# Monthly Archives: January 2016

## Neil Degrasse Tyson Vs. Rapper B.o.B

Hip-hop artist B.o.B has recently been on a one-man mission on Twitter and other social media platforms trying to tell everyone that the world is flat. Here is Neil Degrasse Tyson’s response on the Nightly Show with Larry Wilmore.

For those of you outside the US that may not be able to view the previous video, here is a slightly curtailed version.

## Science and Hype

In the last few years, the media have picked up on a few physics stories that were later shown to be incorrect. Prominently, in the last couple years, stories of superluminal neutrinos and evidence for cosmic inflation at BICEP2 flooded the internet. Premature media coverage of high-temperature superconductivity in $H_2S$ and the Higgs boson also occurred, but these findings stood up to the peer review process. Most recently, a rumor was started about the detection of gravitational waves at LIGO. There is an interesting take on these events, focusing on the already-infamous LIGO tweet, in a Physics Today piece by Stephen Corneliussen. I recommend reading it.

It is my personal opinion that the aforementioned scientific discoveries should not have been reported to the media until they stood up to the peer review process. This point of view is not meant to blame the media; we physicists are in fact more responsible for alerting the media than they are for reporting the findings. (Do you really expect a reporter not to report a story? They are just doing their job after all!) Of course, the peer review process is itself far from perfect (just think of the absurd case of Schon for instance) but at least it provides an extra layer of assurance concerning new results.

This is not a cut-and-dry issue, and I wholly acknowledge this, but I do think that we can do better than the current state of affairs.

## Origin of the solitary wave

Back in 1834, a Scottish engineer named John Scott Russell noticed a strange kind of wave that propagated away from a boat that had made a sudden stop. He describes how he chased the wave on horseback along the river for about two miles! Here is his account from this manuscript:

“I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped – not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation”.

What he called a “Wave of Translation” is now known as a solitary wave or a soliton. Here is an excellent Youtube video of a solitary wave that has been generated in a lab setting. Remarkably, in the video, when two solitary waves meet, they appear to pass right through one another.

The strange thing about the solitary wave, is that it can propagate for miles at a time without breaking or disappearing, i.e. it is much more stable than the garden-variety (ocean-variety?) wave.

There are two concepts that are necessary to go over in order to understand the solitary wave. These are (i) non-linearity and (ii) dispersion.

Dispersion occurs when the phase velocity and the group velocity of a wave differ. If one forms a Gaussian wavepacket from a dispersive medium, then one would expect the amplitude to decrease and for the wave to broaden over time. You can see this effect occurring for the waves in the boat’s wake in this Youtube video. To those more familiar with quantum mechanics than fluid mechanics (which is usually the case for most physics students!), the same thing happens for a Gaussian wavepacket for an electron wavefunction, but does not occur for a wavepacket of light. It is important to note that this broadening and disappearing can happen in a medium that is not viscous. Viscosity has nothing to do with the dispersion of the water wave nor of the electron.

Non-linearity in water waves, on the other hand, is essentially what causes waves to “break”. The water at larger heights moves at a faster speed than the water at lower heights which results in a multi-valued function. Here is an instructive image which depicts well what I’m trying to get across (taken from here):

Also, here is Youtube video of breaking ocean waves to just soothe your soul.

So why is the solitary wave so stable and what is preventing it from breaking or disappearing? It turns out that the dispersion and non-linearity amazingly cancel each other out in order to sustain the solitary wave and allow it to propagate much further than regular waves.

Feynman said in his Lectures on Physics:

[water waves] that are easily seen by everyone and which are usually used as an example of waves in elementary courses […] are the worst possible example […]; they have all the complications that waves can have.

While this is true, it is these complications that give rise to some startling phenomena, including the solitary wave.

## An Integral from the SSH Model

A while ago, I was solving the Su-Schrieffer-Heeger (SSH) model for polyacetylene and came across an integral which I immediately thought was pretty cool. Here is the integral along with the answer:

$\int_{0}^{2\pi} \frac{\delta(1+\mathrm{tan}^2(x))}{1+\delta^2\mathrm{tan}^2(x)}\frac{dx}{2\pi} = \mathrm{sgn}(\delta)$

Just looking at the integral, it is difficult to see why no matter what the value of $\delta$, the integral will always give +1 or -1, which only depends on the sign of $\delta$. This means that if $\delta=1,000,000$ or if $\delta=0.00001$, you would get the same result, in this case +1, as the answer to the integral! I’ll leave it to you to figure out why this is the case. (Hint: you can use contour integration, but you don’t have to.)

