Note: the following is the fourth in a series of five blog post that I was invited to contribute to the blog ribbonfarm on the topic of particles and fields. You can go here to see the original post, which contains some discussion in the comments.
Let’s start with a big question: why does science work?
Writ large, science is the process of identifying and codifying the rules obeyed by nature. Beyond this general goal, however, science has essentially no specificity of topic. It attempts to describe natural phenomena on all scales of space, time, and complexity: from atomic nuclei to galaxy clusters to humans themselves. And the scientific enterprise has been so successful at each and every one of these scales that at this point its efficacy is essentially taken for granted.
But, by just about any a priori standard, the extent of science’s success is extremely surprising. After all, the human brain has a very limited capacity for complex thought. We human tend to think (consciously) only about simple things in simple terms, and we are quickly overwhelmed when asked to simultaneously keep track of multiple independent ideas or dependencies.
As an extreme example, consider that human thinking struggles to describe even individual atoms with real precision. How is it, then, that we can possibly have good science about things that are made up of many atoms, like magnets or tornadoes or eukaryotic cells or planets or animals? It seems like a miracle that the natural world can contain patterns and objects that lie within our understanding, because the individual constituents of those objects are usually far too complex for us to parse.
You can call this occurrence the “miracle of emergence”. I don’t know how to explain its origin. To me, it is truly one of the deepest and most wondrous realities of the universe: that simplicity continuously emerges from the teeming of the complex.
But in this post I want to try and present the nature of this miracle in one of its cleanest and most essential forms. I’m going to talk about quasiparticles.
This past week, certain portions of the internet worked themselves into a tizzy over a math problem about a crocodile.
Specifically, this problem:
As with most word problems, I’m sure that what confounded people was making the translation between a verbal description of the problem and a set of equations. The actual math problem that needs to be solved is pretty standard for a calculus class. It just comes down to finding the minimum of the function (which is one of the things that calculus is absolutely most useful for). In practical terms, that means taking the derivative and setting it equal to zero.
But it turns out that there is a more clever way to solve the problem that doesn’t require you to know any calculus or take any derivatives. It has fewer technical steps (and therefore comes with a smaller chance of screwing up your calculation somewhere along the way), but more steps of logical thinking. And it goes like this.
The problem is essentially asking you to find the path of shortest time for a crocodile moving from one point to another. If the crocodile were just walking on land, this would be easy: the quickest route is always a straight line. The tricky part is that the crocodile has to move partly on land and partly on water, and the water section is slower than the land section.
If you want to know the speed of the crocodile on land and on water, you can pretty much read them off directly from the problem statement. The problem gives you the equation . The quantity in that equation represents the path length for the on-land section, and the quantity is the path length for the water section. (That square root is the length of the hypotenuse of a triangle with side lengths and — apparently the river is 6 meters wide.) Since , this means that the on-land speed is m/s, and the water velocity is m/s (remember that the units of time were given in tenths of a second, and not seconds — not that it matters for the final answer).
That was just interpreting the problem statement. Now comes the clever part.
The trick is to realize that the problem of “find the shortest time path across two areas with different speed” is not new. It’s something that nature does continually whenever light passes from one medium to another:
I’m talking, of course, about Fermat’s principle: any time you see light go from one point to another, you can be confident that it took the shortest time path to get there. And when light goes from one medium to another one where it has a different speed, it bends. (Like in the picture above: light moves slower through the glass, so the light beam bends inward in order to cross through the glass more quickly.)
The bending of light is described by Snell’s law:
where and are the speeds in regions and , and and are the angles that the light makes with the surface normal.
Since our crocodile is solving the exact same problem as a light ray, it follows that its motion is described by the exact same equation. Which means this:
Here, m/s is the crocodile speed on land, and m/s is its speed in the river. The sine of is , and the sine of is .
So in the end the fastest path for the crocodile is the one that satisfies
If you solve that equation (square both sides and rearrange), you’ll get the correct answer: m.
So knowing some basic optics will give you a quick solution to the crocodile problems.
