## July 28, 2008

### Light Mills

#### Posted by John Baez

Vaughan Pratt asked me some questions about the physics FAQ on light mills. I’ve become quite puzzled. So, it’s time to revive the long-dormant thread on ‘gnarly issues in physics’.

A light mill is also known as a Crookes radiometer:

It seems like a simple thing: an evacuated glass bulb with some vanes that can spin around, black on one side and white on the other. When you shine light on it, it spins. Look which way it spins.

It seems the light pushes harder on the black side of each vane than on the white side. That makes sense, right? After all, light is known to carry momentum. So when light gets absorbed by the black vanes, its momentum pushes them.

This explanation was good enough to satisfy Maxwell… for a while.

But wait: black absorbs light, but white reflects it! A ball bouncing off a wall transfers twice as much momentum to the wall as a ball of equal mass and velocity that hits the walls and sticks. So, light should transfer more momentum to the white faces of the vanes, making the light mill spin the other way.

But that’s not what we see. So what’s going on?

It seems that traces of left-over air in the supposedly ‘evacuated’ bulb play a key role. If you work really hard to remove as much air as you can, the light mill doesn’t spin at all. According to the Wikipedia article:

The effect begins to be seen at partial vacuum pressures of a few mm of mercury (torr) , reaches a peak at around $10^{-2}$ torr and has disappeared by the time the vacuum reaches $10^{−6}$ torr. At these very high vacuums the effect of photon radiation pressure on the vanes can be observed in very sensitive apparatus (see Nichols radiometer) but this is insufficient to cause rotation.

(If you’re not up on your units of pressure, I’ll remind you that standard atmospheric pressure is 760 torr — enough to push mercury 760 millimeters up an evacuated tube.)

Some very good physicists have spent time pondering the light mill: notably Reynolds, Maxwell and Einstein. Here’s a quick history of their explanations, taken from the physics FAQ:

When sunlight falls on the light-mill the vanes turn with the black surfaces apparently being pushed away by the light. Crookes at first believed this demonstrated that light radiation pressure on the black vanes was turning it round just like water in a water mill. His paper reporting the device was refereed by James Clerk Maxwell who accepted the explanation Crookes gave. It seems that Maxwell was delighted to see a demonstration of the effect of radiation pressure as predicted by his theory of electromagnetism. But there is a problem with this explanation. Light falling on the black side should be absorbed, while light falling on the silver side of the vanes should be reflected. The net result is that there is twice as much radiation pressure on the metal side as on the black. In that case the mill is turning the wrong way.

When this was realised, other explanations for the radiometer effect were sought and some of the ones that people came up with are still mistakenly quoted as the correct one. It was clear that the black side would absorb heat from infrared radiation more than the silver side. This would cause the rarefied gas to be heated on the black side. The obvious explanation in that case, is that the pressure of the gas on the darker size increases with its temperature creating a higher force on that side of the vane. This force would push the rotor round. Maxwell analysed this theory carefully — presumably being wary about making a second mistake. He discovered that in fact the warmer gas would simply expand in such a way that there would be no net force from this effect, just a steady flow of heat across the vanes. So it is wrong, but even the Encyclopaedia Britannica gives this false explanation today. As a variation on this theme, it is sometimes said that the motion of the hot molecules on the black side of the vane provide the push. Again this is not correct and could only work if the mean free path between molecular collisions were as large as the container, but in fact it is typically less than a millimetre.

To understand why these common explanations are wrong, think first of a simpler set-up in which a tube of gas is kept hot at one end and cool at the other. If the gas behaves according to the ideal gas laws with isotropic pressure, it will settle into a steady state with a temperature gradient along the tube. The pressure will be the same throughout otherwise net forces would disturb the gas. The density would vary inversely to temperature along the tube. There will be a flow of heat from the hot end to the other but the force on both ends will be the same because the pressure is equal. Any mechanism you might conjecture that would give a stronger force on the hot end than on the cool end with no tangential forces along the length of the tube cannot be correct since otherwise there would be a net force on the tube with no opposite reaction. The radiometer is a little more complex but the same principle should apply. No net force can be generated by normal forces on the faces of the vanes because pressure would quickly equalise to a steady state with just a flow of heat through the gas.

Another blind alley was the theory that the heat vaporised gases dissolved in the black coating which then leaked out. This outgassing would propel the vanes round. Actually, such an effect does exist — but it is not the real explanation, as can be demonstrated by cooling the radiometer. It is found that the rotor then turns the other way. Furthermore, if the gas is pumped out to make a much higher vacuum, the vanes stop turning. This suggests that the rarefied gas is involved in the effect.

For similar reasons, the theory that the rotation is propelled by electrons dislodged by the photoelectric effect is also ruled out. One last incorrect explanation which is sometimes given is that the heating sets up convection currents with a horizontal component that turns the vanes. Sorry, wrong again! The effect cannot be explained this way.

The correct solution to the problem was provided qualitatively by Osborne Reynolds, better remembered for the “Reynolds number”. Early in 1879 Reynolds submitted a paper to the Royal Society in which he considered what he called “thermal transpiration”, and also discussed the theory of the radiometer. By “thermal transpiration” Reynolds meant the flow of gas through porous plates caused by a temperature difference on the two sides of the plates. If the gas is initially at the same pressure on the two sides, there is a flow of gas from the colder to the hotter side, resulting in a higher pressure on the hotter side if the plates cannot move. Equilibrium is reached when the ratio of pressures on either side is the square root of the ratio of absolute temperatures. This is a counterintuitive effect due to tangential forces between the gas molecules and the sides of the narrow pores in the plates […]

The vanes of a radiometer are not porous. To explain the radiometer, therefore, one must focus attention not on the faces of the vanes, but on their edges. The faster molecules from the warmer side strike the edges obliquely and impart a higher force than the colder molecules. Again these are the same thermomolecular forces that are responsible for thermal transpiration. The effect is also known as thermal creep since it causes gases to creep along a surface where there is a temperature gradient. The net movement of the vane due to the tangential forces around the edges is away from the warmer gas and towards the cooler gas with the gas passing round the edge in the opposite direction. The behaviour is just as if there were a greater force on the blackened side of the vane (which as Maxwell showed is not the case), but the explanation must be in terms of what happens not at the faces of the vanes but near their edges.

Maxwell refereed Reynolds’s paper, and so became aware of Reynolds’s suggestion. Maxwell at once made a detailed mathematical analysis of the problem, and submitted his paper, “On stresses in rarefied gases arising from inequalities of temperature”, for publication in the Philosophical Transactions; it appeared in 1879, shortly before his death. The paper gave due credit to Reynolds’s suggestion that the effect is at the edges of the vanes, but criticised Reynolds’s mathematical treatment. Reynolds’ paper had not yet appeared (it was published in 1881), and Reynolds was incensed by the fact that Maxwell’s paper had not only appeared first, but had criticised his unpublished work! Reynolds wanted his protest to be published by the Royal Society, but after Maxwell’s death this was thought to be inappropriate.

It is worth comparing the Wikipedia version of the story, which differs in significant ways. For one thing, it brings Einstein into the game. It also provides two effects that could explain the rotation, and leaves open the question which is stronger:

• Crookes incorrectly suggested that the force was due to the pressure of light. This theory was originally supported by James Clerk Maxwell who had predicted this force. This explanation is still often seen in leaflets packaged with the device. The first experiment to disprove this theory was done by Arthur Schuster in 1876, who observed that there was a force on the glass bulb of the Crookes radiometer that was in the opposite direction to the rotation of the vanes. This showed that the force turning the vanes was generated inside the radiometer. If light pressure was the cause of the rotation, then the better the vacuum in the bulb, the less air resistance to movement, and the faster the vanes should spin. In 1901, with a better vacuum pump, Pyotr Lebedev showed that in fact, the radiometer only works when there is low pressure gas in the bulb, and the vanes stay motionless in a hard vacuum. Finally, if light pressure were the motive force, the radiometer would spin in the opposite direction as the photons on the shiny side being reflected would deposit more momentum than on the black side where the photons are absorbed. The actual pressure exerted by light is far too small to move these vanes but can be measured with devices such as the Nichols radiometer.
• Another incorrect theory was that the heat on the dark side was causing the material to outgas, which pushed the radiometer around. This was effectively disproved by both Schuster’s and Lebedev’s experiments.
• A partial explanation is that gas molecules hitting the warmer side of the vane will pick up some of the heat, bouncing off the vane with increased speed. Giving the molecule this extra boost effectively means that a minute pressure is exerted on the vane. The imbalance of this effect between the warmer black side and the cooler silver side means the net pressure on the vane is equivalent to a push on the black side, and as a result the vanes spin round with the black side trailing. The problem with this idea is that while the faster moving molecules produce more force, they also do a better job of stopping other molecules from reaching the vane, so the net force on the vane should be exactly the same — the greater temperature causes a decrease in local density which results in the same force on both sides. Years after this explanation was dismissed, Albert Einstein showed that the two pressures do not cancel out exactly at the edges of the vanes because of the temperature difference there. The force predicted by Einstein would be enough to move the vanes, but not fast enough.
• The final piece of the puzzle, thermal transpiration, was theorized by Osborne Reynolds, but first published by James Clerk Maxwell in the last paper before his death in 1879. Reynolds found that if a porous plate is kept hotter on one side than the other, the interactions between gas molecules and the plates are such that gas will flow through from the cooler to the hotter side. The vanes of a typical Crookes radiometer are not porous, but the space past their edges behaves like the pores in Reynolds’s plate. On average, the gas molecules move from the cold side toward the hot side whenever the pressure ratio is less than the square root of the (absolute) temperature ratio. The pressure difference causes the vane to move cold (white) side forward.

Both Einstein’s and Reynolds’s forces appear to cause a Crookes radiometer to rotate, although it still isn’t clear which one is stronger.

If you’re not completely confused yet, let me point out something else.

Even the light mill in complete vacuum is trickier than I’ve let on so far! Following the physics FAQ, I said:

But wait: black absorbs light, but white reflects it! A ball bouncing off a wall transfers twice as much momentum to the wall as a ball of equal mass and velocity that hits the walls and sticks. So, light should transfer more momentum to the white faces of the vanes…

But as Vaughan Pratt pointed out to me, this is oversimplified! When the black faces absorb visible light, they get hot. After a while, they’ll re-radiate energy in the form of infrared light. Heat can also be conducted through the vane, but let’s ignore that and assume the vanes are perfect thermal insulators — life is complicated enough already. How does re-radiation from the black faces affect the problem?

I found a paper that tackles this question:

• M. Goldman, The radiometer revisited, Phys. Educ. 13 (1978), 427–429. (Available here to those with magic powers.)

