In one of this blog’s more popular posts, I made an analogy between a basketball offense and a traffic network. The rudimentary analysis involved suggested that taking the best shot each time down the court is not the same as running the most efficient possible offense. It’s an idea that caused a minor stir, I guess because people like basketball and counterintuitive phenomena.
The post did draw some criticism, though, largely for being too vague. And rightly so, I think; the post was a little bit of a tease. For example, I made this statement:
In the study of traffic, this was Braess’s Paradox: closing a road can improve traffic. In basketball, it gets called “The Ewing Theory”, and its implication is this: eliminating a scoring option can improve the efficiency of your offense.
But I never actually demonstrated the “Ewing Theory” in action. That is, I never gave a plausible example of an offense that gets better when a player is removed.
In this post I want to rectify that situation. I’ll try to diagram a particular play, and show how the effectiveness of the play might improve when one of the key players is removed. Doing so will further highlight the idea of the “price of anarchy”: that making the best choice at each step in the game is not the same as playing the most efficient possible game.
The drive-and-kick play
Consider the play diagrammed below, which is designed to produce a layup for either the point guard (which I label 
The play starts with the ball in the hands of either (
Next to each pathway is a function that is meant to show the effectiveness of that step of the play, i.e. the probability that the given step will be completed successfully. The variable 
My choice of efficiencies for each step is completely arbitrary, but I imagine the situation like this: the (
The most “resonable” solution
If you look at the play diagram above, and you start to reason about what is the best way to run the play, two things will become apparent:
- The (
- By the same logic, the (
If you buy this reasoning, which seems pretty solid, then you can cross out two edges of the network above. The (
The most straightforward method is to use the “Nash Equilibrium” solution, which goes like this: if (
The final result of this strategy is that the “drive” portion works half of the time, and the layup is successful two thirds of the time, so that the overall success rate of the play is 1/3.
Remove the center
Before I proceed, I should make it clear that in our solution above the center (
But does it? Imagine that the team loses its star center to injury, and a replacement player is used who doesn’t know how to run the play. This replacement center simply stands off to the side every time the play is run, and remains completely uninvolved.
As a result, the diagram of the play becomes something like this:
The pass part of the “drive and kick” is now completely eliminated, since the player who facilitated the pass is gone. Instead, there are now only two options for the play: a drive and layup by the (
This should be a shocking result. We removed the center from the play, thereby eliminating two potential passes and one scoring option. But the efficiency of the play went up, from 33% to 37.5%. This is Braess’s Paradox. Or, to my mind, it is a diagrammatic proof-of-concept for the infamous “Ewing Theory”.
It shouldn’t have come to this
If it seems like there’s something wrong with this result, that’s because there is. Removing elements of a network should never improve its real efficiency. So there must be something wrong with the “reasonable” solution above. Even though the logic seemed relatively unimpeachable, there is in fact a problem with the way we considered the offense one step at a time. For example, just because (
The correct way to optimize this play is to consider all the options simultaneously. That is, start by making a list of all possibilities for the play (there are six). Then write the efficiency of each possibility as a function of the frequency that it (and the other possibilities) are used. If the efficiencies of the six options are 
For this particular play, the optimum efficiency is 42.3%, well above the 33% that we got from the “reasonable” solution. It involves the (
Just to belabor the point, I should make it clear that the true optimum solution has the (
But it isn’t true. The best solution is not the most obvious solution, nor is it the one that seems most fair to the players involved.
My very first day as a graduate student was a pretty embarrassing one. I had arranged to arrive at the university in May, three months before the beginning of my semester, in order to spend the summer as a research assistant for the professor who would later become my adviser. I actually first met my adviser in the men’s room, where I had stopped on my way to his office. I couldn’t handle the uncomfortableness of introducing myself during an unfortunate first meeting in the bathroom, so I pretended not to recognize him. Of course, that strategy worked out fine for the moment, but it made the real introduction two minutes later significantly more awkward.
Anyway, the truly embarrassing part came during the discussion that followed my introduction. My adviser started asking me physics questions to see how well I understood various topics. I don’t really remember any of the questions, but I do remember that I did an absolutely horrible job of answering them. He was talking about topics I had learned — mostly classical physics, E&M, and quantum mechanics — but he talked about them in a way that seemed surprising and foreign to me. Clearly, he had a more advanced way of thinking about these things, one which allowed him to manipulate them conceptually in his mind without doing the kind of formal solutions that always accompany intro-level physics classes. At the end of our discussion he pronounced his verdict in a painfully forthright way: “Clearly you are not ready for research yet.” My adviser has never been much for tactful withholding of opinion.
The following month proceeded something like this: I would go to his office in the morning and he would give me a problem to work on. I would spend the afternoon and evening reading through my old textbooks and lecture notes and trying to piece together a reasonable solution, usually using the direct and formal mechanisms I had learned as an undergraduate. The next morning I would bring my big solution to his office and present my result (the usual evaluation: you’re kind of right, but not completely right). Then he would show me how to solve the problem with one diagram and three lines of algebra, give me another problem to work on, and the cycle would repeat.
If I have any “intuition” in physics, it was only developed through a painful and somewhat embarrassing process like this one. Sometimes I think that to be a scientist is to live in constant fear of embarrassing yourself.
Anyway, in this post I want to discuss one of my very first problems from that summer. It’s kind of fun, in the sense that the solution is clever, widely applicable, and doesn’t take a lot of specialized knowledge to understand. It’s also the problem that introduced me to one of the most important strategies in physics: solving by analogy.
The bug zapper problem
Here is the problem, as my adviser posed it to me:
Some very large volume is filled with small, randomly-diffusing objects. In the middle of the volume is a big sphere. Whenever one of the random diffusers happens to run into the sphere, it gets absorbed. If the density of diffusers far from the sphere is
, what is the rate at which diffusers are absorbed into the sphere?
Just to give you a better way of imagining the problem, here is how I re-stated the problem to myself:
A spherical bug zapper is hanging from a tree in the middle of the jungle. The jungle has some density of mosquitoes
It was only later that I discovered that this is actually sort of a famous problem: the answer is called the Smoluchowski diffusion rate. It is used for things like predicting reaction rates in living cells, where randomly-diffusing proteins need to find their target sites within the cell.