This happy coincidence brings to mind a great Richard Feynman quote: “Nature uses only the longest threads to weave her patterns, so each small piece of her fabric reveals the organization of the entire tapestry.” It turns out that this particular tapestry had both light rays and crocodiles in it.
By the way, this post has a very cool footnote. It turns out that ants frequently have to solve a version of this same problem: they need to make an efficient trail from the anthill to some food source, but the trail passes over different pieces of terrain that have different walking speeds.
Suppose someone told you the following:
You are invited to teach a class to a group of highly-motivated high school students. It can be about absolutely any topic, and can last for as little as 5 minutes or as long as 9 hours.
What topic would you choose for your class?
As it happens, this is not just a hypothetical question for me at the moment. In November MIT is hosting its annual MIT Splash event, and the call for volunteer teachers is almost exactly what is written in the quote above. Students, staff, and faculty from all over MIT are invited to teach short courses on a topic of their choosing, and the results are pretty wild.
A few of my favorite courses from last year:
- The History of Video Game Music
- How to Create a Language
- Cryptography for People Without a Computer
- Build a Mini Aeroponic Farm
- Calculating Pi With a Coconut
- Advanced Topics in Murder
So now I would like to turn to you, dear blog readers, for help.
What should I teach about? Please let me know, in the comments, what you think about either of these two questions:
- If you were a high school student, what kind of class would you want to go to?
- If you were in my position, what kind of class would you want to teach?
The two ideas that come to mind immediately are:
- Quantum Mechanics with middle school math
Use Algebra 1 – level math to figure out answers to questions like: What is wave/particle duality? How big is an atom? How do magnets work? What is quantum entanglement?
- The Math Behind Basketball Strategy
Learn about some of the difficult strategic decisions that basketball teams are faced with, and see how they can be described with math. Then solve a few of them yourself!
Imagining yourself as a high school student, which of those two sounds better to you? Any suggestions for alternative ideas or refinements?
UPDATE: You can find my courses listed on the MIT Splash catalog here. Thanks for all your helpful comments, everyone! This should be a lot of fun.
Note: the following is the third in a series of five blog post that I was invited to contribute to the blog ribbonfarm on the topic of particles and fields. You can go here to see the original post, which contains some discussion in the comments.
I was 12 years old when I first encountered this quote by Samuel Beckett:
“Every word is like an unnecessary stain on silence and nothingness.”
That quote impressed me quite a bit at the time. It appeared to my young self to be simultaneously profound, important, and impossible to understand. Now, nineteen years later, I’m still not sure I understand what Beckett meant by that short sentence. But I nonetheless find that its dark Zen has worked itself into me indelibly.
The Beckett quote comes to mind in particular as I sit down to write again about quantum field theory (QFT). QFT, to recap, is the science of describing particles, the most basic building blocks of matter. QFT concerns itself with how particles move, how they interact with each other, how they arise from nothingness, and how they disappear into nothingness again. As a framing idea or motif for QFT, I can’t resist presenting an adaptation of Beckett’s words as they might apply to the idea of particles and fields:
“Every particle is an unnecessary defect in a smooth and featureless field.”
Of course, it is not my intention to depress anyone with existential philosophy. But in this post I want to introduce, in a pictorial way, the idea of particles as defects. The discussion will allow me to draw some fun pictures, and also to touch on some deeper questions in physics like “what is the difference between matter and antimatter?”, “what is meant by rest mass energy?”, “what are fermions and bosons?”, and “why does the universe have matter instead of nothing?”
Let’s start by imagining that you have screwed up your zipper.
A properly functioning zipper, in the pictorial land of this blog post, looks like this:
But let’s say that your zipper has become dysfunctional, perhaps because of an overly hasty zip, and now looks more like this:
This zipper is in a fairly unhappy state. There is zipping a defect right in the middle of it: the two teeth above the letter “B” have gotten twisted around each other, and now all the zipper parts in the neighborhood of that pair are bending and bulging with stress. You could relieve all that stress, with a little work, by pulling the two B teeth back around each other.
