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Bug zappers and electrostatics: the Smoluchowski diffusion rate

November 19, 2009

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.

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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 n_{\infty}, 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 n_{\infty}, which fly around randomly (and are not particularly attracted to the bug zapper).  Whenever a mosquito runs into the bug zapper, it is killed.  On average, how many mosquitoes are killed every minute?

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.

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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 r from the center of the bug zapper, there will be some average density n(r) of mosquitoes.  Far from the bug zapper, this density approaches n_{\infty}.  In the immediate vicinity of the bug zapper, the density of mosquitoes should be somewhat smaller, since mosquitoes in that area are being killed off.  You can also think of it like this: at some point immediately to the right of the bug zapper, there are no mosquitoes approaching from the left.  So the number of mosquitoes at this point must be somewhat reduced, since at most points there are mosquitoes approaching from all directions.

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 n(r) continuously approaches zero as you come to the surface of the bug zapper.

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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 n(r) is the concentration of randomly-moving objects at some point r, then whenever there is a concentration gradient \nabla n(r) there will be a current \bold{J} equal to

\bold{J} = -D \bold{\nabla} n(r).       (1)

The variable D here is called the “diffusion coefficient”; it describes how quickly the randomly-diffusing objects are moving and changing direction.  In three dimensions, D = v^2 t/3, where v is the average speed of the diffusing objects and t is the average time those objects move before changing their direction (say, by 90 degrees).  The minus sign in front of D is there to remind you that the current moves toward lower density.

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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 \nabla \cdot  .  So to say that “there are no sources of new mosquitoes” is to say

\bold{\nabla} \cdot \bold{J} = 0.       (2)

When I have taught physics in the past, one of the hardest parts is to convince people that intimidating expressions like \nabla \cdot \bold{A} stand for relatively simple ideas.

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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 \bold{E} based on the electric potential \phi:

\bold{E} = -\nabla \phi.

And equation (2) looks like Gauss’s law, which says that electric field is not created or destroyed in the absence of electric charges:

\nabla \cdot \bold{E} = 0.

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 \phi_{\infty}.  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 \phi with D n.

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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 \phi_{\infty}, then the charge Q on the sphere must satisfy\phi_{\infty} = Q/4 \pi \epsilon_0 R, where R is the radius of the sphere.  The electric flux into the sphere can be found simply from Gauss’s law: (electric flux) = Q/\epsilon_0 = 4 \pi R \phi_{\infty}.

So that’s it.  Now we just need to replace (electric flux) with (flux of mosquitoes) and \phi_{\infty} with D n_{\infty} and we have the answer.  It looks like this:

Death rate of mosquitoes = 4 \pi R D n_{\infty}.

This is the famous Smoluchowski diffusion rate.

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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 = C D n_{\infty}/\epsilon_0,

where C is the electrical capacitance of a piece of metal with that particular shape and size.  So now if some smart-aleck asks me “well, what if the bug zapper is a cube?”, or a cylinder, or a dodecahedron, I just need to look up already-known values for the capacitance of such objects and I’ll immediately know the answer.  This is the best form of cheating that I know: you show that the hard problem someone is asking you is identical to a hard problem that someone else has already solved, and you look up the answer.  You come away looking clever without having to do much difficult work at all.

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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.

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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.

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