Believe it or not, the “bug zapper problem” is a problem about fields. Specifically, it is a problem about a “field” of mosquitoes which fills the space around the bug zapper. At a given distance 
I imagine the “mosquito field” to look like this:
Here, the vertical direction (and the color) represents the mosquito density as a function of position (the horizontal directions). The circle in the center of the plot is where the bug zapper is: the density of mosquitoes is zero there. The overall density
Fick’s Law
“Nature abhors a vacuum” is a pretty famous rule-of-thumb in science. It means that whenever a region of space is devoid of some material that the surrounding space is filled with, you can expect that the empty region won’t stay empty for long. For example, if you somehow moved all the air molecules in your room so that they were up against the left wall, and then you released them, you could expect that there would be a sudden rush of air from the left wall toward the empty right side. The resulting wind would probably even blow you over. This doesn’t happen because of some mysterious force that pushes air molecules to the right; it’s just the most likely outcome when there are more randomly-moving objects on one side of the room than the other.
In our case we should expect a similar phenomenon: “mosquitoes abhor a vacuum”. Whenever some portion of space has a higher mosquito density than a neighboring portion, you can expect to see an overall migration of mosquitoes from the more filled place toward the emptier place. So as the bug zapper removes mosquitoes from the air, there will be a corresponding flux of other mosquitoes radially inward.
In fact, the most general statement of this “abhorring a vacuum” tendency is called “Fick’s Law“. It gets stated like this: if 
The variable 
There is one other important law at play here, and this one is even simpler than Fick’s law. It is this: mosquitoes are not created or destroyed, except at the bug zapper surface. In other words, there are no sources of new mosquitoes anywhere in the area surrounding the zapper. Every region in space (except one that includes the bug zapper surface) has as many mosquitoes coming in as going out; there are no nearby sources of mosquito “current”. Mathematically, the way we express the idea of “sources of current” is by the divergence operator 
When I have taught physics in the past, one of the hardest parts is to convince people that intimidating expressions like 
How is a mosquito like the electric potential?
If I had been clever the first time I did this problem, I would have written down equations (1) and (2) and immediately realized that the problem was solved. That’s because equations (1) and (2) look extremely familiar to anyone who has taken an intermediate-level course on electricity and magnetism.
Equation (1) looks a whole lot like the definition of an electric field 
And equation (2) looks like Gauss’s law, which says that electric field is not created or destroyed in the absence of electric charges:
The fact that these equations are so similar is more than just a cute coincidence. In fact, the equations are exactly the same (aside from one multiplicative constant and some different variable names). This means that I can use everything I know about electricity and magnetism to help me solve the bug zapper problem. If I can just translate the bug zapper problem into an analogous electric field problem, then I can look up the answer or solve the simple electric field problem and then translate back to get the real answer to the bug zapper problem.
Let me be more concrete. The density of mosquitoes is like the electric potential: it fills all of space surrounding an electric charge (the bug zapper). The current of mosquitoes is like the electric field: it flows from areas of high potential (mosquito density) to areas of low potential. The absorbing bug zapper is like a big sphere of negative electric charge: it absorbs a current of mosquitoes (electric field). The rate at which mosquitoes get killed is like the “electric flux”, the total flow of electric field into the sphere.
So I can write the bug zapper problem as an electric field problem like this:
A metal sphere hangs in the middle of a large volume. Very far from the sphere there is some constant potential
. At the surface of the sphere the potential goes to zero. What is the flux of electric field into the sphere?
If the problem had been posed to me in this way originally, I would have solved it in 45 seconds. The tricky part was realizing that the original problem was the same thing. Now that we know the analogy, we can just solve the electric field problem and make the substitutions according to our analogy. Once we have the answer, we can replace “electric flux” with “flux of mosquitoes into the bug zapper” and the potential 
I won’t go into a lot of detail about how to solve the electric field problem. It’s actually a pretty straightforward application of Gauss’s law: if the difference in potential between the sphere surface and infinity is 
So that’s it. Now we just need to replace (electric flux) with (flux of mosquitoes) and 
Death rate of mosquitoes = 
This is the famous Smoluchowski diffusion rate.
I should admit here that, if you are reasonably adept at math, this problem probably wouldn’t have been too hard to solve even without the electric field analogy. But the analogy is still quite powerful. For example, if the object had been more complicated than a sphere, the math would have gotten hard very quickly. But the analogy suggests that there is a simple relation between the absorption rate of some particular geometric shape and its capacitance: the amount of charge that can be stored on it at a given electric potential. In fact, the most general expression for the Smoluchowski diffusion rate is
Diffusion rate = 
where 
In which I claim that all knowledge is interconnected… or something
There is sort of a frightening pressure in graduate school toward extreme specialization. After all, as a student you’re supposed to write an original thesis on some very advanced and very technical topic. What this means is that any student who wants to graduate in a reasonable amount of time ends up being hurried to learn as much as possible about their narrow field of study, at the expense of almost all other topics. The rationalization is that these other topics aren’t relevant somehow to becoming a “professional” in your thesis field.
But I am becoming an increasingly firm believer that learning any piece of physics makes you better prepared to do almost any other piece of physics. Science, after all, is an enormously complicated network of ideas. Every theory has a delicate relationship with a myriad of others. Either it relies upon them for its formulation, or it reproduces them in some limit, or it loses its validity and must yield to them in some other limit. No idea in this network is allowed to be contradictory with any other idea; they must all fit together like puzzle pieces or threads in an elaborate rug design. So if you can really understand any part of the network, it will almost certainly give you important clues for better understanding any other part.
In my more grandiose moments, I like think of myself as trying to piece together some enormous puzzle of interconnected ideas, one whose totality constitutes a full description of the universe we live in. In my less-motivated moments, I remind myself that my adviser needs me to understand things like fluid mechanics and nuclear physics in order to talk about semiconductors.
UPDATE: It occurs to me that Richard Feynman communicated this same sentiment in a much more poetic way in his lecture series The Character of Physical Law:
Nature uses only the longest threads to weave her patterns, so each small piece of her fabric reveals the organization of the entire tapestry.
I recently read Thomas Levenson’s Newton and the Counterfeiter. It wasn’t the most fast-paced book, and the message that got across to me most clearly was that 17th century London was a terrible, terrible place (apparently 2 in 3 children died before age 5). But it still had some pretty good stories about Isaac Newton.