But maybe you don’t want to fix it. You could, instead, push the two teeth labeled “A” around each other, and similarly for the two “C” teeth. Then the zipper would end up in a state like this:
Now you may notice that your zipping error looks not like one defect, but like two defects that have become separated from each other. The first defect is a spot where two upper teeth are wedged between an adjacent pair of lower teeth (this is centered more or less around the number “1”). The second defect is a spot where two lower teeth are wedged between two upper teeth (“2”).
You can continue the process of moving the defects away from each other, if you want. Just keep braiding the teeth on the outside of each defect around each other. After a long while of this process, you might end up with something that looks like this:
In this picture the two types of defects have been moved so far from each other that you can sort of forget that they came from the same place. You can now describe them independently, if you want, in terms of how hard it is to move them around and how much stress they create in the zipper. If you ever bring them back together again, though, the two defects will eliminate each other, and the zipper will be healed.
My contention in this post is that what we call particles and antiparticles are something like those zipper defects. Empty space (the vacuum) is like an unbroken zipper, with all the teeth sewed up in their proper arrangement. In this sense, empty space can be called “smooth”, or “featureless”, but it cannot really be said to have nothing in it. The zipper is in it, and with the zipper comes the potential for creating pairs of equal and opposite defects that can move about as independent objects. The potential for defects, and all that comes with them, is present in the zipper itself.
Like the zipper, the quantum fields that pervade all of space encode within themselves the potential for particles and antiparticles, and dictate the rules of how they behave. Creating those particles and antiparticles may be difficult, just as moving two teeth around each other in the zipper can be difficult, and such creation results in lots of “stress” in the field. The total amount of stress created in the field is the analog of the rest mass energy of a particle (as defined by Einstein’s famous , which says that a particle with large mass takes a lot of energy to create). Once created, the particles and antiparticles can move away from each other as independent objects, but if they ever come back together all of their energy is released, and the field is healed.
Since the point of this post is to be “picture book”, let me offer a couple more visual analogies for particles and antiparticles. While the zipper example is more or less my own invention, the following examples come from actual field theory.
Imagine now a long line of freely-swinging pendulums, all affixed to a central axle. And let’s say that you tie the ends of adjacent pendulums together with elastic bands. Perhaps something like this:
In its rest state, this field will have all its pendulums pointing downward. But consider what would happen if someone were to grab one of the pendulums in the middle of the line and flip it around the axle. This process would create two defects, or “kinks”, in the line of pendulums. One defect is a 360 degree clockwise flip around the axis, and the other is a 360 degree counterclockwise flip. Something like this:
As with the zipper, each of these kinks represents a sort of frustrating state for the field. The universe would prefer for all those pendulums to be pulled downward with gravity, but when there is a kink in the line this is impossible. Consequently, there is a large (“rest mass”) energy associated with each kink, and this energy can only be released when two opposite kinks are brought together.
(By the way, the defects in this “line of pendulums” example are an example of what we call solitons. Their motion is described by the so-called Sine-Gordon equation. You can go on YouTube and watch a number of videos of people playing with these kinds of things.)
In case you’re starting to worry that these kinds of particle-antiparticle images are only possible in one dimension, let me assuage your fears by offering one more example, this time in two dimensions. Consider a field that is made up of arrows pointing in the 2D plane. These arrows have no preferred direction that they like to point in, but each arrow likes to point in the same direction as its neighbors. In other words, there is an energetic cost to having neighboring arrows point in different directions. Consequently, the lowest energy arrangement for the field looks something like this:
If an individual arrow is wiggled slightly out of alignment with its neighbors, the situation can be righted easily by nudging it back into place. But it is possible to make big defects in the field that cannot be fixed without a painful, large-scale rearrangement. Like this vortex:
Or this configuration, which is called an antivortex:
The reason for the name antivortex is that a vortex and antivortex are in a very exact sense opposite partners to each other, meaning that they are created from the vacuum in pairs and they can destroy each other when brought together. Like this:
(A wonky note: this “field of arrows” is what one calls a vector field, as opposed to a scalar field. What I have described is known as the XY model. It will make an appearance in my next post as well.)