Goldman writes:

We deal with a perfect two-vaned radiometer: there is no flow of heat through the device. The black face is taken as perfectly black and the silver face as perfectly silver… It is also assumed that the blackened vane instantly reaches an equilibrium temperature on exposure to the light beam.

Consider light incident horizontally, striking the radiometer with its vanes set at an angle $\phi$ to the direction of the light. The silvered face reflects specularly all the incident light…

For the blackened face the torque from the incident light is the same, but the light is reradiated in all directions following Lambert’s law

Given these assumptions he calculates the torque on the vanes. He then goes on to consider the case of incoming light that is not perfectly horizontal. He concludes, among other things:

It is clear, therefore, that for an ideal radiometer it is not true that the silvered side always does the pushing: it is only true when the light source is sufficiently close to the horizon and then only if the radiometer does not get trapped into simple harmonic motion. A simple calculation shows that on a typical British summer day, when the sky is a uniform grey (equally luminous all over), the torque from the black and silver faces exactly balance, so that for a perfect radiometer no motion would be possible. In fact even bright sunlight carries very little momentum: the maximum energy received is $\sim 10^3$ watts/meter${}^3$; that is a pressure of about 3 millipascals. For all reasonable specifications of radiometer this is too feeble to achieve any motion; more intense sources would have to be used.

He then goes on to tackle the far more challenging case of a light mill in an imperfect vacuum, focusing on the case where the vacuum is close to perfect, so the mean free path of the air molecules is about the size of the vessel. According to the physics FAQ, this is not the case we usually see. Goldman admits that with more gas left, the problem becomes much harder.

But here’s my main point. Did you notice something funny about the quote above?

No, not the wisecrack about ‘a typical British summer day’.

It’s the word ‘silvered’. It’s fine and dandy to study a radiometer with vanes that are silvered on side, but I’ve never actually seen one. The ones I’ve seen are always white on one side. For these we need to redo Goldman’s analysis, even for a light mill in a perfect vacuum!

Can you figure out what a light mill in a perfect vacuum should do, if its vanes are white on one side and black on the other?

And what about a more realistic light mill — the kind we actually see in stores, with an imperfect vacuum? Does anyone really understand how this thing works?

Posted at July 28, 2008 9:39 AM UTC

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### Re: Light Mills

Redoing the analysis for white/black vanes (in hard vacuum) seems simple:

Both sides are perfect Lambertian reflectors, so they radiate equally in all directions. For each photon, energy is proportional to momentum (E=hv and p=hv/c), so the momentum transferred depends only only radiated power, not on the wavelength of the light. Once the system reaches thermal equilibrium, the emitted power will exactly equal the incoming power.

So the fact that one side radiates in infrared and one in visible is irrelevant, and the mill will behave exactly as if both were white or both were black, i.e. it will not turn.

Posted by: Vilhelm S on July 30, 2008 4:54 PM | Permalink | Reply to this

### Re: Light Mills

Yay! I was getting very depressed at how nobody was trying to answer this question. Ask people what they think about quantum gravity — or a theory of everything based on the Lie group $E_8$ — and everyone has a strong opinion. But ask them what’ll happen when you shine light on a gadget with vanes that are black on one side and white and the other, in a perfect vacuum, and suddenly they’re completely quiet!

Yes, I think you’re right. A white surface with a matte finish (as opposed to a glossy one) is a Lambertian reflector. In other words, the radiant intensity of reflected light is proportional to the cosine of the angle between the line of sight and a vector perpendicular to the surface:

This precisely compensates the foreshortening of the surface as viewed from an angle, so the surface has the same apparent brightness no matter which angle you view it from!

Similarly, a perfect blackbody radiates thermally in a Lambertian way.

So, in a Crookes radiometer in a perfect vacuum, with vanes that are matte white on one side and matte black on the other, the light hitting either side of the vanes will be re-emitted in a Lambertian way. The only difference is that on the white side it’s re-emitted at different frequencies than on the black side.

Once both sides are in thermal equilibrium, all the energy coming in will be radiated back out. So, both sides will radiate the same amount of energy — hence momentum, since photons have $E = h \nu$ and $p = h \nu/c$. And, since they’re both Lambertian, this momentum will be radiated in the same angular distribution on each side. So, by symmetry, the total torque should be zero.

It should not turn!

That’s what Vaughn Pratt thinks, and that’s what I think too…

Posted by: John Baez on July 30, 2008 6:47 PM | Permalink | Reply to this

### Re: Light Mills

For the record, I suspect others, like myself, access the Cafe via the RSS comments feed and only see a new post AFTER someone makes a comment :)

Thanks for making me think. I always assumed the “bouncing ball” argument. This sounds like a good submission to Myth Busters :)

Posted by: Eric on July 30, 2008 7:09 PM | Permalink | Reply to this

### Re: Light Mills

*wandering in months later*

Yes, I agree that the momentum exchange from the photons would impart no net effect at thermal equilibrium, but if the assembly is away from equilibrium to begin with, has nonzero mass, and rotates frictionlessly, it should have a nonzero net impulse, and be rotating at constant speed either away from the white side (if it was cold) or away from the black side (if it was hot).

Posted by: Robin Z on November 26, 2008 5:09 PM | Permalink | Reply to this

### Re: Light Mills

An additional factor to take into account is the light source. If it is a uniform light field, as with ambient light (a likely case for a radiometer in the window of a toy shop) then neither side of the vane is going to change that, regardless of specularity, and the behavior will be Lambertian throughout.

Another factor is that the vane is turning steadily (when the effect is operational). Hence the vane simulates a Lambertian reflector averaged over a revolution, regardless of whether the source is collimated (provided arriving normal to the axis of rotation) or Lambertian and whether the reflector is specular or Lambertian. The case of collimated light not normal to the axis of rotation in combination with a Lambertian reflector should be the only (rotating) case asymmetric between the incident and departing rays.

Posted by: Vaughan Pratt on August 1, 2008 5:01 AM | Permalink | Reply to this

### Re: Light Mills

There’s something funny about this. Both of the accepted contributing factors — the direct transfer of heat energy to gas particles, and the increased pressure on the hotter side due to transfer of particles from the colder side — are what us physicists call ‘edge effects’, according to the Wikipedia quote above. So why doesn’t anybody sell a Super Light Mill, with the vanes full of holes?!

Posted by: Jamie Vicary on July 30, 2008 7:55 PM | Permalink | Reply to this

### Re: Light Mills

Jamie wrote:

Both of the accepted contributing factors — the direct transfer of heat energy to gas particles, and the increased pressure on the hotter side due to transfer of particles from the colder side — are what us physicists call ‘edge effects’, according to the Wikipedia quote above. So why doesn’t anybody sell a Super Light Mill, with the vanes full of holes?!

Ah, good. Having shot the usual story about the light mill in a perfect vacuum full of holes, let’s move on to the really interesting case, where there’s a bit of gas left in the bulb!

On December 13, 2006, wirelessguy wrote:

I think the Radiometer might be an excellent subject to bring up for a project, I just wouldn’t try to go answering any questions.

I recently read an article by Mike Ivsin and believe that he’s correct (well at least his questions seem to lead down the right path).

In short, he thinks that the latest theory regarding thermal transpiration happening within the Radiometer is indeed a myth. You can read more here:

I still don’t fully understand how the mill works, but if you think about it, the mill should NEVER rotate in reverse since either the black side would be pushing (heat pressure) or the white side pulling (lack of heat pressure).

Additionally, Scandurra has a patent which works on the theory that thermal creep happens around the ‘edges’ and that increasing the surface area of the edges (i.e. punching extra holes in the vanes) should speed up the mill.

I had a unit built for me (but I can’t swear to it’s accuracy). It moves significantly SLOWER than my identical reference mill, so at present I’m inclines to agree with Mike as far as thermal transpiration goes.

I’m no scientist so I’d love to hear more on the subject from people who can explain more than I can.

Unfortunately the website by Mike Ivsin has a high crackpot index. For example, he says that NASA scientists refused to test the existence of light pressure: “Over the years and into the present, government scientists publish the results of pretend physics from which they fund their wishful thinking.” He claims the bending of light that passes by the Sun is due to the solar corona — not general relativity. And, he has a book Quantum Pythagoreans that explains the truth.

However, Ivsin may be correct in his claim that a light mill put into a freezer runs backwards. I’ve never tried it myself, but the Wikipedia article claims the effect is real!

Unfortunately, this section gives no references. And, it claims that the reversed rotation is due to black-body radiation from the black sides of the vanes. Goldman’s article suggests that this radiation would be too weak.

Can anyone out there do the experiment? I had a light mill when I was a kid, but my mom threw it out.

Anyway… you should read Scandurra’s paper! He does a bunch of interesting calculations and concludes: “The force should be enhanced when the vane is perforated with a large number of small holes.”

Then he makes me skeptical by saying: “In particular nanotechnology could permit for an enhancement of the force by a factor of $10^7$ with respect to the typical order of magnitude. Such force could be observable at atmospheric pressure.” He also shows no sign of having carried out an experiment.

So, I think this issue is up for grabs.

Posted by: John Baez on July 31, 2008 8:41 AM | Permalink | Reply to this

### Off-white solar sails; Re: Light Mills

Fascinating puzzle.

“NASA scientists refused to test the existence of light pressure.” Dude, we took light pressure into account at JPL way back when I was Mission Planning Engineer on Voyager’s Uranus encounter.

See also Dr. Geoffrey Landis and K. Eric Drexler on non-white solar sail performance.

And the book that I assistant edited with the late Sir Arthur C. Clarke and the very alive Dr. David Brin, with great stuff by Bob Forward and Ray Bradbury and Larry Niven and others:

Project: Solar Sail. Edited by Arthur C. Clarke, David Brin, and Jonathan Post. Penguin Books, 1990. ISBN 0-451-45002-7.

wiki.solarsails.info/index.php?title=Project_Solar_Sail

Posted by: Jonathan Vos Post on July 31, 2008 5:27 PM | Permalink | Reply to this

### Re: Off-white solar sails; Re: Light Mills

Jonathan, guys and dudes
There is a big difference between assuming and measuring. Yes, you might have considered light’s (photon) pressure in your work but the paragraph on NASA in the ‘About Light’ section starts with:

“NASA scientists would not direct a beam of laser light against a mirror to confirm the presence or absence of light’s pressure.”

That is the issue here. Theories are okay but can anyone point to NASA or other source that actually measured light’s (laser’s) pressure ON A MIRROR?

BTW, the light mill reversal procedure is on ‘Stump Your Teacher’ page:
www.HyperFlight.com/oh-teacher.htm

Have a good one.