As a young man, Newton was apparently best described as eccentric and relentlessly curious. The book recounts some of his many experiments, most of which are pretty cool and a couple of which are a little alarming. Like the time he inserted a bodkin (a long, flat needle) behind his eye and observed what effect the distortion of his eyeball shape had on his vision. (You can read the description for yourself on the right. Apparently sticking a needle behind your eye causes you to see colored circles.)
For me, though, the absolute highlight of the book was the “debtor’s ledger of sins” that Isaac Newton meticulously maintained during his college years. Newton was apparently a very god-fearing man — at least he spent a lot of time thinking and worrying about god — and as a young man he felt the need to catalog his sins as a form of penitence.
It feels a little like voyeurism, but this list is too great not to share. Here is part of it, as presented in Newton and the Counterfeiter, in Isaac Newton’s own words:
The young Isaac Newton’s “debtor’s ledger of sins”:
- Stealing cherry cobs from Eduard Storer
- Denying that I did so
- Robbing my mothers box of plums and sugar
- Calling Derothy Rose a jade
- Punching my sister
- Striking many
- Wishing death and hoping it to some
- Threating my father and mother Smith to burne them and the house over them
- Striving to cheat with a brass halfe crowne
- Making pies on Sunday night
- Squirting water on Thy day
- Not turning nearer to Thee according to my belief
- Setting my heart on money learning pleasures more than Thee
- having uncleane thoughts words and actions and dreamses
UPDATE: A more complete list has been compiled by The Newton Project, here.
It is the view of modern physics that there is no such thing as truly empty space. When I first heard this, I thought that the person saying it was some kind of crackpot. Didn’t we move past the aether theory in the 19th century? But apparently it is the honest belief of most professional physicists that what we call empty space, or “vacuum”, is really some kind of infinite, space-filling “fabric” upon which ripples can be created that carry force from one object to another. This is the idea of the quantum field. And in a sense, it’s a ridiculous idea, but it’s one that developed slowly through many painful years of puzzling over strange experimental results. The properties of the vacuum — of this strange quantum fabric — are confusing and hard to decipher. A lot of nearly-impenetrable (to me) mathematics has been created for the purpose of their description.
What makes everything so confusing is that apparently the quantum field is always “boiling”. It is filled with a certain dense energy, one which allows particles to spontaneously pop in and out of existence. In fact, our most detailed and accurate description of forces is that they involve the transfer of energy across the quantum field via short-lived excitations which can be called “particles” or “virtual particles”. It all sounds a little like black magic.
From Richard Mattuck's "Guide to Feynman Diagrams in the Many-Body Problem"
The difficulty physicists have in describing empty space is actually the subject of a running joke, which goes like this: In the days of Isaac Newton, people struggled to understand how to predict the combined motions of three interacting objects: the “three-body problem”. However, scientists were pretty sure that they had definite predictive power in the “two-body problem”. Over time, as relativity and quantum field theory were developed, we realized that the two-body and even the one-body problems were much harder than we expected, and we lost the ability to perfectly predict what was happening. Currently, we have to say that we don’t even understand the zero-body problem! We can’t say for sure anymore what “empty space” is like, so apparently we have been making negative progress over the past 300 years.
Joking aside, talking about the boiling of empty space brings up a serious question. If the space around me is filled with bubbling energy, why doesn’t it push me around?
Well, the short answer to that question is the same as the answer to “why don’t I get pushed around by the energetic air molecules surrounding me?” Namely, the air molecules are all very small and they surround me equally, so that on the whole I don’t get pushed in any particular direction.
But there is such a thing as air pressure, and it actually can actually push you around under the right conditions. So is there such a thing as “vacuum pressure”? Can something be pushed around by empty space?
The answer, surprisingly, is yes. It’s called the Casimir Effect, and it usually gets stated like this: two metal plates, sitting next to each other in an absolute vacuum, will be attracted to each other as they are pushed by the “boiling” vacuum energy.
The quantum field in one dimension
Just to make things easier to visualize, I’m going to reduce everything down to one dimension. That is, I’m going to pretend that all of space consists of a single line, along which objects can move. I imagine it like a tightly-stretched string that objects can slide across. Objects living in this string-world can exert a force on each other by making vibrations of the string, which propagate down the line and disturb the motion of others.
In a sense, this is how I think about the electric field in the real world. Consider the repulsion between electrons, which we normally say is mediated by the electric field. In our analogy, two electrons would be dwellers on the “quantum string”. As they sit on the string, the continually disturb it, sending out packets of vibration in either direction. Something like this:
As the disturbance created by each electron hits the other, the electrons end up pushing each other in opposite directions. In our analogy, the disturbance of the string is what we call the electric field.
The picture is complicated by the fact that even in the absence of electrons, the quantum string is always vibrating. In fact, to the best of our knowledge, it is vibrating quite violently, in a way that can best be described as “white noise“: all possible frequencies of oscillation are equally represented. If you imagined the quantum string to have some finite length, its disturbance in the absence of electrons (its “vacuum oscillations”) might look like this:
the indelible oscillations of the quantum field
When electrons are present, the electric field they create — the disturbances they send down the line — must travel on top of all this noise. But as long as the string is well-behaved this isn’t a problem: the extra disturbance just propagates down the line without any trouble.
One thing should bother you, though: when I say that the string vibrates with all possible frequencies, it implies that the string has infinite energy. For every mode of oscillation, there is a certain energy corresponding to that mode. The “white noise” picture above is actually a combination of many different oscillation frequencies, each with its own contribution to the energy. Like this:
Oscillations with shorter wavelength (higher frequency) have higher energy
The vacuum oscillations of our quantum string somehow have contributions from all of these modes, which means the string contains the sum of all their energies, plus infinitely more. If you think it sounds silly to say that vacuum oscillations have infinite energy, so that all of space is filled with an infinite density of energy, then I would agree with you. But that is precisely the case in our best description of the vacuum.
What happens when you block the field?
Apparently creating an electric field between two objects is a matter of disturbing the vacuum that separates them. But there are materials that can block electric field. What effect do such materials have on the vacuum itself?