Now that you have some pictures, let me use them as a backdrop for some deeper and more general ideas about QFT.
The first important idea that you should remember is that in a quantum field, nothing is ever allowed to be at rest. All the pieces that make up the field are continuously jittering back and forth: the teeth of the zipper are rattling around and occasionally twisting over each other; the pendulums are swinging back and forth, and on rare occasion swinging all the way over the axle; the arrows are shivering and occasionally making spontaneous vortex-antivortex pairs. In this way the vacuum is never quiet. In fact, it is completely correct to say that from the vacuum there are always spontaneously arising particle-antiparticle pairs, although these usually annihilate each other quickly after appearing. (Which is not to say that they never make their presence felt.)
If you read the previous post, you might also notice a big difference between the way I talk about fields here and the way I talked about them before. The previous post employed much more pastoral language, going on about gentle “ripples” in an infinite quantum “mattress”. But this post uses the harsher imagery of “unhappy defects” that cannot find rest. (Perhaps you should have expected this, since the last post was “A children’s picture book”, while this one is “Samuel Beckett’s Guide”.) But the two types of language were actually chosen to reflect a fundamental dichotomy of the fields of nature.
In particular, the previous post was really a description of what we call bosonic fields (named after the great Indian physicist Satyendra Bose). A bosonic field houses quantized ripples that we call particles, but it admits no concept of antiparticles. All excitations of a bosonic field are essentially the same as all others, and these excitations can blend with each other and overlap and interfere and, generally speaking, happily coincide at the same place and the same time. Equivalently, one can say that bosonic particles are the same as bosonic antiparticles. In a bosonic field with many excitations, all particles are one merry slosh and there is literally no way of saying how many of them you have. In the language of physics, we say that for bosonic fields the particle number is “not conserved”.
The pictures presented in this post, however, were of fermionic fields (after the Italian Enrico Fermi). The particles of fermionic fields — fermions — are very different objects than bosons. For one thing, there is no ambiguity about their number: if you want to know how many you have, you just need to count how many “kinks” or “vortices” there are in your field (their number is conserved). Fermions also don’t share space well with each other – there is really no way to put two kinks or two vortices on top of each other, since they each have hard “cores”. These properties of fermions, together, imply that they are much more suitable for making solid, tangible matter than bosons are. You don’t have to worry about a bunch of fermions constantly changing their number or collapsing into a big heap. Consequently, it is only fermions that make up atoms (electrons, protons, and neutrons are all fermions), and it is only fermions that typically get referred to as matter.
Of course, bosonic fields still play an important role in nature. But they appear mostly in the form of so-called force carriers. Specifically, bosons are usually seen only when they mediate interactions between fermions. This mediation is basically a process in which some fermion slaps the sloshy sea of a bosonic field, and thereby sets a wave in motion that ends up hitting another fermion. It is in this way that our fermionic atoms get held together (or pushed apart), and fermionic matter abides.
Finally, you might be bothered by the idea that particles and antiparticles are always created together, and are therefore seemingly always on the verge of destruction. It is true, of course, that a single particle by itself is perfectly stable. But if every particle is necessarily created together with the agent of its own destruction, an antiparticle, then why isn’t any given piece of matter subject to being annihilated at any moment? Why do the solid, matter-y things that we see around us persist for so long? Why isn’t the world plagued by randomly-occurring atomic blasts?
In other words, where are all the anti-particles?
The best I can say about this question is that it is one of the biggest puzzles of modern physics. (It is often, boringly, called the “baryon asymmetry” problem; I might have called it the “random atomic bombs” problem.) To use the language of this post, we somehow ended up in a universe, or at least a neighborhood of the universe, where there are more “kinks” than “antikinks”, or more “vortices” than “antivortices”. This observation brings up a rabbit hole of deep questions. For example, does it imply that there is some asymmetry between matter and antimatter that we don’t understand? Or are we simply lucky enough to live in a suburb of the universe where one type of matter predominates over the other? How unlikely would that have to be before it seems too unlikely to swallow?