Posted by: Mike Ivsin on August 1, 2008 2:39 PM | Permalink | Reply to this

### Re: Off-white solar sails; Re: Light Mills

Regarding prior work measuring light pressure via mirrors, this is a standard approach, see e.g. P.N. Lebedev’s 1901 paper “Experimental examination of light pressure” or the setup described in Ditchburn’s 1952 Optics textbook. Speaking as a taxpayer I would hope that yet another verification of light pressure using mirrors is very low on NASA’s list of priorities.

Posted by: Vaughan Pratt on August 3, 2008 5:43 AM | Permalink | Reply to this

### Re: Light Mills

Instead of increasing the effect with holes, how about decreasing it by stopping the white paint a millimeter or so short of the edge? There will still be a hot-to-cold transition region with air sliding over it, but it will away from the edge and hence the torque developed by that force will be reduced to almost zero. The region of influence of the force is very unlikely to extend out a millimeter and reach the edge, and in any event will be maximal in a place where it can exert no torque.

So many different experiments have been done with variations on the original theme, many by Osborne Reynolds, that I would be most surprised if this had not already been done and simply forgotten in the meantime.

Posted by: Vaughan Pratt on August 1, 2008 5:14 AM | Permalink | Reply to this

### Re: Light Mills

Would you believe the Boston Museum of Science’s gift shop is out of radiometers? I live just down the street, so I walked over and browsed their shelves. Not finding the toy I wanted, I asked a clerk, who asked another clerk, who looked it up in their computer — “We should have sixty-one in stock” — and went off to the basement storeroom to check. No luck!

Two light mills are winging their way to me via Amazon’s two-day shipping. I hope to be sticking one in a bucket of ice water quite soon.

Posted by: Blake Stacey on August 10, 2008 5:19 PM | Permalink | Reply to this

### Re: Light Mills

I hope to be sticking one in a bucket of ice water quite soon.

I hope you try putting one in a freezer — people say that works.

My own experimental skills are so bad I’ve never been able to verify that the little light in the refrigerator actually turns off when you shut the door.

Posted by: John Baez on August 10, 2008 8:05 PM | Permalink | Reply to this

### On the balkony

One could wait for a nice and clear winter day in some northern country, go out and do the measurement.

A reference measurement could be made indoors. The illumination would be similar, only the temperature being different.

Nature would provide temperature variation during the winter, if one wants to investigate the temperature dependence.

Here, in Finland, the best time would be in February. I’d guess some physics teachers could be interested to offer this as a project.

Posted by: Kari Maijala on August 1, 2012 6:00 AM | Permalink | Reply to this

### Re: Light Mills

My name is tim weeks. I’ve been deeply confused by this light mil phenomenon business. The thing is, when the mill is new, (and the vacuum’s at it’s optional operating parameters) then new it spins in the logical way, i.e. while white is pushed away, the black absorbs. And thats how they start. Wait a few years and the vacuum looses its vacuum, well suddenly the mill spins the other way. Black reject the light and the white seems to absorb it. I’m too stupid to follow the scientific explantation but what I did notice was that no mention of the reversing factor in peoples observations. This reversal question still eludes me. My sister is the owner of the thirty year old mill. My one broke so couldn’t show anyone. When I saw my sister last year I was able to show everyone that indeed the thing had reversed it’s rotation. Everyone was stupefied because we all had one in the family. Only my sisters still exists. 30/35 years and the change is real. Also the black side is carbon cos when I broke it I had a black finger, not paint.

Posted by: timothy Weeks on June 13, 2011 3:26 PM | Permalink | Reply to this

### Re: Light Mills

Recently I got to watch one of these light mills as a salesman tried to convince me to purchase his company’s brand of “low E” windows. These windows are coated with a metallic oxide that is designed to let visible light through but to partially block both the UV and infrared. The salesman had a very bright lamp to imitate the sun, and the demonstration consisted of showing how much more slowly the light mill turned when the low E glass was interposed between it and the source. Does this effect have any bearing on the issue? What would happen in the reverse situation, if just infrared, or just UV, or a combination of those were used? Here is just about
everything
you need to know about low E windows.

Posted by: Stefan on July 31, 2008 5:34 PM | Permalink | Reply to this

### Re: Light Mills

The “issue” presumably is that low E glass blocks damaging UV and warming IR without interfering with esthetic visible light. The problem with the saleman’s demonstration interpreted naively is that it only demonstrates blocking, not selective blocking.

However when taken in conjunction with a reasonably accurate assessment of visible blocking (which should be part of his sales pitch if the pitch is to be meaningful, since the customer can at least make that assessment) it becomes a very reasonable demonstration. Were there no significant change in the scenery, yet the radiometer slowed significantly, it would demonstrate selective IR blocking.

Unfortunately it says nothing about damaging UV, which unfortunately can blow out the pigments in your valuable furnishings and paintings at a warming level too low for the radiometer to notice. Low-E done right should be good against UV anyway, but the radiometer can’t demonstrate that.

Posted by: Vaughan Pratt on August 1, 2008 5:31 AM | Permalink | Reply to this

### Re: Light Mills

I agree with John that the soft vacuum case is the (very!) interesting one. However there’s a couple more things to say about the hard vacuum case.

1. Almost all successful attempts to measure light pressure in this way have succeeded by introducing an asymmetry, namely not shining an equal amount of light on both sides of the vane, accomplished in part by not spinning it but instead measuring displacement. Once one realizes that Crookes’s idea is badly flawed, it is natural to turn to asymmetric radiometers. Simply blasting light at one side of a mirror suspended from a long thread is a straightforward approach. A more delicate one is to put a mirror at each end of a horizontal rod suspended by a thread and blast light at each mirror (in opposite directions, thus developing a torque) periodically, with a period matching that of the natural oscillation of the setup so as to gradually build up oscillation, as when pumping a swing or a laser. Even better is to blast the opposite sides of the mirrors alternately with that period so as to keep both sides equally warm, eliminating the asymmetry in whatever warming effect might be resulting from the illumination. Doing this in a hard vacuum is considered mandatory for meaningful results here, though enough symmetry should compensate for a less-than-perfect vacuum. There seem to be a number of measurements incorporating some if not all of these ideas.

2. With the requisite scroll of enchantment you can find much interesting information about radiometers in Frank Crawford’s one-page 1985 AJP paper “Running Crooke’s radiometer backwards” here.

One error Crawford makes (in my opinion) is to echo Ditchburn and Britannica in claiming that “molecules in the residual gas that drift up against a black (hotter) side of a vane therefore get a stronger kick than from a white side, and the corresponding stronger recoils drive the rotation with the black sides receding.” My take on the situation is that this supposedly stronger kick is even weaker than the edge effects of Reynolds, Maxwell, and Einstein, and that the explanation has to lie elsewhere. However he does cite what one presumes are experiments he did with a radiometer in a fridge, which made it run backwards for 5-10 minutes until equilibrium was reached. He also warmed it up in an oven, let it settle into equilibrium, then took it out and observed the same behavior as in the fridge: backwards rotation.

None of that is hard-vacuum stuff; however Crawford also cites Blum and Roller, Physics Vol 2 (Holden-Day SF 1982) who claim someone was able to demonstrate the effect with a highly evacuated “standard” Crookes’s radiometer (the toy-store kind, one hopes, not the non-spinning kind already known to do the job), but says “I have not been able to find an original reference on this.” (Anyone know what this refers to?)

This seems hard to believe given that the possibility of demonstrating the effect in a hard vacuum with a standard (spinning) Crookes’s radiometer would seem to have been completely dashed by the above discussions.

Nevertheless there still remain loopholes. In particular Crookes’s radiometer could work in a hard vacuum if the emissivity of either side is frequency dependent.

Suppose the black side is actually white at far infrared, around 1000 cm-1 (10 microns, the peak radiation of a black body at 300 K, room temperature), thus blocking reradiation on the black side. This could happen if covered with an ordinary glass cover slip (as used with microscope slides), which is transparent at 10000 cm-1 (1 micron) but I think opaque at 1000 cm-1 (certainly if as thick as a slide itself).

Suppose further that the white side is actually black at far infrared, which could happen if the white pigment were anodized on, which should be thin enough to be transparent at 1000 cm-1.

Then inbound light incident on both sides will be either reflected or reradiated, in both cases entirely from the side that looks white but is actually black at far IR. The net effect should be to drive the vane towards the black side.

Posted by: Vaughan Pratt on August 1, 2008 6:37 AM | Permalink | Reply to this

### Re: Light Mills

Brilliant! What we need is a team of top-notch experimentalists to take a break from particle physicists and try out some of your suggestions.

With any luck, you could then design three Crookes radiometers that look identical to the careless eye, but exhibit different behavior in ordinary conditions. One turns black side forwards, one turns white side forwards, and one doesn’t turn at all.

If these were made affordable, they would do a much better job of teaching physics than the darn thing has done so far.

Posted by: John Baez on August 1, 2008 9:03 AM | Permalink | Reply to this

### Re: Light Mills

It’s a long time since I wrote that article for the FAQ and it’s funny to see it being discussed again.

One thing about the article that was always lacking was any quantitative formula to back up the qualitative explanation given. What does the force on the vanes depend on? Does it depend on the area of the vanes, the perimeter, the thickness? How does it vary with pressure and temperature gradient? There should be some kind of formula. I always thought I (or someone else) would find it later and add it in, but that never happened, It’s a bit odd.

The theory that it is related to transpiration applied to the edges of the vane suggests that it would work better if the vanes had thicker edges and a longer perimeter, or even holes as suggested above, but I suspect that this is wrong otherwise they would already be made that way.

One thing I realised when I researched the article was that late 19th century physicists knew an awful lot about the kinetic theory of gases. Maxwell, Rayleigh and Crookes were real experts on it. It was the frontline of research at the time and they spent a lot of time thinking about it. I think they must have been right about their final explanation, but perhaps there are other ways of looking at it that would help us understand better.

Posted by: PhilG on August 1, 2008 11:16 AM | Permalink | Reply to this

### Re: Light Mills

Phil wrote:

It’s a long time since I wrote that article for the FAQ and it’s funny to see it being discussed again.

Hi! Long time no see!

I would like to update this FAQ, though it’s hard to know how, since so much seems to be unresolved and controversial.

At the very least, I would like to have the history include the work of Einstein and his ‘other’ explanation of how the device works in a soft vacuum.

(I put ‘other’ in quotes because even this is controversial: Scandurra says that Einstein’s explanation subsumes the thermal creep explanation — see section 3 of his paper.)

I would also like to raise Vaughan Pratt’s objections to the popular story of how the radiometer would work in a hard vacuum, were it not limited by friction. Goldman’s paper is also relevant here.