Consider a slab of metal. As a rule, the electric field inside a piece of metal is zero. This is because of all the mobile charges the metal has: if electric field is applied to the interior of the metal, then all the electrons inside it are pushed around until they achieve a configuration that cancels the field. The idea behind the Casimir effect is this: maybe a piece of metal can block not just the electric field made by charges, but the vacuum oscillations themselves. Perhaps the mobile charges within the metal respond to oscillations in the vacuum in the same way that they responded to normal electric field: by rearranging themselves to cancel it out.
If that’s the case, then the vacuum between two pieces of metal should be somewhat different than the vacuum outside. Normally, the vacuum can have arbitrarily low-energy oscillations (if nothing is bounding it, it can have infinite wavelength). But between two pieces of metal, it cannot oscillate with a wavelength longer than twice the distance between the plates. So there is a certain amount of energy outside the two metal plates that is not present inside. This is the main idea behind the Casimir effect. The vacuum is oscillating with more energy outside the plates than it is inside, and as a result the plates get pushed together. It’s kind of like having an air-tight box with some of the air removed from the inside. There is a difference between the amount of energy outside the box (in the form of energetic air molecules) and the amount inside the box, so the walls of the box get pushed inward. The same thing is happening here: the surrounding “medium” (the vacuum) has more energy outside the plates than inside, so the plates get pushed together.
To make it concrete, you can think again about our one-dimensional “quantum string”, but this time we put three metal plates along the string. If you arrange them so that the distance between the left plate and the middle plate is half the distance between the middle plate and the right plate, you get a situation like this:
That extra mode on the right makes a difference. It means that there is a higher energy density on the right than there is on the left. As a result, the middle plate gets pushed to the left.
And of course, the closer the two plates on the left are to each other, the greater the difference between the energy densities. As a result, the apparent pressure pushing the plates together gets larger as the distance between them gets smaller. In 1D the pressure is proportional to 
Visualizing the Casimir effect
On Wikipedia, the Casimir effect is illustrated like this:
I guess those big, blue bubbles stand for vacuum oscillations. It’s a pretty good picture; you can see that the longer-wavelength oscillations exist outside the plates but not inside. But I think an animation is in order, where you can actually watch the torturous twitching of the vacuum. I made my own attempt at a visualization in the video below.
The two plates on the left start out slightly closer together than the ones on the right, so there is an imbalance in pressure that causes the plates to come together. If you look closely, you can see that low-frequency oscillations start getting “squeezed out” near the end of the video.
I admit, I’ve never really understood the allure of Twitter. It seems like a version of AOL instant messenger that only allows you to leave away messages.
But apparently it is convenient for some people. And so I created a Gravity and Levity Twitter account (username “gravity_levity”), which will automatically announce any blog updates. If you have questions, comments, or suggestions for future posts, you can of course send me a Twitter message and I’ll respond. Anything that encourages discussion about physics and other cool things in the universe is a plus for me. Just don’t expect inane hourly updates, endless RT-ing, or barrages of sarcastic witticisms.
A great story in physics can change the way you see the world. If physics is worth all the terrible effort (for some of us, anyway), it’s because once you’ve learned some part of it — really learned it — it changes the way you think about the universe around you. Sometimes when I’m walking down a hallway, for example, I’ll think about how both my feet and the floor they’re pressing against are really 99.9999999% empty space, but still I don’t fall through because of mysteriously strong electrical forces and an even more mysterious Pauli Exclusion Principle. Or I’ll imagine the compression and rarefraction of air molecules that comprises the music I’m listening to, and how those pressure waves get transduced via oscillating hairs in my ears into electrical signals that please my brain. Or, if I’m feeling particularly outlandish, I’ll imagine my body as an entangled mass of oscillations and “defects” propagating through a boiling, bubbling quantum sea.
I’m sure there are easier ways to stimulate your imagination than through the intensely confusing study of physics, but for someone who was born short of creativity (I had little to zero tolerance for “pretend” games as a kid), these images can be quite awe-inspiring.
The subject of the last post, I think, constitutes a great story in physics. I only recently realized how it changed the way I look at the world around me, when my wife and I went on vacation in the Caribbean. (You might want to read the last post, by the way, or this one might sound like gibberish).
A particularly pretty rock
We were walking on the beach one afternoon when my wife found this beautiful piece of coral for me:
There’s something aesthetically pleasing about this rock in general, I think. It has a nice, regular pattern, of the sort that could be made into a carpet or a wallpaper. Stare at it a little longer, though, and you’ll start to see clear “lattice lines” connecting the circular skeletal marks that once housed individual coral polyps, and regions where the marks form a surprisingly regular hexagonal lattice.
After a while I found myself tracing out the lines, and looking for the “quasiparticle” dislocations, like this one:
And here’s a place where the dislocations came as a pair — a “lower energy excitation”:
Maybe this seems like a cute analogy — something I liked just because it reminded me of something else. But there are very few true coincidences in physics, so if a piece of coral has the same sort of structure that a two-dimensional electron gas does, then they probably have some physics in common.
I don’t actually know much about coral
But I know this much: the appearance of dislocations as “quasiparticles” only requires two things to be true of the individual objects that comprise the field
- Individuals must have an effective repulsion from each other
- Individuals must have an effective attraction to the landscape they sit on
These two facts are certainly true of coral polyps, who grow as clones of their neighbors on an ever-expanding skeleton. Coral polyps repel each other because they must compete for nutrients. A coral polyp survives by sifting nutritious material from the water that washes past it, so it has a large incentive to not overlap strongly with another polyp that would steal all its food. Coral polyps have an effective attraction to the “bed” they sit on because it requires a finite amount of resources to build additional coral skeleton. Even though polyps repel each other, it would be too expensive for the organism to keep them very separated, because growing that much extra bone requires a lot of food input. So the coral tries to build up its polyps with a certain density while keeping them as far apart as possible. That how you get a hexagonal lattice, with “dislocations” as imperfections. I would imagine that the harder-pressed the coral is for survival, the more carefully it will build its lattice of polyps (like a low-temperature electron gas). A coral that is thriving and has no scarcity of food, on the other hand, will probably have a very disordered lattice (like a high-temperature electron gas).
Of course coral polyps don’t shift about the way an electron gas does, so coral “quasiparticles” don’t actually attract or repel each other in real-time. But still, the description of both coral and electrons has some common language, because the two systems share some common physics.