And, for that matter, are we even allowed to use the fact of our own existence as evidence for a physical law? After all, if matter were equally common as antimatter, then no one would be around to ask the question.
And perhaps Samuel Beckett would have preferred it that way.
As part of a friendly blog post exchange, I just contributed a guest post to the condensed matter physics blog This Condensed Life. My goal was to explain how to think about Cooper pairs, which are the paired states of electrons that enable superconductivity.
You can read the post here.
As a teaser, I’ll give you my preferred picture of a Cooper pair:
which I think is an upgrade over the typical illustrations.
Editor’s note: The following is a guest post contributed by Anshul Kogar. Anshul is a postdoctoral researcher in experimental condensed matter physics who currently splits his time between Argonne National Laboratory and the University of Illinois at Urbana-Champaign. He maintains the blog This Condensed Life, which discusses conceptual ideas and recent developments in condensed matter physics.
This may surprise you, but the answer to the question in the title leads to some profound quantum mechanical phenomena that fall under the umbrella of the Berry or geometric phase, the topic I’ll be addressing here. There are also classical manifestations of the same kind of effects, a prime example being the Foucault pendulum.
Let’s now return to the original question. Ultimately, whether or not one returns to oneself after walking in a closed loop depends on what you are and where you’re walking, as I’ll describe below. The geometric phase is easier to visualize classically, so let me start there. Let’s consider a boy, Raj, who is pictured below:
Now, Raj, for whatever reason, wants to walk in a rectangle (a closed loop). But he has one very strict constraint: he can’t turn/twist his body while he’s walking. So if Raj starts his walk in the top right corner of the rectangle (pictured below) and then walks forward normally, he has to start side-stepping when he reaches the bottom right corner to walk to the left. Similarly, on the left side of the rectangle, he is constrained to walk backwards, and on the top side of the rectangle, he must side-step to the right. The arrow in the diagram below is supposed to indicate the direction which Raj faces as he walks. When Raj returns to the top right corner, he ends up exactly in the same place that he did when he started — not very profound at all! But now let’s consider the case where Raj is not walking on a plane, but walking on the surface of a sphere:
Again, the arrow is supposed to indicate the direction that Raj is facing. This time, Raj starts his trek at the north pole, heads to Quito, Ecuador on the equator, then continues his walk along the equator and heads back up to the north pole. Notice that on this journey, even though Raj obeys the non-twisting constraint, he ends up facing a different direction when he returns to the north pole! Even though he has returned to the same position, something is slightly different. We call this difference anholonomy.
Why did anholonomy result in the spherical case and not the flat rectangle case? Amazingly, it turns out to be related to the hairy ball theorem (don’t ask me how it got its name). Crudely speaking, the hairy ball theorem states that if you have a ball covered with hairs, you can’t comb the hairs straight without leaving at least one little bald spot or tuft. In the image below you can see a little tuft on both the top and bottom of the sphere:
The sphere which Raj traversed had a little bald spot at the north pole, leading to the anoholonomy.
Now before moving onto the Foucault pendulum, I want to explicitly state the items that were critical in obtaining the anholonomy for the spherical case: (i) The object that is transported must have a direction (i.e. be a vector); (ii) The object must be a transported on a surface on which one cannot properly comb hair.