I agree that a bit more quantitative stuff would be nice, though it should probably be presented as theory, not confirmed fact.

Maybe it would also be good to include suggestions for a number of experiments! Then someone reading the FAQ might try some of these.

I’d be glad to work with you on rewriting this FAQ. I’d be even happier to have you do all the work.

Posted by: John Baez on August 1, 2008 1:51 PM | Permalink | Reply to this

### Re: Light Mills

Hi! Sounds like I am talking myself into some work.

Now that I see how Maxwell’s theory works qualitatively I would like to add something to the FAQ for that. To counter your argument that this is “just a theory” I would have to find out what experimental measurements had been made.

I’ll leave the hard vacuum case to you. :-)

Posted by: PhilG on August 1, 2008 9:05 PM | Permalink | Reply to this

### Re: Light Mills

I have just found out that if you search on google books for Maxwell’s paper “On stresses in rarefied gases arising from inequalities of temperature” you find a copy of it in his collected papers.

Maxwell gives a formula for the pressure due to variations of temperature. It is inversly proportional to density which is why the radiometer works for rarefied gasses. The pressure as (a symmetric tensor) depends on the second derivatives of the temperature. In a static situation the temperature satisifies the same equations as an electic potential. For a vane in the radiometer the temperature on the black side is higher than the ambient temperature so there is a temerature gradient out from the vane. Away from the edges the gradient is uniform so its second derivative is small. That is why they say that the force is around the edges of the vanes.

It should be possible to solve the equations given by Maxwell for some vane shapes (e.g. a disk with a hole in it) and work out the total force.

Posted by: PhilG on August 1, 2008 12:31 PM | Permalink | Reply to this

### Re: Light Mills

Why not go with the flow? Craft and polish thin vanes from unetched channel plate, blacken one side in a reducing atmosphere, etch the channels, and build the radiometer. If thin channel gas diffusion across a temp gradient is the boojum, that should be a real whizzer of a radiometer!

For smaller holes, cleverly blacken one side of Anopore filters without plugging the nm-diameter holes.

Posted by: Uncle Al on August 1, 2008 7:57 PM | Permalink | Reply to this

### Re: Light Mills

Al, your “cleverly” seems spot on. I asked Anopore, “Is color variation possible? In particular is it possible to blacken the material without blocking the pores, e.g. by anodizing or ion implantation?”

Anopore’s Andy Blackman responded as follows.

“I am not aware of any way to produce color in Anopore filters. The process that makes the filters is very similar to anodizing. You can’t anodize the filter, because the filter has already been anodized in its manufacture. Conventional “black” anodizing processes work on “wet” surfaces directly after the formation of the anodic film; the Anopore membranes are rather thoroughly dry.”

Since Andy didn’t rule out ion implantation immediately, that would be the direction I think I’d pursue: bombard the anopore in an e-beam or sputtering chamber with ions selected for their blackening effect.

An easier way of developing a temperature gradient across the anopore might be simply to put it in a much stronger temperature gradient. Although the anopore itself will act as a thermal short-circuit across the gradient, the adjacent air on each side will still be at very different temperatures, which is all you really care about here.

One possible result is that the .2 um pores may be so thin as to slow the passage of air, so you may end up trading off a higher volume of movement for a lower velocity. If the point of small pores is to maximize their surface area relative to that of the vane, there may well be some optimum pore size that maximizes the effect by restoring some of that lost velocity.

Amplifying the edge effect in this way to the point where it can be felt strongly should allow working backwards to infer the strength of the effect for a standard radiometer. I’ve been thinking about analogous ways of amplifying the effects I’ve been blaming for the radiometer’s behavior, which I now think may be practical to predict theoretically with the available data on radiative line forcing.

My prediction based on the absence of any credible confidence from any source that any of Reynold’s, Maxwell’s, or Einstein’s versions of the edge effect are strong enough to explain the radiometer inclines me to predict that the translation starving mechanism I proposed this morning will turn out to carry the day. But of course everyone is bullish on their favourite theories. :)

Posted by: Vaughan Pratt on August 4, 2008 5:51 PM | Permalink | Reply to this

### Re: Light Mills

Phil, good call going to the source. The trouble I find with Google books however is that you only get glimpses (Google is good about the glimpses but evil about the omissions, so much for “intellectual property”), so I borrowed the Physics library’s copy. (I also looked for Einstein’s 1924 paper but they only had it in German which I’m painfully rusty on. The collected translated works of Einstein seem to stop at 1920, anyone know of translations of his later works?)

The Maxwell-Reynolds shoot-out is every bit as fascinating as that between De Morgan and William Hamilton (the Edinburgh one, not the Irish physicist) 30 years earlier. The editors’ introduction to this volume of Maxwell’s works (the third and last in a series, this one titled not “Vol 3” but rather “On `Avoiding All Personal Enquiries’ of Molecules”) devotes a dozen pages to JCM’s last paper and this episode with OR.

An important change that Maxwell made to Reynold’s edge-oriented account was to suggest that sliding induced forces via roughness of the surface. Reynold’s idea was that at this pressure, mean free path (on the order of 1 mm at 60 mTorr = 8 pascals) started to approach the scale of a vane, being about 10% of a 1 cm vane. Maxwell’s stated objection to this was that Reynold’s calculations showing the relevance of this effect were unreasonably complicated, and he proceeded to obtain his second-derivative approach in its place, which others have since complained (Sutherland 1896 for example) was also quite complicated.

However Maxwell may also have objected under his breath to Reynolds’ introduction of mean free path as being an irrelevant factor, though he appears not to have raised this explicitly. As hinted at by his “no personal enquiries” philosophy alluded to in the volume’s title, Maxwell appears to have an unswerving faith in the ideal gas laws, reminiscent of Einstein’s faith in the determinacy of physics—-one imagines Maxwell’s slogan being “God does not play bowls.”

Nontrivial mean free path invalidates the ideal gas laws by the following contradiction. In any temperature gradient away from a surface (i.e. with the isothermals parallel to the surface) molecules will arrive at the surface from an expected distance from the surface that is on the order of the mean free path. At the start of that trip one supposes the molecules to be at a lower temperature than molecules closer to the surface—-those closest to the surface are assumed in the ideal gas theory to be at the same temperature as the surface, but this will not hold for molecules further out.

It is these long-distance travelers that undermine the ideal gas laws. They arrive with an expected velocity less than the closer molecules, hence a lower temperature than the surface, and hence receive a nontrivial boost in velocity from their encounter with the surface, which speeds them up in proportion to the length of their journey (for any given direction of arrival). This is an actual kick (and I take back my earlier objection to Ditchburn and Britannica, not because they’re necessarily right but because I no longer have the argument I thought I had for their being wrong). This kick is not offset by an equal and opposite kick on the cold side of the vane because it is presumably closer to the temperature of the container; instead momentum is conserved with the help of the container, the wall nearer the hot side of the vane absorbing the reacting kick.

This should be contrasted with molecules arriving at a squishy virtual surface in the interior of a gas, where they typically are scattered. At such virtual surfaces the comings and goings of molecules obey the ideal gas laws due to scattering (it is a nice exercise to show that if all collisions are head-on the gas laws are extremely badly violated, by far worse than mere constant factors). Sufficiently hard and smooth surfaces however do not scatter but rather reflect, like a mirror. (Rough surfaces are in between, acting neither as mirrors nor as virtual surfaces in the interior of the gas; unless they act like virtual surfaces roughness does not save the ideal gas laws.) This does not make any difference in the absence of a temperature gradient but does (in the case of the standard radiometer) for temperature gradients of more than say a millidegree per mean free path.

This nonzero momentum gain creates a real pressure on the vane above that in the gas near the vane. Now this is not to say that the gas can’t reorganize itself to relieve that surplus pressure, but rather that the ideal gas laws don’t show how that organization might happen. In other words one cannot dive right in and calculate the force on a vane based on this insight, but one must at least redo the dynamics in light of it. This Maxwell does not do, or even acknowledge (but then no one challenged him on this point at the time). This is the contradiction, i.e. inconsistency, I wanted to point out.

What is needed next here is to work out what the gas actually does to resolve this inconsistency. To that end I settled down with Schey’s “Div, grad, curl, and all that” to refresh my several decades old acquaintance, but I’m really rusty on it. If there’s someone on this list who’s already up to speed on “all that” please feel free to dive in with the appropriate equations describing this situation (large temperature gradient relative to the mean free path, in the vicinity of .01 degrees per mean free path or more, in the neighborhood of a hard smooth surface) so we can all play around with their solutions either empirically or analytically.

If this has already been done, e.g. for plasma physics, an obvious customer for such equations, where one can imagine encountering hundreds of degrees per mean free path, so much the better.

Posted by: Vaughan Pratt on August 3, 2008 7:34 AM | Permalink | Reply to this

### Re: Light Mills

This is fascinating, Vaughan.

One reason it’s fascinating is that your arguments take us into the same tricky realm of ‘deriving continuum equations from statistical reasoning about particles’ that Boltzmann encountered when deriving his H-theorem from his Stosszahlansatz. Perhaps some of the same confusions that afflict disputes about the ‘arrow of time’ show up in analyzing the Crookes radiometer. But perhaps in a way that can be more easily tested experimentally.

What is needed next here is to work out what the gas actually does to resolve this inconsistency. To that end I settled down with Schey’s “Div, grad, curl, and all that” to refresh my several decades old acquaintance, but I’m really rusty on it.

It actually sounds like you — and me too, but I don’t want to commit myself to anything — need to read about the kinetic theory of gases: stuff that picks up where the Boltzmann equation leaves off by providing a specific formula for the collision term, and also a description of how molecules bounce off the vanes.

For starters, I think it would be very good for us to learn about the Chapman–Enskog equation.

Posted by: John Baez on August 3, 2008 9:41 AM | Permalink | Reply to this

### Re: Light Mills

One advantage we have today that the greats did not have back then is that we can write and implement computer simulations on large scales fairly simply. It doesn’t seem to be that difficult to simulate the thermo-mechanical-electromagnetic dynamics. If I had time, which I don’t, I’d do it myself.

Posted by: Eric on August 3, 2008 10:54 PM | Permalink | Reply to this

### Re: Light Mills

I had a terrible time getting to sleep last night (and for that matter staying asleep) because my brain refused to stop thinking about the problem.

In the morning I hoped I could write down the equations describing what was going on at the surface of the radiometer vane, with conservation of momentum and energy being the only two conservation laws I thought the problem ought to need. But they kept not making sense.