PS
If you stare at the coral above for long enough, you’ll see that something weird is happening to the lattice lines near the middle of the rock: they get bent as they pass near the center. This is a fairly serious defect that can’t be explained as a dislocation or any combination thereof (it’s actually called a “disclination”: a rotational shift of lattice lines). It isn’t surprising that the center was also the place where the surface was the least flat. There’s a sort of a gentle lump in the center of the coral, and this seems to encourage the formation of defects in the lattice. There is a lesson here for electron gases as well: topological or electrostatic defects in the “bed” that the electron sit on can disrupt or destroy their order, even when there is very little thermal energy to jostle them around.
Imagine, if you will, that you are an alien from some advanced and distant civilization. You find yourself fascinated by humans, whom you observe from your own planet through an ultra-high-powered telescope. As individuals, you think you know what humans are like: at least you have a sense of their characteristic size and patterns of motion. But you are puzzled by the behavior of large groups of humans. You therefore decide to make a study entitled “the properties of large, densely-packed groups of humans”. You begin your study by turning the gaze of your telescope to the biggest, densest group of humans you can find: the crowd at a football stadium.
The collection of humans inside the football stadium seems at first to be an enormous, chaotic, impossibly-complex collection of individual movements. But after a long period of observation, you see something truly remarkable: the humans begin doing “the wave”. What a startling observation this would be! From 80,000 humans packed together and moving around in a hopelessly complicated mess arises something remarkably simple: a single wave, which moves around the stadium with its own characteristic size and speed. You complete your study by observing “the wave”, writing down laws that describe its size and speed, and trying to predict when and where it will occur in the future.
The reason I tell this funny story is to make the following point: sometimes doing physics is like being an alien and watching “the wave”. In physics we examine large collections of objects, which we think we understand individually, and try to make sense of what they are doing collectively. If you’re a scientist, and you get to observe the emergence of a simple and unexpected thing from the complex group, then that’s a beautiful moment. And if you can somehow understand how it is that all the individual objects created, in concert, the simple emergent thing, then that’s a really beautiful moment. In physics we live for beautiful moments.
Unfortunately, the language physicists use to describe things is usually pretty boring (see the Calvin and Hobbes comic below), so a lot of the great stories in physics sound a lot more boring than they are. A surprisingly simple, emergent thing that arises from the collective interactions of a huge group of individual things usually gets called a “quasiparticle”. It’s a boring name that describes a fairly fascinating concept: a quasiparticle is an object that moves and responds as if it were its own independent particle, even though it is really the result of the collective motion of lots of individuals. Identifying and describing quasiparticles is one of the more fun things that physicists do.
In this post I want to tell the story of a particular kind of quasiparticle: a dislocation in a two-dimensional electron gas. It’s a particularly satisfying story, I think, because it can be understood and described without a lot of complicated math, and because it leads directly to an audacious line of imaginative further questions.
The field
A common theme among quasiparticles is that they arise from a large “field” of constantly shifting, jostling individuals. “The wave” at a football game, for example, was made from the “field” of people packed inside the stadium. For the quasiparticles I want to describe now, the field is what we call a two-dimensional electron gas. Basically, you can think of it as a bunch of charged balls (electrons) which slide around on top of a large, flat plate of opposite charge. Like this:
Cartoon version of a two-dimensional electron gas: negative electrons (blue) sit on top of a positively charged plate (red).
Individual electrons are attracted to the plate, but are repelled from each other. As a result, they try to space themselves out evenly in order to neutralize the plate. That way that there are no regions of the plate with too much positive charge (a scarcity of electrons) or too much negative charge (electrons crowded together). Of course, unless the electron gas is at absolute zero, the electrons will have some thermal energy and will jostle back and forth a little bit as they slide around on top of the positive plate. If you start cooling everything down, though, and thereby rob the electrons of their thermal motion, the electrons slowly settle into an organized and very regular configuration. They take on the form of a hexagonal (or honeycomb) lattice. Like this:
This picture, and the next few, are adapted from an image here: http://chemed.chem.purdue.edu/genchem/topicreview/bp/materials/defects3.html . Please excuse the graininess.
This honeycomb structure is the arrangement that keeps the electrons furthest separated from each other while still maintaining the density required to neutralize the positive plate. It is the perfect balance between the electrons’ attraction to the positive plate and their repulsion from each other. We call this arrangement the “ground state”; it minimizes the total energy of the system. It’s funny how something with six-fold symmetry can arise spontaneously from the spherically-symmetric force between electrons. But there it is.
You may notice that the electrons in the lattice above are arranged in orderly, parallel lines which run in three directions: horizontally, slanted to the left, and slanted to the right. Here is that same lattice, with the lines of symmetry shown in red:
The orderliness of the lattice can be characterized by how regular these “lattice lines” are: whether they maintain their spacing and orientation over long distances.
Defects in the field
Now that you know what a perfectly-ordered electron gas looks, you can ask what happens when we add back a little thermal energy. Naturally, the electrons start kicking around again and they lose their perfectly ordered structure. In other words, we start to see regions of higher- and lower- than average electron density. Something like this, for example:
If you look closely, you’ll see that the area in the center of this figure has a little hole in it. That is, the jostling of individual electrons has created a region where the electrons are more sparse than would be expected. This disrupts the lattice lines. If you try to draw them all out, you’ll find that one of the lines (going down to the right) ends right at the defect region. Like this:
This defect is called a “dislocation”. It is a spot where one of the lattice lines comes to an end. And since the electrons are all moving about, the dislocation is not fixed in space; it can move around as individual electrons shift. Imagine, for example, if the electron immediately to the right of the dislocation moved a little to the left; this would complete the broken lattice line but break the one immediately to the right, thereby moving the dislocation one space to the right. In this way the dislocation can glide around the field of electrons, just like “the wave” moves through a stadium. The random, individual motions of electrons will send the dislocation moving randomly around through the field. The dislocation is destroyed only when it comes into contact with another dislocation pointing the opposite direction, i.e. when two broken planes meet each other at their loose ends. And in fact, dislocations arise from the ground state only in opposite pairs.
