mercury is from the sun), about 10% bigger than Jupiter. In addition to the star dimming by 2% every orbit, it also dims by 0.3%  at another point in the orbit, 180 degrees of rotation later. Why does this happen? When the planet is in front of the star, it blocks light from star from reaching Earth. When it is behind the planet, the star blocks light from the planet from reaching Earth. When the planet is "next to" the star, from our perspective, we receive light from both.

Brightness of the star over time. a. planet goes in front of star. b. star goes in front of planet.

Knutson made a really detailed observation of the brightness of the star, taking 278,528 images over a 33 hour period. The star is far enough away that it just looks like a blob of bright pixels with no structure, known as a point-spread function. There are some exoplanets that can be directly imaged, but not this one. They monitored the brightness of the star as the planet went and emerged from behind the star. The planet is rotating as it revolves, so each point in time is looking at a different hemisphere that is facing Earth, and by monitoring the different brightnesses at different points in the planet's orbit, they could measure how much brightness the planet was beaming at the Earth.

Cartoon of my understanding of how this analysis is done. The planet is a bright side that always faces the star, and a dark side facing away. In the top image, the light beamed from the planet to the Earth comes from both the bright and the dark sides. In the second, it has rotated a bit more and now more of the bright side is beaming to Earth, so the planet's contribution increases. When the planet goes behind the star we can no longer see it. Sizes and distances are surprisingly not to scale.

This is in effect, a proxy measurement of the average temperature of each hemisphere that is pointed at Earth at a given time. This can be processed into a map of the temperature across the surface of the planet!

The surface temperature across the planet! This image is from the Wikipedia article on this planet, it is a different visualization of the same data that is found in the paper.

The heat map looks cool, but it's important to look at the raw data.

The planet is close enough to the star that it is tidally locked, meaning that the same face is always pointed at the star, same as how the same face of the moon is always pointed at the Earth. This means that on the planet's surface, the star always appears to be in the same point in the sky, and there is one point on the equator where the sun is directly overhead. Logically, one would expect this to be the hottest. However, they found that the hottest point was actually 30 degrees East of the sub-stellar point, while the coldest was 30 degrees West of the darkest spot on the planet. The temperature ranged from about 1200 Kelvin down to 970 Kelvin. For comparison, the hottest part of Mercury is about 700 Kelvin.

Why is the hottest part of the planet not the place with the most sun? The authors conclude that there is an atmospheric wind on the planet, blowing East at hundreds of meters per second, essentially sweeping the heat over a bit.

In 2012, one of the initial authors and two others improved on these methods, using much more sophisticated techniques, and gained the ability to measure temperature as a function of longitude as well as latitude. They improved the accuracy of the latitude of the hotspot to 21 degrees, and concluded that the hottest point is within 11 degrees of the equator. This was expected, but now it is measured.

Even though the methods are coarse, I think it's awesome that we can map the surfaces of planets around other stars.

Deadlifts and Animal Speeds: Post on PhysicsForums

If anyone regularly checks this blog after three posts, know that I have written an article on PhysicsForums, where more of them may be appearing!

https://www.physicsforums.com/insights/scaling-laws-speed-animals/

Does the squishy affect the stretchy?

For my Ph.D. I wanted to compare my experimental measurements to a theoretical model. Unfortunately, the model I needed did not yet exist. This is the story of the collaboration that brought that new model into existence.

My graduate years were spent looking at DNA molecules squished into funny shaped tubes. They feature a very narrow slit, narrow enough to make the molecule spread out in two dimensions, and are filled with little pits, that the molecule can fall into to increase entropy. When a molecule falls into a certain number of pits, it looks like little tetris pieces. My master's was spent studying how DNA behaves in systems like this. My Ph.D. was spent using these systems to learn more about polymer physics.

Let the cavalcade of gifs begin! Each little bright square is a little less than a micron.

It's a big molecule but a small cavalcade.

DNA is a polymer. DNA is the best polymer.

A polymer is basically a molecular chain or string, with a path that curves around randomly. Polymers are generally in a reservoir with some temperature (in my case, in room temperature water), so they change orientations very quickly and randomly as water molecules bump into different parts of the chain. Entropy is effectively the number of ways a system can be rearranged and still look the same, and in the case of a polymer it's the number of ways you can reorient the path of the chain and still have the whole thing stay the same size. The maximum entropy configuration is what is called a "random coil." I would call it a blob, but that refers to something else.

When I study DNA stretched between these pits in a narrow slit, there are a few important things I have to take into account:

DNA is Stretchy

If you take the two ends of a polymer chain and hold them fixed, there are a finite number of ways to reorient the chain and still have the ends the same distance apart. If you pull them farther apart, there are fewer ways: you have decreased the entropy. Because the temperature of the system makes it constantly re-orient itself**, the ends will move closer together and entropy will increase. This is called an entropic force. This video paints a pretty good picture of initially stretched chains recoiling to their higher-entropy random coil configuration. Here is a gif from one of the first papers to visualize DNA doing it.

One of the first visualizations of stretched DNA relaxing to increase entropy, from Science vol. 264, 6 May 1994.

DNA is Squishy

So that's what happens when you stretch it. What happens when you squeeze it? The random coil has some preferred size, and when you put it in a tube smaller than that size, it has to squish itself in, and this again prevents certain configurations from forming, again lowering the entropy. This video (unfavourable format warning) shows this phenomenon: first DNA is driven into a narrow region where it is forced to adopt a low-entropy state, and then it recoils from the narrow region back into the reservoir to increase its entropy. Below is another nice video (that I did not make) of DNA being stretched from its random coil configuration then getting squished into a tube. To put it in human terms, each of our cells contains about 4 meters of coiled DNA. We have 100 trillion cells. There is enough DNA inside us to stretch to the sun 2000 times.