It turns out that the result actually has some interesting topological implications for the SSH model, as there are fractional statistics associated with the domain wall solitons. I guess it’s not so surprising that an integral that possesses topological properties would show up in a physical system with topological characteristics! But I thought the integral was pretty amusing anyhow, so I thought I’d share it.

Aside: For those who are interested in how I arrived at this integral in the SSH model, here are some of my notes. (Sorry if there are any errors and please let me know!) Also, the idea of solitons in the SSH model actually bears a strong qualitative resemblance to the excellent zipper analogy that Brian Skinner used on his blog.

## Envisioning the Future Technological Landscape

I recently read the well-written and prescient piece entitled As We May Think by Vannevar Bush, which was published in The Atlantic magazine in July of 1945. With World War II coming to a close, and with many physicists and engineers involved in the war effort, Bush outlines what he sees as the future work of physical scientists when they return to their “day jobs”. Many of his predictions concentrate on technological advancements. Reading it today, one is struck by how visionary this article has turned out to be (though it may be argued that some of the prophesies were self-fulfilling). It should be pointed out that this article was written before the discovery of the transistor, which Bardeen and Brattain discovered in 1947.

The most stunning of his predictions to my mind were the following:

1. Personal computers
2. Miniature storage capable of holding vast amounts of data (including encyclopedias)
3. Something akin to digital photography, which he calls dry photography
4. The internet and world wide web
5. Speech recognition (though he foresaw people using this more widely than is currently used)
6. Portable or easily accessible encyclopedias with hyperlinked text
7. Keyboard- and mouse-controlled computers

Reading about how he saw the future makes it less surprising that Bush was Claude Shannon‘s thesis advisor. For those of you who don’t know, Shannon’s work gave rise to the field now known as information theory and also to the idea that one could use transistors (or binary logic/Boolean algebra) to implement numerical relationships. His ideas underpin the language of the modern computer.

It is amazing the clarity with which Bush saw the technological future. I heartily recommend the article as some eye-opening bedtime reading, if that makes sense.

## Reflecting on General Ideas

In condensed matter physics, it is easy to get lost in the details of one’s day-to-day work. It is important to sometimes take the time to reflect upon what you’ve done and learned and think about what it all means. In this spirit, below is a list of some of the most important ideas related to condensed matter physics that I picked up during my time as an undergraduate and graduate student. This is of course personal, and I hope that in time I will add to the list.

1. Relationship between measurements and correlation functions
2. Relationship between equilibrium fluctuations and non-equilibrium dissipative channels (i.e. the fluctuation-dissipation theorem)
3. Principle of entropy maximization/free-energy minimization for matter in equilibrium
4. Concept of the quasi-particle and screening
5. Concept of Berry phase and the corresponding topological and geometrical consequences
6. Broken symmetry, the Landau paradigm of phase classification and the idea of an order parameter
7. Sum rules and the corresponding constraints placed on both microscopic theories and experimental spectra
8. Bose-Einstein and Cooper Pair condensation and their spectacular properties
9. Logical independence of physical theories on the theory of everything
10. Effects of long-range vs. short-range interactions on macroscopic properties of solids
11. Role of dimensionality in observing qualitatively different physical properties and phases of matter

The first two items on the list are well-explained in Forster’s Hydrodynamics, Fluctuations, Broken Symmetry and Correlation Functions without the use of Green’s functions and other advanced theoretical techniques. Although not yet a condensed matter phenomenon, Bell’s theorem and non-locality rank among the most startling consequences of quantum mechanics that I learned in graduate school. I suspect that its influence will be observed in a condensed matter setting in due time.

## Sounding Out Krakatoa

I recently watched an interesting documentary on Krakatoa, which is what inspired this post.