Quasielectrons and quasipositrons
You can see now why a dislocation is a little bit like a real particle: it arises as part of a spontaneously-created pair and glides randomly around the field until it annihilates (peacefully) with an opposite. But the truly surprising “particleness” comes from the way in which it attracts and repels other dislocations.
A dislocation creates strain in the field around it. The electrons immediately surrounding the dislocation are highly strained: they get pushed into each other or rarified from one another in a way that costs energy. Electrons farther from the dislocation are only mildly strained, and the amount of strain in the field decays with distance. There is actually a neat way of figuring out the degree of strain in the field as a function of distance that is very similar to Gauss’s Law:
A path that would normally be closed will fail to return to itself by one "unit" if it encloses a dislocation. So the perimeter of each closed loop shares "one unit" of strain. Image taken from this paper: http://prola.aps.org/abstract/RMP/v60/i1/p161_1
I won’t go too far into the specifics, but the upshot is that the “strain field” surrounding a dislocation is very much like the electric field that surrounds a real, two-dimensional electron. When two same-direction dislocations are brought close to each other, the strain in the field becomes very large, and as a result the dislocations are pushed apart. If two opposite dislocations are brought close to each other, the strain in the field pulls them together.
It’s worth pausing here to emphasize the real point of this post: just like “the wave”, the emergence of these quasiparticles would be a surprising and beautiful thing to observe. Imagine it like this: you decide to study a complicated jumble of electrons, which you watch through your high-powered measurement devices. After observing for a while, you see something remarkable: a “defect” arises in the field of electrons, which travels around with its own characteristic size and speed. Even more astonishing, this defect behaves as if it were a real electron (call it a “quasielectron”). It repels other quasielectrons in exactly the same way that real electrons repel each other, and it attracts “quasipositrons” (opposite-facing dislocations). The two can even annihilate each other, or arise spontaneously from the field, just like real electrons and positrons do.
I should mention, as a side note, that the main features of this story would be completely unchanged if we replaced the electrons with something else. As long as we have a collection of objects that are repelled by each other and attracted to the space they sit on, the same sorts of quasiparticles will arise. There would still be a hexagonal lattice, with opposite-facing dislocations that appear spontaneously, and a “Gauss’s Law” analogy that determines how dislocations strain the field around them. In other words, the existence of “quasielectrons” and “quasipositrons” does not require the existence of real electrons and positrons.
Impudent Questions
It only takes a little bit of impudence to hear this story and want to ask the following question: what if real electrons actually are “quasiparticles” in some field that’s too small to see? What if an electron is really some kind of defect in a vast, space-filling lattice of tiny, jostling objects? What if all the matter and energy we observe in the universe is really a collection of defects and ripples propagating through the cosmic “fabric”?
If the startling similarities between real particles and quasiparticles are enough to convince you that these “cosmic fabric” ideas must be true, then that makes you a crank. But if this story doesn’t get your imagination going, then I don’t know what to tell you. Because these are the beautiful moments that physicists live for.
I like to think that a story like this one must have been the inspiration for quantum field theory, which describes all matter as a disturbance in some space-filling field. Outlandish as it may sound, it is arguably the most successful physical theory we have. I only wish that my quantum field theory class hadn’t looked like this all the time.
One of my favorite posts on this blog so far has been the one about my eight lifetimes. In it, I postulated that we measure time relative to our age, and as a result each length of time wherein our age doubles carries an equal psychological weight. There’s nothing scientific about this discussion — it’s just an idea. But it has absolutely changed the way I think about my life and the process of getting older. At the risk of being a shameless self-promoter, I highly recommend reading it.
The argument about “my eight lifetimes” can be summarized this way: in your life you will undergo roughly eight major transformations. That is, you get eight “lifetimes” during which you become a new and different person. If you’re reading this blog, chances are that you’re already on your sixth or seventh. This is not to say you have one foot in the grave: having an entire lifetime ahead of you is still a big deal. For me, a 25-year-old, the two remaining lifetimes are the transformation from a 20-something-year-old to a middle aged man, and then from middle age to an old man. Both of those time periods are a big deal, and each of them contains plenty of living to be done. I just imagine that the time period between age 6 and age 12 was a similarly big deal.
All of this brings me back to the Gompertz Law of human mortality. The Gompertz Law is already an extraordinarily fair statement: no one escapes mortality, which becomes exponentially more probable in old age. But if you subscribe to my idea of time progressing relative to itself (of life being composed of “lifetimes”), then the consequence of the Gompertz Law is an almost extreme level of fairness. Look at it this way: about 96% of people survive to age 48 (the beginning of the “eighth lifetime”), but only 4% make it to age 96 (the end of the eighth lifetime). If you try to come up with an equation for probability of survival vs. number of lifetimes lived, you get an almost absurd exponential within an exponential within an exponential. And the result looks like this:
That, in my book, is extreme fairness. Virtually all of us get to live to the end of our seventh lifetime, but almost none of us get to complete the eighth.
It may seem that this argument is somehow making light of mortality among older people. That I am claiming the difference between living to age 55 and living to age 80 doesn’t matter because it only constitutes half a “lifetime”. This is certainly not my intention. The difference between living to 55 and living to 80 is a huge deal. In my way of thinking, each of us is allotted nearly eight lifetimes before we die, and when someone is deprived of some number of those eight, it is a tragedy.
Of course, what this post really highlights is that there is nothing more tragic than death during childhood. If a 40-year-old man dies, he is deprived of the chance to live out his life and become a new kind of person. When an infant dies, he or she is deprived of seven such chances. My wife is working in a pediatric hospital these days, and the stories she tells about shaken baby syndrome are almost cripplingly sad.
I strongly suspected that I was a nerd during my early teenage years, but I wasn’t really sure of it until I took Calculus. Calculus was by far my favorite class in high school. I loved it because questions that seemed like they should be impossible to answer were suddenly made solvable. And not only were they solvable, but I could figure them out myself. It was an amazing sense of power.
To show you how big a deal this was to my adolescent self, let me give you an example of a question that the pre-calculus me would have thought was impossible while the post-calculus me could solve. And, just to be weird, I’ll phrase the question in the form of a conversation with myself.
- me #1: “I’ve been driving for the past hour. How much gas have I burned?”
- me #2: “I don’t know. My speed has been changing.”