The Mystery of the Squishy Stretch

The question I wanted to know the answer to was: if DNA is squished into a tube, does that affect how hard it is to stretch? A general reason this might be the case is that the entropic force arises to resist the loss of entropy, but if entropy is already reduced by confinement, this restoring force might be less. In addition to genuine curiosity, I also needed a good model of stretched confined DNA to fully understand my measurements, which had some curiosities that I thought might be related to the stretchy-squishy conundrum.

When I was at the APS March Meeting, the largest yearly physics conference, in 2012 in Boston, I saw a talk from an Ottawa physicist Martin Bertrand about confined polymer simulations. I spoke to him, and told him about my experimental system, and asked if he was interested in simulating it, basically to verify the model I was using (I was still in a master's mode of thought). He then introduced me to Hendrick de Haan, also at Ottawa (now at Oshawa), who was actually interested in working with me to do simulations. I have been collaborating with him since. Initially we spoke just about doing some simulations of a molecule hopping between pits, but around this time my spring woes started to develop, so I asked him if he could figure out the problem I've described in the preceding paragraph.

I liked the idea of this project, because it would lead to a new idea that was interesting independently of my experiments. If Hendrick devoted his time and energy to simulating molecules hopping between pits {which he is more than happy to do :) }, and my experiments ended up going nowhere, the world would not benefit from any new ideas. If he figured out the stretchy-squishy theory, the world would gain knowledge whether my experiments worked or not.

It took a few years (it was not the main priority for either of us), but he figured it out, with colleagues Tyler Shendruk and David Sean. I will give a brief rundown of the theoretical insights that lead to this. There are two complementary methods to theoretical physics: pencil and paper derivations, which you use to get dandy expressions and equations for your theory, and numerical simulation, which you use to see how the big picture emerges from the fine details. Hendrick and friends used a combination of both, and the two techniques guided each other well.

Simulations are good for giving you the right answer if you ask the right question. Hendrick's molecular dynamics simulations basically simulated a chain of beads connected by springs with a certain bending rigidity. The beads receive random agitation from the "temperature" of the system, and everything moves around according to the thermal agitation and the forces acting on it. He simulated this for a range of heights and a range of forces, enough to get data of the force-extension relationship for all relevant scenarios. These serve as computational "data" against which a pencil-and-paper theory can be tested (real data is also nice, but experiments are not quite at the point of directly measuring this).

A stretched chain in a slit. If it looks rough it's because I copied a .mov from Hendrick's powerpoint slide then used a website to convert it to .gif and then used a different website to make the file smaller. Thanks Hendrick!

So how do we*** write down a theory that describes the squishy's effect on the stretchy? The Marko-Siggia equation gives the relationship between the force exerted on a polymer and the distance between the ends. In the limit of low forces, it behaves like a harmonic spring, and in the limit of high forces it behaves like a fluctuating string.

The Marko-Siggia formula. F is force, p is persistence length (50 nm for DNA), x is how far apart the ends are, Lc is the total length of the chain. If the units don't make sense, divide F by kT.

This equation describes how a stretched chain behaves in three dimensions, and someone had also derived how it behaves in two dimensions. Tyler and Hendrick focused on what happens between two and three dimensions, writing down an expression that simplifies to the 2D or 3D Marko-Siggia equations in the correct limit. Obviously we do not live in a fractional-dimension'ed universe, so what is the meaning of this fractional dimension?

To summarize, a polymer chain is half as floppy in two dimensions as in three, because there are less ways for its path to fluctuate. As the walls of a slit start to dictate the motion of a chain, they make it effectively stiffer as the chain is less likely to veer off its current path. So, this effective dimensionality is essentially a measure of how much less floppy the chain becomes as it is confined.

This is where the numerical simulations help guide the theory. By measuring the effective stiffness of an unstretched chain and various levels of confinement, they found a relationship for the effective dimensionality as a function of slit height. My experiments take place in slits between 50 and 200 nanometers, which corresponds to roughly 2.24 and 2.78.

The modified Marko-Siggia equation, interpolating between the 2D and 3D limits in a slit. d is effective dimensionality and h is slit height. This is actually the low force limit, read the paper to learn more!

So finally, there is an expression for a modified Marko-Siggia formula that describes a confined stretched chain. Does it work? The simplest thing to do is compare it to the numerical simulations, and we see that it does.

The simulation data matches the theory, and interpolates well between the 2D and 3D limits. Image taken from MacroLetters paper.

There is an additional subtlety however. If a chain is strongly stretched, its behaviour doesn't change that much when it is confined in a slit (imagine a chain stretched so much that it would never bounce into one of the walls...it would be like the walls aren't even there). To describe the stretchy-squishy interaction under high forces, they looked at it in terms of how the walls restrict certain Fourier modes of a vibrating string, and found a way of characterizing the effective dimensionality in terms of both height and stretch.

The paper was published on arXiv and then in the journal MacroLetters, which is inexplicably unavailable to Canadians. I am grateful that the editors and referees convinced the authors to switch from theorist units to actual units. I started using their version of the Marko-Siggia equation as part of my data analysis, and ultimately got some nice results that we published in a paper together. Even though I was not directly involved in the stretchy-squishy paper, I am proud to have been part of the process that lead to its creation.

I suppose I didn't actually answer the question in the title: yes, squishing DNA makes it easier to stretch.

**I don't feel the need to be consistent with my hyphenation.

***Here we refers to the writer guiding the readers through a process.