The 1883 eruption of Krakatoa, a volcanic Indonesian island, was one of the largest in recorded history, killing between 30,000 – 100,000 people. Wikipedia gives a good overview of its remarkable destructive power. The sound that emanated from the eruption was perhaps the loudest in recorded history — reports suggest that sailors ruptured their eardrums, and subsequently went deaf, up to 80 miles away from the island. The eruption was heard across huge distances, from Sri Lanka to Australia (see pg 80-87). People on Rodrigues Island, close to Madagascar, were reported to have heard the eruption from across the Indian Ocean. Rodrigues Island is about 5,700 km or 3,800 miles from Krakatoa. Here is a Google map showing their separation (click to enlarge):

In this post, I intend to make a couple calculations to discuss the following:

1. The approximate sound level at Rodrigues Island
2. The maximum distance at which the volcano was probably heard
3. The possibility of acoustic circumnavigation of the world

There are three facts that are needed to discuss the above points:

1. Reports suggest that the sound level of the eruption was approximately a whopping 175 dB at 100 miles from the volcano.
2. Sound intensity falls as $1/r^2$.
3. Acoustic damping in air is generally lower for lower frequencies. (A great little applet where one can calculate the sound absorption coefficient of air can be found here.)

In addition, we can make some decent estimates by using the sound level formula:

$SL (dB) = 10*Log_{10}(I/I_0) - \alpha*r$,

where $I_0$ is $10^{-12} W/m^2$ is the threshold of human hearing, $\alpha$ is the coefficient of sound absorption in $dB/m$ and $r$ is the distance in $m$.

Here is a plot for the sound level in dB as a function of distance from the eruption site with Rodrigues Island marked by the dots (click to enlarge):

There are immediately a couple things to note:

1. I have plotted three curves: the blue curve is a calculation that does not consider any damping from air at all, the yellow curve considers a low but audible frequency sound that includes damping, while the green curve considers an infrasonic sound wave that also includes damping. (The damping coefficient was obtained from this link assuming a temperature of 20C, pressure of 1atm and humidity of 75%).
2. Close to the volcano, this calculation gives us an unrealistically large value for the sound level. It turns out that sound cannot exceed ~194dB because this is the sound level at which rarefaction of air corresponds to a vacuum. Values greater than this correspond to a shock wave.

Keeping these things in mind, the yellow curve probably is the best estimate of the sound level on Rodrigues Island, since it includes damping for an audible signal (humans can’t hear below about 20 Hz). Therefore, we can estimate that the eruption was heard with about 70 dB on Rodrigues Island! This is approximately the sound level of a noisy restaurant.

If we follow the yellow line to about the 40 dB mark, which is an approximate value where someone may still notice the sound, this would be at a distance of about 4,800 miles! This is approximately the distance from Cape Town, South Africa to Baghdad, Iraq.

The last point to address is the seven-time infrasonic global circumnavigation. It turns out that if one follows the green line on the plot out to where it reaches the 0 dB level, this would be approximately at a distance of about 11 million meters. The earth’s circumference is approximately 40 millions meters, however, and if we were to circumnavigate the world seven times, the required distance of travel would be 280 million meters. What went wrong in the calculation?

There is one major factor to consider. Very low frequency sound basically propagates with very little damping through air. For sub-Hz infrasonic sound, values for the absorption coefficient don’t seem to be very easy to find! (If you know of a database for these, please share and I’ll update this post). Let us then consider the case of no damping (the blue curve). The blue curve actually crosses the 0 dB mark at a distance of approximately 85 trillion meters. This way over-steps the mark (corresponding to circumnavigation 2.125 million times!). Even though this is a ridiculous estimate, at least it beat the 280 million meter mark, which suggests that with the right absorption coefficient, we may be in the right ballpark. A quick calculation shows that for realistic values of the absorption coefficient (about half the value of the 5Hz sound absorption coefficient), we would be very close to the 280 million meter mark (in fact, I get about 225 million meters for this absorption coefficient). This tells us that it is indeed possible for low frequency sound to circumnavigate the planet this way!

Interestingly, we can learn quite a bit concerning the sound propagation of the Krakatoa eruption using relatively simple physics.

Note: Throughout, we have neglected one very important effect — that of reflection. Anyone who has been inside an anechoic chamber will be acutely aware of the effects of sound reflection. (In an anechoic chamber, one can actually stand at different spots and hear the interference pattern when playing a sine wave from a pair of speakers). Even with this oversight, it seems like we have been able to capture the essential points, though reflection probably had a non-trivial effect on the acoustic propagation as well.