Suddenly it dawned on me: no wind! Molecules are busily diffusing all over the place, but none of them are blowing steadily in one direction, at least not in the manner of an ordinary wind. This means that with all the diffusion there is a third conservation principle: conservation of mass across any boundary.

This supplies the missing equation. I will show that when the mean free path is significant, the ratio of “purposeful” motion to Brownian motion rises to a level where a new law kicks in: temperature is inversely proportional to the square of the pressure.

Before proving this, let me convince you it’s true with a simple little experiment (well, it wasn’t simple to do but interpreting the results is simple). Plot the pressure and temperature of the earth’s atmosphere. This is already done for you in the nice chart at http://en.wikipedia.org/wiki/Image:Atmosphere_model.png from the Wikipedia article on Earth’s atmosphere. The critical point seems to be at an altitude of around 70-80 km, the onset of the thermosphere (appropriately named as we’ll see in a moment).

Below this point the temperature of the atmosphere approximates the effective temperature of the Earth, 248 K. (This is most readily obtained as follows. Start with the Earth’s albedo of .367, subtract that from 1 giving .633, take the square root of that to get .7956, multiply that by the radius .6955 megameters of the Sun giving .5533, divide that by the diameter 149.6 megameters of the Earth’s orbit giving .00185, take the square root of that to get .043, and multiply that by the Sun’s temperature of 5778 K to get 248.5 K. The often-seen figure of 255 K is due entirely to using the poor approximation of .3 for the albedo as you can check for yourself by using it in the above calculation, which is identically equivalent to other more complicated formulas for effective temperature of a planet.)

At sea level the atmosphere is warmed by the Earth’s molten core and perhaps other effects such as tidal friction. At 10 km those contributions vanish and the effective temperature is reached and even overshot slightly (one plausible explanation sometimes given being that vertical diffusion has to overcome gravity). Then it heats up again at 50 km, presumably because this is the region that burns off the incoming meteoric detritus and anthropogenic space junk (the latter less significant than the former to date). Above that it goes back to the effective temperature.

But then at 70 km it dives steeply below the effective temperature. I attribute this to the mean free path there. At sea level this is 70 nm, essentially a crowded cocktail party where even with a lot of energy it takes time to work your way across the room. At 70 km the pressure has dropped four orders of magnitude, bringing the mean free path up to .7 mm, readily visible to the naked eye and over a million times larger than an air molecule. The hitherto-confined kinetically dancing 244 K molecules can now start to stretch their legs, darting between levels of significantly different pressure (relative to before).

This new freedom is a release for them that has the effect of dropping the pressure more sharply than before. I’m guessing that at this level this sudden escape initially drops the temperature, much as CO2 escaping from a cylinder becomes dry ice.

But as the pressure continues to drop, long-distance travel between collisions starts to really dominate, at which point a new effect kicks in: temperature rises quadratically with decreasing pressure. You can see this effect very graphically in the right half of the graph, where the slope of the green line is very close to twice that of the blue line!

With the rapidly rising temperature the precipitous drop in pressure resulting from this new-found freedom of the molecules is offset by the pressure created by this temperature, which brings back some of the cocktail party atmosphere, but now more in the manner of an aerial dogfight than a living room as the molecules furiously crash into each other at much higher speed than at sea level.

Ok, so how does T grow as 1/P2? (Actually mP2/4dR to be precise where m = .02897 kg, the molar mass of air, d is number of degrees of freedom of an average air molecule, and R is the gas constant.) We can see this as follows.

Claim 1. The mass rate of flow of gas across any collision-free surface of unit area is P/2v, pressure divided by twice the velocity. This is because P = 2mv at such a surface, proved by making the surface solid and noting that the pressure must be double the rate mv of momentum bouncing off that surface (double because of the reversal).

By conservation of matter throughout the atmosphere, P/2v must be the same over the entire range of this effect, from 80 km onwards. This is the crucial point I’d been missing before. (Although matter is also conserved below this level, v is relatively independent of P, being as constant as T, which holds fairly steady when descending to ground while P increases under the increasing burden of the atmosphere.)

At this point we have an obvious inconsistency between the ideal gas law PV = nRT and the more fundamental E = dRT/2 where E is the energy of a mole of gas of temperature T and d is the number of degrees of freedom of one molecule of that gas. Guess which one loses? The ideal gas law only holds in the limit, and with molecules jumping more than a millimeter between collisions and the pressure dropping by a factor of two every five kilometers we’re now very far from that limit.

Now E = 1/2 mv2, everywhere including the center of the earth. Eliminating E gives T = mv2/dR = mP2/4dR, definitely not at the center of the earth, but throughout the thermosphere and beyond. QED

Claim 2. Crookes’s radiometer works essentially like the earth’s atmosphere, despite being on so vastly smaller a scale.

If you bought Claim 1, the proof of Claim 2 should be pretty obvious. With the constants all clearcut from the formulas (assuming I didn’t forget one) it should be straightforward to calculate radiometer behavior analytically and check it against the known radiometer behavior, known for the past 135 years (more than twice my age).

Now this theory may well be all hogwash, since I just thought of it, and everything prior to it that I “just thought of” turned out not to work. However I must say I find very reassuring that in the thermosphere the temperature is found empirically to slope upwards at twice the rate the pressure is falling (on its log-linear scale).

If it holds up then a huge advantage of this explanation over those of Reynolds, Maxwell, and Einstein is that it can be derived in a high school physics class, with more students following than for some of the other laws. The burgeoning interest in climatology these days should make this eminently accessible phenomenon of strong interest to the general public.

(In 1876, at the height of the radiometer excitement, Maxwell was summoned by Queen Victoria to explain it to her. His letter to Robert Cay of 15 May 1876 indicates that he was noncommittal about the explanation. With a simple explanation he would presumably not have needed to be so coy.)

Incidentally this relationship at the top of any astronomical body’s atmosphere might provide a more satisfactory explanation of the long-standing mystery of the excess heat of the Sun’s corona than the (very new) explanation at http://www.sciencedaily.com/releases/2008/05/080529120710.htm , which has something of the flavor of the Reynold’s-Maxwell-Einstein explanation of the radiometer effect: “what else could it possibly be, all other explanations have been ruled out.” In this case it might be that both effects are at work; if they were equally strong that would be particularly interesting.

Posted by: Vaughan Pratt on August 3, 2008 10:59 PM | Permalink | Reply to this

### Re: Light Mills

Before I got to the meat of this, I spotted this:

At sea level the atmosphere is warmed by the Earth’s molten core and perhaps other effects such as tidal friction.

This sounds extraordinarily wrong to me. At sea level, the atmosphere is warmed by the ground, which is heated by the sun. Notice the way it gets colder outside at night? And the way it’s cooler in the shade? Those are clues …

In a deep mineshaft, geological heat makes a bigger contribution than sunlight, but that’s not relevant to the atmosphere generally.

I’m pretty sure tidal friction makes a completely negligible contribution to heating. You need a lot of friction to get significant heating. I don’t think the Thames feels cooler at slack water than it does in flood …. And windy days don’t feel exceptionally warm.

Posted by: Tim Silverman on August 4, 2008 8:31 PM | Permalink | Reply to this

### Re: Light Mills

Tim is right. The Earth’s surface is mainly warm because of the Sun, the way the Earth’s surface absorbs light and re-radiates infrared, and the way the atmosphere is less transparent to infrared light than visible light. Tidal friction and geothermal heating are irrelevant except as tiny corrections.

Vaughan wrote:

Then it heats up again at 50 km, presumably because this is the region that burns off the incoming meteoric detritus and anthropogenic space junk (the latter less significant than the former to date).

That also sounds wrong to me. I bet the main source of energy is the Sun: visible light, and further up also things like ultraviolet, X-rays, and maybe radio waves and the solar wind. The Earth’s magnetic field is also relevant (cf aurora borealis), but I’m not sure how much energy gets transferred from this to the ionized upper atmosphere.

Posted by: John Baez on August 4, 2008 9:46 PM | Permalink | Reply to this

### Re: Light Mills

Hmm. I too have known those frenzied, sleepless nights, turning a problem over and over in my mind, trying out solutions, but, as Wordsworth almost said, physics is motion recalculated in tranquility. For the benefit of the confused, let’s redo some of those calculations carefully.

We’ll assume all molecules have the same mass $m$. Let the gas have a mass density $\rho$ (hence a number density $\frac{\rho}{m}$). They’re striking the surface from all directions, with a Maxwell-Boltzmann distribution of velocities.

In a time $\delta t$, molecules moving at an angle $\theta$ to the normal to our surface will move closer to the surface by a distance $v cos\theta \delta t$. So over an area $A$, a volume $A v cos\theta \delta t$ will have its molecules strike the surface. If all the molecules were moving at speed $v$, that would be a total of $\frac{\rho}{m}A v cos\theta \delta t$ molecules. Each will transfer a momentum $2 m v cos\theta$. So the pressure due to molecules in these circumstances would be

$\frac{\rho}{m}v cos\theta \cdot 2 m v cos\theta$

$=2\rho v^2 cos^2\theta$.

However, the molecules don’t all have the same speed. They are distributed in velocity space with the probability distribution

$\left(\frac{m}{2\pi k T}\right)^{\frac{3}{2}}e^{-\frac{m v^2}{2k T}}v^2 sin\theta dv d\theta d\phi$

Let’s temporarily use $a$ to denote the quantity $\frac{m}{kT}$. So the Maxwell-Boltzmann distribution is

$\left(\frac{a}{2\pi}\right)^{\frac{3}{2}}e^{-\frac{a v^2}{2}}v^2 sin\theta dv d\theta d\phi$

Integrating our pressure contribution, we get

$\int_0^{2\pi}\int_0^{\frac{\pi}{2}}\int_0^\infty 2\rho v^2 cos^2\theta\left(\frac{a}{2\pi}\right)^{\frac{3}{2}}e^{-\frac{a v^2}{2}}v^2 sin\theta dv d\theta d\phi$

$=2\rho \left(\frac{a}{2\pi}\right)^{\frac{3}{2}}\left(\int_0^\infty v^4 e^{-\frac{a v^2}{2}}dv\right)\left(\int_0^{\frac{\pi}{2}}cos^2\theta sin\theta d\theta\right)\left(\int_0^{2\pi}d\phi\right)$

$=2\rho \frac{a}{2\pi}\sqrt{\frac{a}{2\pi}}\left(\int_0^\infty v^4 e^{-\frac{a v^2}{2}}dv\right)\Big[\frac{-1}{3}cos^3\theta\Big]_0^{\frac{\pi}{2}}[\phi]_0^{2\pi}$

$=2\rho \frac{a}{2\pi}\sqrt{\frac{a}{2\pi}}\left(\int_0^\infty v^4 e^{-\frac{a v^2}{2}}dv\right)\frac{1}{3}2\pi$