Now, how does the Foucault pendulum get tied up in all this? Well, the Foucault pendulum, swinging in Paris, does not oscillate in the same plane after the earth makes a full 24-hour rotation. This difference in angle between the original and the next-day plane of oscillation is also an anholonomy, except the earth rotates instead of the pendulum taking a walk. Check out this great animation from Wikipedia below:
If the pendulum was at the north (south) pole, the pendulum would come back to itself after a 24-hour rotation of the earth, (). If the pendulum was at the equator, the pendulum would not change its oscillation plane at all (). Now, depending on the latitude in the northern hemisphere where a Foucault pendulum is set up, the pendulum will make an angle between and after the earth makes its daily rotation. The north pole case, in light of the animation above, can be inferred from this image by imagining the earth rotating about its vertical axis:
The concept on anholonomy in the quantum mechanical case can actually be pictured quite similarly to the classical case described above. In quantum mechanics, we describe particles using a wavefunction, which in a very basic sense is also a vector. The vector does not exist in “real space” but what physicists refer to as “Hilbert space”. Nonetheless, the geometrical game I played with classical anholonomy can also be played in this abstract “Hilbert space”. The main difference is that the anholonomy angle becomes a phase factor in the quantum realm. The correspondence is as so:
The expression on the right is precisely the Berry/geometric phase. In the quantum case, regarding the two criteria above, we already have (i) a “vector” in the form of the wavefunction — so all we need is (ii) an appropriate surface on which hair cannot be combed straight. It turns out that in quantum mechanics, there are many ways to do this, but the most famous is undoubtedly the case of the Aharonov-Bohm effect.
In the Aharonov-Bohm experiment, one prepares a beam of electrons, splits the beam and passes them on either side of the solenoid, recombining them on the other side. A schematic of this experimental steup is shown below:
In the image, B labels the magnetic field and A is the vector potential. While many readers are probably familiar with the magnetic field, the vector potential may not be as household a concept. The vector potential was originally thought up by Maxwell, and considered to be a mathematical oddity. He realized that one could obtain B by measuring the curl of A at each point in space, but A was not given any physical meaning.
Now, it doesn’t immediately seem like this experiment would give one a geometric phase, especially considering the fact that the magnetic field outside the solenoid is zero. But let’s take look at the pattern for the vector potential outside the solenoid (top-down view):
Interestingly, the vector potential outside the solenoid looks like it would have a tuft in the center! Criterion number (ii) may therefore potentially be met. The one question left to be answered is this: can the vector potential actually “rotate” the electron wavefunction (or “vector”)? The answer to that question deserves a post to itself, and perhaps Brian or I can fill that hole in the future, but the answer seems to be emphatically in the affirmative.
The equation describing the relationship between the anholonomy angle and the vector potential is:
where is the rotation (or anholonomy) angle, the integral is over the closed loop of the electron path, and is just a proportionality constant.
The way to think about the equality is as so: is an infinitesimal “step” that the electron takes, much in the way that Raj took steps earlier. At each step the wavefunction is rotated a little compared to the previous step by an amount , dictated by the vector potential. When I add up all the little rotations caused by over the entire path of the electrons, I get the integral around the closed loop.
Now that we have the anholonomy angle, we need to use the classical quantum relation from above. This gives us a phase difference of between the electrons that go to the right and left of the solenoid. Whenever there is a non-zero phase difference, one should always be able to measure it using an interference experiment — and this is indeed the case here.
An experiment consisting of an electron beam fired at double-slit interference setup coupled with a solenoid demonstrates this interference effect most profoundly. On the setup to the left, the usual interference pattern is set up due to the path length difference of the electrons. On the setup to the right, the entire spectrum is shifted because the extra phase factor from the anholonomy angle.
Again, let me emphasize that there is no magnetic field in the region in where the electrons travel. This effect is due purely to the geometric effect of the anholonomy angle, a.k.a. the Berry phase, and the geometric effect arises in relation to the swirly tuft of hair!
So next time you’re taking a long walk, think about how much the earth has rotated while you’ve been walking and whether you really end up where you started — chances are that something’s just a little bit different.
Let’s start like this: think of some number that describes nature, or any object in it. It can be any mathematical or physical constant or measurement, in any system of units.
I predict, using my psychic powers, that you were much more likely to have thought of a number that begins with 1, 2, or 3 rather than a number that begins with 7, 8, or 9.