Maybe this seems like a boring problem, but what a shock it was to me that such questions could be answered. In order to calculate how much gas you burned, you need to know how fast you were going (a graph like the one on the right would do the trick). But, during the last hour, you weren’t moving at one speed; you were moving at various times with a whole bunch of different speeds. It seems like the question should be unanswerable.
Enter Calculus, which tells you how to deal with things that are changing, even if they are changing continuously. Now you can solve the unsolvable problem. Amazing.
I’m telling you this story because a similar thing happened to me as a graduate student. A question that seemed like it should be absolutely impossible to address turned out to be solvable. Let me illustrate what kind of question it was, and how impossible it seemed, by returning to the dialogue above, this time as the post-Calculus version of myself:
- me #1: “I’ve been driving for the past hour. How much gas have I burned?”
- me #2: “Okay, I can figure that out, even though your speed has been changing. Just tell me your speed as a function of time during the last hour.”
- me #1: “I have no idea.”
Now that is a question that seems unanswerable. In order to calculate the amount of gas burned, you need to know your speed during the last hour — the “history” of your last hour of driving. But what if you don’t know that history? It seems like you would be up the proverbial creek without a proverbial paddle.
Enter a new kind of Calculus, one that allows you to make calculations that assess the present or predict the future, even when you don’t know the history leading up to it. In physics this math is known as the “path integral” approach, and more generally it’s called functional integration. It was astonishing to me in the way that Calculus had been during high school.
Now, I don’t have the qualifications to teach path integrals to a general audience. But I can give an example that will perhaps illustrate what the path integral approach is all about, and hint at its power in solving “impossible” problems.
A fun problem for you, and a matter of life and death for an ant
A (female) soldier ant is returning home from a raid on an enemy colony about 1 meter away from her own anthill. This ant, like the majority of members of many ant species, is completely blind, so finding her way home has been something of an ordeal. An even bigger danger is awaiting her this very moment, however, at the entranceway to her own home. Our soldier ant, during the course of her surprisingly violent battling, has been drenched in the pheremones of her enemy victims. If these pheremones have not worn off by now, then the sentries guarding the entranceway (who are also blind) will mistake her for an enemy. And ants are not known for their patience with enemies.
The question of “will our soldier be mistakenly killed by her own kind” depends at this moment very critically on how long it took her to walk home. The longer her path was, the better the chance that the smell of her enemies has dissipated, and the better are her odds of surviving her encounter with the sentries. If she walked at a constant speed, then we can assume the enemy pheremones dissipate exponentially (as is the case with evaporation/dissipation of chemicals on a surface). Just to make it concrete, I’ll say that her chance of being killed follows the following law:
where L is the total length of her trip home, from enemy anthill to her own. This formula implies that if our ant took a very circuitous route home — say, 20 meters long — then she has only about a 13% chance of being killed by the sentries. But if she took the most direct route — L = 1 meter — then there’s about a 90% chance of her being dismembered on the spot.
If we knew what path she took, we could use Calculus to figure out her path length L and predict what’s going to happen. But we, the observers, are only watching this moment, as the soldier returns home; we have no idea what path she took. Can we still say anything about what’s going to happen to her?
One possible path the ant may have taken, from enemy den to home
The answer, surprisingly, is yes. Usually, people use Calculus for summing (integrating) over all possible values of some quantity. The Calculus of path integrals is similar: it sums (integrates) over all possible paths (or histories) that connect one moment and another. And it allows us to predict the future of this ant. In evaluating whether our ant will live or die, we need to integrate over all possible paths that she could have walked along. Something like this:
100 possible paths the ant could have taken. (If the image isn't moving anymore, refresh your browser window to see the animation)
If we want to know her chance of living or dying, we must average over each of these separate paths, plus an infinite number of others.
I’m only going to say here that it’s possible. Usually, there’s no simple generic solution to a path integral. In most cases you either need a computer or a lot of patience to come up with a result. And you need some way of weighting the different paths, i.e. deciding how likely the different paths are. Here I am just going to use a very simple assumption: that the ant doesn’t stray more than 4 meters from the straight line connecting her origin and destination, and that she doesn’t change her direction more often than every 8 centimeters or so. These are completely arbitrary assumptions, but they allow for a finite result. I won’t go into much detail here, but feel free to ask questions about the particulars of the calculation and I’ll respond.
Here’s the surprisingly definite result: our ant’s average path length was 9.9 meters, and as a result she has a 62% chance of surviving. Hooray for her. And hooray for us, because what what we just did is amazing to me: we made a definite prediction for the future despite a complete ignorance about the past.
Electrons are like blind ants
I talked about ants in my example because I happen to think they are fascinating. But I could have talked about electrons. In fact, the kind of approach I’m talking about here is a fundamental part of modern quantum mechanics. Physicists frequently ask questions about fundamental particles that are similar to my little interior monologue at the beginning of this post:
- physicist #1: “I need to know what properties an electron will have when it gets to point B, given that it started at point A”
- physicist #2: “Okay, what path does it take to get there?”
- physicist #1: “I have no idea”
Somehow physics moves on, in the face of these impossible problems. And I guess that’s why it’s so much fun.
What do you think are the odds that you will die during the next year? Try to put a number to it — 1 in 100? 1 in 10,000? Whatever it is, it will be twice as large 8 years from now.
This startling fact was first noticed by the British actuary Benjamin Gompertz in 1825 and is now called the “Gompertz Law of human mortality.” Your probability of dying during a given year doubles every 8 years. For me, a 25-year-old American, the probability of dying during the next year is a fairly miniscule 0.03% — about 1 in 3,000. When I’m 33 it will be about 1 in 1,500, when I’m 42 it will be about 1 in 750, and so on. By the time I reach age 100 (and I do plan on it) the probability of living to 101 will only be about 50%. This is seriously fast growth — my mortality rate is increasing exponentially with age.