$=\frac{2}{3}\rho a\sqrt{\frac{a}{2\pi}}\left(\int_0^\infty v^4 e^{-\frac{a v^2}{2}}dv\right)$

(integrating by parts:)

$=\frac{2}{3}\rho a\sqrt{\frac{a}{2\pi}}\left(\bigg[\frac{-1}{a}v^3 e^{-\frac{a v^2}{2}}\bigg]_0^\infty+\frac{1}{a}\int_0^\infty 3 v^2 e^{-\frac{a v^2}{2}}dv\right)$

$=\frac{2}{3}\rho\sqrt{\frac{a}{2\pi}}\int_0^\infty 3 v^2 e^{-\frac{a v^2}{2}}dv$

(integrating by parts again:)

$=2\rho\sqrt{\frac{a}{2\pi}}\left(\bigg[\frac{-1}{a}v e^{-\frac{a v^2}{2}}\bigg]_0^\infty+\frac{1}{a}\int_0^\infty e^{-\frac{a v^2}{2}}dv\right)$

$=2\rho\sqrt{\frac{1}{2\pi a}}\int_0^\infty e^{-\frac{a v^2}{2}}dv$

$=2\rho\sqrt{\frac{1}{2\pi a}}\frac{1}{2}\int_{-\infty}^\infty e^{-\frac{a v^2}{2}}dv$

$=2\rho\sqrt{\frac{1}{2\pi a}}\frac{1}{2}\sqrt{\frac{2\pi}{a}}$

$=\frac{\rho}{a}$

Recalling that $a=\frac{m}{k T}$, we have

$P=\frac{\rho}{m}k T$

Now, $\frac{\rho}{m}$ is the number density. So denoting total volume by $V$, number of moles by $n$, and Avogadro’s number by $N_A$, we have $P=\frac{n N_A}{V}k T$. Since $N_A k=R$, we get

$PV=n R T$.

The moral of which is:

Effects like this aren’t simple

For the phenomena at hand, the mean free path is almost certainly going to enter in via non-equilibrium fluxes.

The atmosphere is complicated. Radiative transfer, incoming and outgoing fluxes, conduction, hydrodynamics, variable composition, phase transitions … or, to say the same thing less mathematically, wind, sunshine, clouds and rain. The thermal structure of the atmosphere is not really an elementary physics problem, though one can find elementary physics problems in there.

Posted by: Tim Silverman on August 4, 2008 11:07 PM | Permalink | Reply to this

### Re: Light Mills

Correction to my previous message: the diameter of the Earth’s orbit is 299.2 megameters (within .3% of 300). The figure I gave of 149.6 is the radius. The other numbers are all correct, in particular the quotient of .5533 by the diameter is indeed .00185.

The more easily remembered approximations of .37 for albedo, .7 for the Sun’s radius, 300 for the diameter of the Earth’s orbit, and 5800 K for the temperature of the Sun, give 249.6 K, a lot closer to the correct answer of 248.5 K than the oft-mentioned 255 K.

A Latex’d version of the formula I used is at http://en.wikipedia.org/wiki/Talk:Effective_temperature . Its content is simply that the fourth power of the ratio of the planet’s effective temperature to that of the illuminating star equals the fraction of the sky (all 4π steradians of it) occupied by the Sun, corrected for the planet’s albedo.

Posted by: Vaughan Pratt on August 4, 2008 4:04 AM | Permalink | Reply to this

### Re: Light Mills

I’m afraid this theory too is hogwash. I just realized that I was treating the temperature on the plot of Earth’s pressure and temperature as logarithmic, justifying the comparison with pressure. Unfortunately for my theory it is linear, making the fact that the slope is twice that of the pressure immaterial. The reason it is twice is the scientifically irrelevant one that the temperature scale was chosen to make the temperature fit on the same plot.

That temperature is linear with altitude in the thermosphere is very interesting, but unfortunately the proportionality of P and v does not predict it. Empirically P decreases exponentially with increasing T, but I have no explanation.

This relationship does however strongly violate the ideal gas law PV = nRT, so at least my point about it being violated remains valid.

Posted by: Vaughan Pratt on August 4, 2008 5:08 AM | Permalink | Reply to this

### Re: Light Mills

I was so sold on the slope argument that I did not check the algebra carefully enough. Another error (much more minor): only 3 of the d degrees of freedom of a molecule contribute to its velocity. Hence T = mv2/3R, not mv2/dR, a small change in constant factor.

Also the occurrence of m in v = P/2m was lost when substituting for v in T = mv2/3R, which should have given P2/4dRm, though this too is only a constant factor off given that the mass flow m (in kg/s/m2) is constant over the whole atmosphere by the argument in the paragraph following my Claim 1. (Also since m is a mass rate, E wherever I used it denotes power, not energy.)

This 3R denominator is itself suspect outside the domain of applicability of the ideal gas laws, since we can no longer assume that all directions are equally likely. So T as a function of v could be as much as 3 times the above: T = mv2/R, if the motion were somehow constrained to be largely vertical.

Worse yet, if say solar radiation were to heat the molecules almost entirely in their vibrational and rotational modes (a very reasonable assumption given the heating mechanism involved), v could be arbitrarily small relative to √T. If furthermore the density of the gas decreased even faster than the pressure, the rise in T would not be contradicted by the rapidly falling P because the slow-moving but high-energy molecules would encounter the (virtual) surface of unit volume so infrequently as to develop very little pressure, even though the energetically vibrating molecules will give the (real) surface of any barometer quite a bang on those extremely rare occasions (due to the extremely low velocity) when they do touch it.

In this scenario the ideal gas law P = ρRT does not disappear altogether but instead morphs into the form P = α(h)ρRT where α(h) < 1 is a dilution factor depending on altitude h and reflecting the greatly reduced rate of arrival of molecules at the barometer’s surface due to their vanishingly small velocity. So we need α(h) = P/ρRT, i.e. something of the form α(h) = ch/bh for suitable constants b > 0 and c < 1 deducible empirically from the atmosphere graph. α(h) gives the amount by which v must be decreased relative to normal gas behavior.

One relationship we can rely on is the simple one between P and the remaining mass of atmosphere above, which P supports. This should permit prediction of the density at h from the empirical graph, noting that with translation contributing far less energy than the other degrees of freedom the relationship between P and T must be diluted along the above lines.

Predicting all this analytically however seems like a big challenge. It seems clear that T must be computed from consideration of absorption of solar radiation without (immediate) reference to v. On the other hand the relationship between P and mass seems clear since P measures the remaining mass above. So that’s one reliable handle.

Well, enough rambling, time to check these and other handles to see which can be used to give a reliable model of what’s going on.

What I do continue to believe however is that the key to solving the radiometer riddle lies in a better understanding of the earth’s atmosphere. With more insight into the ways in which PV = nRT can break down at low pressure and high temperature gradient one stands more of a chance of understanding how the radiometer works. We have a lot more information about the atmosphere than the radiometer to check these kinds of theories against. This is not to say that the radiometer closely mimics the atmosphere but only that the same or similar breakdowns at low pressure seem like promising candidates for exploration.

I would be most surprised if the radiometer’s behavior could be described within the domain of applicability of the ideal gas laws.

Posted by: Vaughan Pratt on August 4, 2008 7:15 AM | Permalink | Reply to this

### Re: Light Mills

I slept on this problem but the solution was so obvious it woke me up after two hours.

Normally the energy of a molecule is equally distributed among its degrees of freedom by frequent collisions. When the source of heating is radiation lines specific to a transition of a molecule from one combination of eigenvalues of rotation and vibrational modes to another, initially the molecule will acquire energy in those degrees of freedom.

Translation does not participate in this initial feeding frenzy because free motion has no frequency-specific resonances and molecules can only be pushed down, which gravity is already doing very nicely thank you.

(Skip this on a first reading, but in deference to this blog’s venue, the n-category cafe, it should be mentioned that this distinction between free and bound behavior is nicely modeled by Pontrjagin duality. According to yin-yang and Cartesian duality, the universe divides up equally into two dual parts. This duality was originally believed by Descartes and his followers to be noninteracting, though Descartes himself, perhaps mindful of Galileo’s recent shameful treatment, allowed an exception for humans, postulating the pineal gland as the seat of that interaction, a ploy that failed to keep his works off the Index Librorum Prohibitorum. Today we (well, some nuts among us) understand the interaction to be via inner product, first recognized in the setting of Hilbert space but subsequently observed in Chu spaces, a much more general setting having a couple of connections with n-categories, one of which is that John Baez and I gave an interleaved series of lectures on the two subjects to a summer school in Coimbra, Portugal in 1999. Inner product is nature’s pineal gland, overcoming the standard objection to dualism that it is a noninteracting and hence inoperative theory. The Pontrjagin duality of the real line with itself expresses free existence, while that of the integers and the unit circle with each other expresses bound existence, with integers modeling the particles and the unit circle their dual waves. The setting for Pontrjagin duality is the category Ab of abelian groups, more precisely topological such, more precisely yet locally compact such. Fields arise as monoids in Ab, and Hilbert space as a vector space arises as a module over a field, typically Q or sometimes R, with an inner product placing it midway in abstractness between a vector space and a coordinate space. That’ll be enough of that now.)

At terrestrial pressures, or in the case of the Sun, at pressures found below the corona, this newly acquired energy is quickly redistributed equitably among all the degrees, thereby feeding translation. The mechanism here is that when two almost stationary but violently vibrating molecules touch, the vibration drives them apart violently, converting vibrational energy into translational.

As the pressure drops, so does the frequency of collisions, starving translation. The knee of the curve describing translation starvation is where the ideal gas laws break down. Molecules slow down relative to the prevailing temperature. This reduces translational momentum, an irrelevancy given that collisions transfer energy whether or not translational momentum is present. More significantly it reduces collision rate, the mechanism underlying translation starvation. Even if you get nothing else out of this extended ramble, at least quote me on that.

A concomitant factor is the early bird getting the worm of radiation in the upper atmosphere. As the worms are eaten the late birds lower down are starved of lines, exponentially with altitude by the Beer-Lambert law. So actually two effects are at work here in governing the knee of the curve, pressure and line absorption, both exponential in altitude.

To summarize in one sentence, the principle here is that although each collision can transfer a lot of energy to translation, the collisions need to happen on a regular schedule for that transfer to be effective, which low pressure prevents, whose effect is magnified by the better access to radiation lines enjoyed by the higher molecules.

I conjecture that this principle is the common underlying mechanism for the radiometer effect, the thermosphere, and the solar corona. In all three it puts temperature out of whack with velocity, allowing the former to rise far beyond what the ideal gas laws permit. The early-bird effect may play less of a role with some of these effects than others, but certainly low pressure is key here.