As it turns out, the probability is about four times higher. In fact, the probability of having a particular first digit decreases monotonically with the value the digit (1 is a more common first digit than 2, 2 is more common than 3, and so on). And the odds of you having picked a number that starts with 1 are about seven times higher than the odds of you having picked a number that starts with 9.
This funny happenstance is part of a larger observation called Benford’s law. Broadly speaking, Benford’s law says that the lower counting numbers (like 1, 2, and 3) are disproportionately likely to be the first digit of naturally-occurring numbers.
In this post I’ll talk a little bit about Benford’s law, its quantitative form, and how one can think about it.
But first, as a fun exercise, I decided to see whether Benford’s law holds for the numbers I personally tend to use and care about.
(Here I feel I must pause to acknowledge how deeply, ineluctably nerdy that last sentence reveals me to be.)
So I made a list of the physical constants that I tend to think about — or, at least, of the ones that occurred to me at the moment of making the list. These are presented below in no particular order, and with no particular theme or guarantee for completeness and non-redundancy (i.e., some of the constants on this list can be made by combining others).
After a quick look-over, it’s pretty clear that this table has a lot more numbers starting with 1 than numbers starting with 9. A histogram of first digits in this table looks like this:
Clearly, there are more small digits than large digits. (And somehow I managed to avoid any numbers that start with 4. This is perhaps revealing about me.)
As far as I can tell, there is no really satisfying proof of Benford’s law. But if you want to get some feeling for where it comes from, you can notice that those numbers on my table cover a really wide range of values: ranging in scale from (the Planck length) to (the sun’s mass). (And no doubt they would cover a wider range if I were into astronomy.) So if you wanted to put all those physical constants on a single number line, you would have to do it in logarithmic scale. Like this:
The funny thing about a logarithmic scale, though, is that it distorts the real line, giving more length to numbers beginning with lower integers. For example, here is the same line from above, zoomed in to the interval between 1 and 10:
You can see in this picture that the interval from 1 to 2 is much longer than the interval from 9 to 10. (And, just to remind you, the general rule for logarithmic scales is that the same interval separates any two numbers with the same ratio. So, for example, 1 and 3 are as far from each other as 2 and 6, or 3 and 9, or 500 and 1500.) If you were to choose a set of numbers by randomly throwing darts at a logarithmic scale, you would naturally get more 1’s and 2’s than 8’s and 9’s.
What this implies is that if you want a quantitative form for Benford’s law, you can just compare the lengths of the different intervals on the logarithmic scale. This gives:
where is the value of the first digit and is the relative abundance of that digit.
If you have a large enough data set, this quantitative form of Benford’s law tends to come through pretty clearly. For example, if you take all 335 entries from the list of physical constants provided by NIST, then you find that the abundance of different first digits is described by the formula above with pretty good quantitative accuracy:
Now, if you don’t like the image of choosing constants of nature by throwing darts at a logarithmic scale, let me suggest another way to see it: Benford’s law is what you’d get as the result of a random walk using multiplicative steps.
In the conventional random walk, the walker steps randomly to the right or left with steps of constant length, and after a long time ends up at a random position on the number line. But imagine instead that the random walker takes steps of constant multiplicative value — for example, at each “step” the walker could have his position multiplied by either 2/3 or 3/2. This would correspond to steps that appeared to have constant length on the logarithmic scale. Consequently, after many steps the walker would have a random position on the logarithmic axis, and so would be more likely to end up in one of those wider 1–2 than in the shorter 8–9 bins.
The upshot is that one way to think about Benford’s law is that the numbers we have arise from a process of multiplying many other “randomly chosen” numbers together. This multiplication naturally skews our results toward numbers that begin with low digits.
By the way, for me the notion of “randomly multiplying numbers together” immediately brings to mind the process of doing homework as an undergraduate. This inspired me to grab a random physics book off my shelf (which happened to be Tipler and Llewellyn’s Modern Physics, 3rd edition) and check the solutions to the homework problems in the back.
So the next time you find yourself trying to randomly guess answers, remember Benford’s law.