And if my mortality rate (the probability of dying during the next year, or during the next second, however you want to phrase it) is rising exponentially, that means that the probability of me surviving to a particular age is falling super-exponentially. Below are some statistics for mortality rates in the United States in 2005, as reported by the US Census Bureau (and displayed by Wolfram Alpha):
This data fits the Gompertz law almost perfectly, with death rates doubling every 8 years. The graph on the right also agrees with the Gompertz law, and you can see the precipitous fall in survival rates starting at age 80 or so. That decline is no joke; the sharp fall in survival rates can be expressed mathematically as an exponential within an exponential:
Exponential decay is sharp, but an exponential within an exponential is so sharp that I can say with 99.999999% certainty that no human will ever live to the age of 130. (Ignoring, of course, the upward shift in the lifetime distribution that will result from future medical advances)
Surprisingly enough, the Gompertz law holds across a large number of countries, time periods, and even different species. While the actual average lifespan changes quite a bit from country to country and from animal to animal, the same general rule that “your probability of dying doubles every X years” holds true. It’s an amazing fact, and no one understands why it’s true.
There is one important lesson, however, to be learned from Benjamin Gompertz’s mysterious observation. By looking at theories of human mortality that are clearly wrong, we can deduce that our fast-rising mortality is not the result of a dangerous environment, but of a body that has a built-in expiration date.
The lightning bolt theory
If you had never seen any mortality statistics (or known very many old people), you might subscribe to what I call the “lightning bolt theory” of mortality. In this view, death is the result of a sudden and unexpected event over which you have no control. It’s sort of an ancient Greek perspective: there are angry gods carousing carelessly overhead, and every so often they hurl a lightning bolt toward Earth, which kills you if you happen to be in the wrong place at the wrong time. These are the “lightning bolts” of disease and cancer and car accidents, things that you can escape for a long time if you’re lucky but will eventually catch up to you.
The problem with this theory is that it would produce mortality rates that are nothing like what we see. Your probability of dying during a given year would be constant, and wouldn’t increase from one year to the next. Anyone who paid attention during introductory statistics will recognize that your probability of survival to age t would follow a Poisson distribution, which means exponential decay (and not super-exponential decay).
Just to make things concrete, imagine a world where every year a “lightning bolt” gets hurled in your general direction and has a 1 in 80 chance of hitting you. Your average life span will be 80 years, just like it is in the US today, but the distribution will be very different:
Your probability of survival according to the "Lightning Bolt Theory"
What a crazy world! The average lifespan would be the same, but out of every 100 people 31 would die before age 30 and 2 of them would live to be more than 300 years old. Clearly we do not live in a world where mortality is governed by “lightning bolts”.
The accumulated lightning bolt theory
I think most people will see pretty quickly why the “lightning bolt theory” is flawed. Our bodies accumulate damage as they get older. With each misfortune our defenses are weakened — a car accident might leave me paralyzed, or a knee injury could give me arthritis, or a childhood bout with pneumonia could leave me with a compromised immune system. Maybe dying is a matter of accumulating a number of “lightning strikes”; none of them individually will do you in, but the accumulated effect leads to death. I think of it something like Monty Python’s Black Knight: the first four blows are just flesh wounds, but the fifth is the end of the line.
Your probability of survival according to the "accumulated lightning bolt" theory
Fortunately, this theory is also completely testable. And, as it turns out, completely wrong. Shown above are the results from a simulated world where “lightning bolts” of misfortune hit people on average every 16 years, and death occurs at the fifth hit. This world also has an average lifespan of 80 years (16*5 = 80), and its distribution is a little less ridiculous than the previous case. Still, it’s no Gompertz Law: look at all those 160-year-olds! You can try playing around with different “lightning strike rates” and different number of hits required for death, but nothing will reproduce the Gompertz Law. No explanation based on careless gods, no matter how plentiful or how strong their blows are, will reproduce the strong upper limit to human lifespan that we actually observe.
The cops and criminals inside your body
Like I said before, no one knows why our lifespans follow the Gompertz law. But it isn’t impossible to come up with a theoretical world that follows the same law. The following argument comes from this short paper, produced by the Theoretical Physics Institute at the University of Minnesota.
Imagine that within your body is an ongoing battle between cops and criminals. And, in general, the cops are winning. They patrol randomly through your body, and when they happen to come across a criminal he is promptly removed. The cops can always defeat a criminal they come across, unless the criminal has been allowed to sit in the same spot for a long time. A criminal that remains in one place for long enough (say, one day) can build a “fortress” which is too strong to be assailed by the police. If this happens, you die.
Lucky for you, the cops are plentiful, and on average they pass by every spot 14 times a day. The likelihood of them missing a particular spot for an entire day is given (as you’ve learned by now) by the Poisson distribution: it is a mere 
But what happens if your internal police force starts to dwindle? Suppose that as you age the police force suffers a slight reduction, so that they can only cover every spot 12 times a day. Then the probability of them missing a criminal for an entire day decreases to 
This is the Gompertz law, in cartoon form: your body is deteriorating over time at a particular rate. When its “internal policemen” are good enough to patrol every spot that might contain a criminal 14 times a day, then you have the body of a 25-year-old and a 0.03% chance of dying this year. But by the time your police force can only patrol every spot 7 times per day, you have the body of a 95-year-old with only a 2-in-3 chance of making it through the year.
More questions than answers
The example above is tantalizing. The language of “cops and criminals” lends itself very easily to a discussion of the immune system fighting infection and random mutation. Particularly heartening is the fact that rates of cancer incidence also follow the Gompertz law, doubling every 8 years or so. Maybe something in the immune system is degrading over time, becoming worse at finding and destroying mutated and potentially dangerous cells.
Unfortunately, the full complexity of human biology does not lend itself readily to cartoons about cops and criminals. There are a lot of difficult questions for anyone who tries to put together a serious theory of human aging. Who are the criminals and who are the cops that kill them? What is the “incubation time” for a criminal, and why does it give “him” enough strength to fight off the immune response? Why is the police force dwindling over time? For that matter, what kind of “clock” does your body have that measures time at all?
There have been attempts to describe DNA degradation (through the shortening of your telomeres or through methylation) as an increase in “criminals” that slowly overwhelm the body’s DNA-repair mechanisms, but nothing has come of it so far. I can only hope that someday some brilliant biologist will be charmed by the simplistic physicist’s language of cops and criminals and provide us with real insight into why we age the way we do.
UPDATE: G&L reader Michael has made a cool-looking (if slightly morbid) web calculator to evaluate the Gompertz law prediction for different ages. If you want to know what the law implies for you in particular, and are not terribly handy with a calculator, then you might want to check it out.