This mechanism in hand, one can proceed with numerical computation of the effects, all of which have been studied sufficiently as to make this feasible, if not all bits equally easy.

The easiest bit should be the mechanics of velocity, pressure, and temperature. Below the knee (but not so far as to approach the liquid phase) this is just PV = nRT. Above it there needs to be a new law. It seems to me that the law PV = α(h)nRT should be the basic form, where α is a suitably parametrized dilution factor. Obviously h can’t be the fundamental parameter for α, which is going to depend rather on molecular mass and radiation parameters. The former is easy, the latter divides up into absorption strength and the early-bird effect, the exponential decay in the strength of absorption.

In view of the atmosphere’s behavior, GHG, dominantly CO2, may be the culprit behind the strange downwards pressure bulge peaking at 113 km. If true this would constitute the GHG bulge in atmospheric pressure. CO2 being 50% heavier than the average air molecule, its proportion in the atmosphere decreases at some point, whence the early-bird effect kicks in for GHG at a lower altitude than the homonuclear molecules. The bulk of the heat absorbed by GHG should be in a region where it is still quite dense relative to air, maybe not 300 ppm but say 100 ppm (pure guess, it may well be much less). In that region the gas law breaks down strongly for GHG but at best weakly for homonuclear. In the absence of the latter the CO2 molecules would slow down, but surrounded by fast-moving homonuclear molecules they are sitting ducks for collisions and hence participate in collisions at the expected rate of 1 in every 10,000 collisions (assuming 100 ppm GHG), which should keep them moving as fast as the homonuclear molecules.

Well, with that back-of-the-envelope calculation I confess I now don’t see how hot GHG in cold homonuclear gas drives things one way or the other, although empirically it looks like at lower altitudes it somehow lowers the pressure, adiabatically for the homonuclear molecules at least, which pushes down the temperature since the dominant homonuclear molecules still constitute an ideal gas. But at higher altitudes the heating effect becomes irresistible, presumably because this is the knee of the curve for the homonuclear molecules judging by the smoothness of the turn-around in temperature at 90 km.

Judging by the downward pressure bulge the GHG contribution to radiative heating may well peak at 113 km. Higher up, that effect dies down, just as the radiometer effect ultimately dies down with decreasing pressure. In other words I’m conjecturing (not particularly strongly for the moment since it all seems so iffy) that the downward pressure bulge in the atmosphere and the upward activity bulge in the radiometer have a similar origin even if not exactly the same; for one thing there is no obvious reason why any pressures should vary exponentially in the radiometer.

The Sun on the other hand should behave much more like the earth. A key difference however is that the early bird effect works the opposite way to Earth because the Sun is a radiation source, not a sink like the Earth. I have no idea how that change in sign might affect things, though it still seems clear that at some point the translation starvation effect will kick in somewhere in the solar corona, heating it above the expectation based on not taking that effect into account.

Regarding the Chapman-Enskog solution to the Boltzmann equation that John mentioned, I don’t see how it helps in the case where translational energy is vastly less than vibration in a given mode. The Burnett equations might be more relevant, but it seems to me that the effect is so extreme that the equations should be relatively straightforward, this breakdown of the gas laws being so gross. The Burnett equations may however describe the knee of the curve, whose location however can surely be pegged more simply from backwards extrapolation of its extensions on either side.

Well, absent more quantitative support this is probably enough qualitative theorizing for now. Not so much TMI as TMT. Back to bed.

Posted by: Vaughan Pratt on August 4, 2008 3:37 PM | Permalink | Reply to this

### Re: Light Mills

Couple more corrections. First, I just noticed another misreading of the NRLMSISE plot at [[Earth’s atmosphere]]. The blue line I was calling pressure is actually mass density in g/cm3. Oops. Below the thermosphere (e.g. in the mesosphere or stratosphere) they’re linearly related to the extent T remains more or less constant, but they diverge badly in the thermosphere.

Second, my use of the effective temperature of the Earth as a predictor for temperature below the thermosphere could be improved. For one thing, at 10 km the factor of 1-A correcting for albedo is not applicable since at that altitude most of the contribution to albedo is lower down and hence not blocking the Sun. Second, the concept of effective temperature of a whole planet should not be applied to objects smaller than the planet and near its surface, since unlike a whole planet they don’t have the night sky on their “dark side” (away from the Sun). Both factors predict a higher temperature for small objects near the planet than the effective temperature of the planet.

Since my previous post I’ve been reading up on aeronomy, aka atmospheric physics. Although I haven’t absorbed their picture fully yet I get the impression they have a good handle on why the thermosphere behaves in the odd way I described, and it seems for essentially my reasons if I’ve understood theirs. On the other hand I haven’t seen them point out that PV = nRT breaks down badly at an altitude of 150 km, so I’m not sure what that means – either I’m misunderstanding something or they’re not seeing any need to point this out. Meanwhile you can find no end of people willing to swear on a stack of bibles that PV = nrT is accurate at low pressure and high temperatures, so something’s wrong somewhere. At some point I may try writing up my explanation of what’s going on in the thermosphere and bounce it off people for their reactions. But not until next week at the earliest–I’m giving a paper at an algebra conference in Denver this week.

I’m assuming the solar corona theorists have already incorporated the known theoretical basis for the Earth’s thermosphere into their (low) prediction of the Sun’s temperature in the corona, and that I don’t have anything to add to their understanding. The anomaly there presumably lies elsewhere, e.g. the June article I mentioned earlier.

Meanwhile I remain convinced that the radiometer effect is essentially the anomalous behavior of the thermosphere in a bottle.

Posted by: Vaughan Pratt on August 5, 2008 3:19 AM | Permalink | Reply to this

### Re: Light Mills

Following up on the last paragraph of my previous post, here’s my explanation of the radiometer effect, based on the present understanding of the thermosphere.

It is well known to atmosphere scientists that the thermosphere violates the equipartition theorem in an ideal gas in favor of bound (vibrational and rotational) modes over translational. In this condition there exists at least one bound degree of freedom for which the contribution kT/2 to the net energy of the molecule greatly exceeds the 3kT’/2 contribution of translation (3 assuming no directional bias in translation, less if there is bias). That is, the molecules are vibrationally hot (temperature T) but translationally cold (temperature T’). Translation contributes twice (multiplicatively) to pressure, once via rate of flow of molecules and once via translational energy (representable as p2/2m where p is the momentum mv) as well as vibrational exchanged in a collision. The energy exchanged can be coupled to the receiving particle both within and between modes. That is, energy exchange at a collision is conserved but can come from either bound or free modes and become either bound or free energy in the receiving molecule.

In this imbalanced condition, energy transfer is accomplished at the normal rate per collision (with the donor molecule contributing largely bound energy, which can become either bound or free), but the rate of collisions is drastically reduced. The upshot is that the apparent pressure is decreased in proportion to the rate reduction. Contrast this with the balanced condition, where the energy transferred per collision remains the same (though a more equitable proportion of it incidentally comes from the translation modes) but the collision rate is much higher, thereby increasing the pressure in proportion to the rate.

The radiometer effect can now be explained as an imbalance gradient between a cold surface and a hot, with the cold end more imbalanced than the hot. The total energy per molecule remains the same along this gradient, but the proportion of translational energy is greater at the hot end. This gradient is created initially by the hot surface contributing more energy than the cold, which is distributed equitably between vibration and translation in the molecules. The effect then runs away as the faster molecules at the hot end meet the hot surface faster than the slower ones at the other end meet the cold surface, suggesting that the gradient may not be linear and that in the middle the imbalance is closer to that of the cold end.

The equilibrium condition for this setup is that all molecules have the same average energy, and exchange the same amount of energy per collision. But due to the imbalance gradient these transfers slow down along the gradient from hot to cold, resulting in a steady dilution of the translational energy along that gradient.

The condition for this imbalance in the radiometer is that the incident radiation include the requisite frequencies to excite the vibrational modes, and that the pressure be low enough that the translation modes are starved of this energy due to the low rate of collisions.

When the pressure is too low there is not enough mass for this effect to create a sufficient pressure difference between the hot and cold sides of the vanes.

Among the more challenging prerequisites for a quantitative analysis of this effect are the line strengths of the gas and the proportion of those frequencies in the incident radiation. Recent tools developed for estimating line driving effects in planetary gases, e.g. http://www.cosis.net/abstracts/EGU2008/06613/EGU2008-A-06613-1.pdf?PHPSESSID= , should help.

In ordinary operation the same radiation is used to excite the gas molecules and heat the black side of the vanes. One way of confirming this explanation experimentally would be to precharge a radiometer with CO2 so that after evacuation the residual gas is CO2 rather than air. CO2 has a family of lines in the vicinity of 2300 cm-1 (4.3 μm) associated with its bending mode and indexed by rotation numbers, the totality of which is far more strongly absorbing than any corresponding family for either O2 or N2. Ordinary light should therefore drive such a radiometer much more strongly, while infrared boosted at that neighborhood should drive it more strongly yet. An ordinary CO2 laser at 945 cm-1 (10.6 μm) would excite the rather weaker lines at that wavelength in a CO2-charged radiometer, but the effect should still be much stronger than with an air-charged radiometer under white light.

(Note that glass strongly absorbs far IR, necessitating a strong source, but not so strong as to overheat and break the glass. An alternative is to use N2 and/or O2; although its absorption lines are weaker they are in a region where glass is transparent, permitting a weaker source for the lines.)

Illumination of the radiometer with two independently controlled radiation sources, a low-power narrow-band one for exciting the gas and a higher-power one with all but the lines of the other for warming the vanes, should allow observation of the interplay of these two components. The effect should drop by much more than a factor of two when either source is interrupted.

If the effect has some other origin such as the Reynolds-Maxwell-Einstein edge effects, no such strong dependence on choice of gas or exciting frequencies should be observed.

If my explanation is correct this will permit a variant of the radiometer serving to illustrate line-specific absorption by gases, of interest to students of both general physics and climatology.

Posted by: Vaughan Pratt on August 5, 2008 7:25 AM | Permalink | Reply to this

### Re: Light Mills

My explanation neglected to stress the point that the hot side of the vane is driven by the greater pressure. The reason the pressure is not distributed uniformly in the radiometer is that the ideal gas law PV = nRT now has the form PV = αRT where α ≤ 1 is the dilution of pressure attributable to the imbalance. What is distributed uniformly is energy; in an ideal gas pressure is proportional to energy and hence is just as uniformly distributed as energy. An imbalance gradient permits a pressure gradient without creating an energy gradient.

Posted by: Vaughan Pratt on August 5, 2008 7:45 AM | Permalink | Reply to this

### Re: Light Mills - A practical experiment

Here’s a more practical experiment. Illuminate a standard radiometer with an ultraviolet lamp and an infrared lamp. These serve to warm respectively the oxygen component of the residual air and the vanes. Shutting off either should stop the radiometer, instantly for the ultraviolet light since the gas has negligible thermal mass, more slowly for the infrared light since the vanes will radiate their heat away relatively slowly while the rarefied gas provides negligible convective cooling.

This depends on oxygen’s absorption being strongest in the ultraviolet, making it the principal contributor to the effect. I would not expect nitrogen to contribute as strongly, making the wavelengths of the oxygen lines the critical frequency-dependent aspect.

Posted by: Vaughan Pratt on August 5, 2008 8:48 AM | Permalink | Reply to this

### Re: Light Mills - A practical experiment

Offhand question: Would the glass of the radiometer bulb absorb UV in the same region as the oxygen inside? That might warm something you’d rather not have warmed.

Posted by: Blake Stacey on August 5, 2008 7:51 PM | Permalink | Reply to this

### Re: Light Mills - A practical experiment

I’d expect not because ordinary glass is transparent to UV, testable by illuminating a piece of glass with a UV lamp and observing its temperature rise.

But you’re right that the bulb temperature is very relevant (as is well known empirically). Anything you can do to cool the bulb should enhance the effect. If you wrapped it up in dry ice, 194.3 K, to cool down the gas as far as possible, and focused the infrared on the center to really heat up the vanes, the effect should be much stronger.

One inconsistency between my theory and experiment is that putting a radiometer in a fridge supposedly runs it backwards. I don’t have an explanation for that.

Posted by: Vaughan Pratt on August 5, 2008 9:57 PM | Permalink | Reply to this

### Re: Light Mills - A practical experiment

Oops, I take that back. I’d been imagining the oxygen lines just below violet, say around 350 nm in wavelength, but now that I look at it the strong ones seem to be more like 100 nm, where ordinary glass is strongly absorbing. In that case it may be a bit tricky to get the ultraviolet in without heating the glass excessively. Have to try it and see. The option of cooling the glass with ice or cold air or dry ice is always available.

Posted by: Vaughan Pratt on August 5, 2008 10:25 PM | Permalink | Reply to this

### Re: Light Mills - A practical experiment

Or use a quartz radiometer! (-:

Posted by: Blake Stacey on August 5, 2008 11:00 PM | Permalink | Reply to this

### Re: Light Mills - A practical experiment

Since the radiometer works with the light that does get in, it should just be a matter of seeing whether that light can separated into the gas-warming and vane-warming contributions. If they overlap too much then not, but if they’re largely disjoint then a near-UV or even blue light might suffice for the gas and a near-IR or even red for the vanes. Ordinary glass is transparent to both near-UV and near-IR. If it runs with blue and red, experiment with their positions (e.g. by moving each away from the radiometer independently until it almost stops) until you find a position where interrupting either color stops the radiometer. Then move both lamps 50% closer and try again, to rule out the possibility that the stopping is merely due to half as much heating.

While the absence of such an effect wouldn’t be instantly fatal to my theory, I’d certainly start to worry. (But I’m in Denver this week and need to work on the slides for my talk so I can’t get too worried just now about this.)

Posted by: Vaughan Pratt on August 6, 2008 7:01 AM | Permalink | Reply to this

### The radiometer as a poor man’s spectrophotometer

Another application is spectrophotometry. If my theory is right, different gases in a radiometer should respond differently to different spectra. There’s a wide range of gases and mixtures thereof, allowing the design of radiometers tuned to pick out specific spectra. This would be a cruder instrument than a real spectrophotometer, but at the price of an incandescent light bulb (which is no easier to make than a radiometer) you get at least what you pay for.

This prompts the interesting question, what is the resolving power of radiometers in the following sense? Given two spectra, does there exist a radiometer, or small set of radiometers if necessary, capable of distinguishing them reliably? Any given criterion for reliability should create a metric d on the space of all spectra and a real r such that two spectra s,t can be reliably distinguished by some radiometer (or set thereof when that’s allowed) if and only
d(s,t) ≤ r. A typical application would be go-no-go spectrophotometry, deciding for example whether a given spectrum was close enough to a desired spectrum by that metric. Not every metric need arise in this way, and you may have to settle for a less desirable metric than you’d like.

This is primarily applicable to 2000 cm-1 and above (5 μm and below) as ordinary glass is opaque to far infrared light, and transparent material such as zinc selenide would be prohibitively expensive, defeating the economic benefit of the radiometer.

Incidentally if my writing seems to have drifted into a 19th century style it’s from reading too many 19th century papers about radiometers. :)

Posted by: Vaughan Pratt on August 5, 2008 5:23 PM | Permalink | Reply to this

### Re: Light Mills

This applet is a very simple toy model that might be useful for exploring some aspects of this. It doesn’t pretend to be anything like a detailed, faithful simulation of all the actual physics, but people might enjoy playing around with it and seeing if anything relevant can be gleaned from it.

Posted by: Greg Egan on August 6, 2008 11:22 AM | Permalink | Reply to this

### Re: Light Mills

Congratulations! Very impressive.

Posted by: Bruce Bartlett on August 6, 2008 12:15 PM | Permalink | Reply to this

### Re: Light Mills

Is the rotor frictionless? I can’t be sure, but its direction of motion seems to persist, although it sometimes turns one way, sometimes the other…

Posted by: Tim Silverman on August 6, 2008 7:58 PM | Permalink | Reply to this

### Re: Light Mills

Yes, the rotor’s frictionless (at its bearings). Sorry I didn’t make that explicit. Of course it still loses momentum to the surrounding particles.

In the absence of the thermal energy flux the rotor would be undergoing a kind of constrained Brownian motion. Given the low number of particles (and their high mass) the behaviour is always very noisy, but with the default parameter choices it usually does seem to settle into persistent positive rotation eventually, after going backwards for a while first.

Posted by: Greg Egan on August 6, 2008 11:33 PM | Permalink | Reply to this

### Re: Light Mills

I’ve tweaked the applet so you can ramp down the energy flux mid-simulation (changing the other parameters restarts the whole thing from scratch), and see what happens when the black and white sides return to equal temperatures. (The rotor does coast for a while, and there’s also ongoing Brownian motion.)

Posted by: Greg Egan on August 7, 2008 12:19 AM | Permalink | Reply to this

### Re: Light Mills

Greg, your applet is really cool. A couple of immediate reactions.

1. Where exactly is the center of the “vane particle” you create to model vane temperature? I can see what it should be to within the diameter of a particle, but that variation can make a huge difference to the outcome of the collision, to the extent of having the incident particle pass through the vane! A logical choice would be to time the collision so that the line between the centers of the colliding particle was normal to the vane, ensuring a mirror-like reflection in the case both particles have the same velocity.

2. The vanes of a typical radiometer don’t extend as far to the center as in your model. The center has more black-white interaction, the outer parts involve the walls more. Could you either provide an additional parameter as a lower bound on how close to the center the vanes go, or just make the vanes 10-20% of how wide they presently are? The former would let us see whether this factor plays any role.

3. Were it not for round-off error, it should be possible to stop your simulation at any point, negate all velocities (particles and vane), and then let it keep running. Since all collisions are elastic the configuration should then return to q0 with the vanes stationary.

IEEE arithmetic is unfortunately not invertible so this presumably can’t be checked empirically (though it might be interesting to see how close it gets after spending the same time running backwards as forwards).

4. As the rotor picks up speed counterclockwise the white faces are going to hit the particles faster and the black faces slower. When this effect balances the temperature imbalance the rotor should be at its limiting angular velocity. In a real radiometer the particles will be moving at several hundred meters per second, presumably a lot faster than the vanes, so it’s unclear whether this is what’s limiting the vane speed in the real case but it seems implausible at first glance.

5. Your simulation would go lickety-split on an SLI-connected NVIDIA 8800GTX or similar running CUDA.

6. The mean free path in your model is at least half the radius of the bulb (you could include that in your virtual instrumentation), way longer than the expected 1 mm mfp in a typical radiometer. On the other hand I’m much less convinced than I was in my post to this blog on Aug. 3 at 7:34 am (paragraphs 5, 6, etc.) that a very large mfp on its own makes any difference — rethinking my reasoning there convinced me it was fallacious. I wish I could go back and color-code the bits I wrote that I now disagree with; as it stands it’s hard to tell from the blog what I believe now from what I used to believe. (This was the reason I gave John originally for withholding some of my ideas, that having to retract some had the bad side effect of undermining the ones I continue to stand by. True to form I got sucked into the discussion anyway…)

Posted by: Vaughan Pratt on August 7, 2008 5:59 AM | Permalink | Reply to this

### Re: Light Mills

Vaughan:

The dummy “vane particle” collides with the incident free particle with both of their disks tangent to the vane surface at the moment of collision. So the impulse is always normal to the vane surface.

I’ve added a choice for the inner radius of the vane (with a gap as the default), and a display of the mean free path.

Posted by: Greg Egan on August 7, 2008 7:41 AM | Permalink | Reply to this

### Re: Light Mills

But now trying to use these, I realize there’s a bug of sorts. After assigning random velocities to the particles, the total angular momentum (which is invariant for lack of friction) is some random quantity. This creates a permanent bias that tends to mask the effect.

One way to solve this would be to give the vane the corresponding opposite angular momentum. While that wouldn’t bother me (the effect is down at the Brownian level), for a cleaner start you might track the angular momentum as it accumulates during particle creation and slightly bias each particle as some function of total AM and number of particles so far that makes the former converge to zero with the last particle while not excessively biasing the first few particles. Or simpler: at the end just divide up the negation of the total angular momentum among the particles (slightly tricky since converting angular to linear momentum depends on distance from the center).

Posted by: Vaughan Pratt on August 7, 2008 12:52 PM | Permalink | Reply to this

### Re: Light Mills

If the inner radius exceeds the outer radius of the vanes, clearly there will be no movement of the vanes. There’s something strange about the case 50% inner, 60% outer however: no vane motion at all, not even Brownian. All other combinations that make the vane nonzero in size (that I had time to check) exhibit Brownian motion. What’s with 50-60? This makes it unclear what to make of the other vane settings.

Posted by: Vaughan Pratt on August 7, 2008 1:42 PM | Permalink | Reply to this

### Re: Light Mills

What’s with 50-60?

That was a bug. I’ve fixed it.

Posted by: Greg Egan on August 7, 2008 3:17 PM | Permalink | Reply to this

### Re: Light Mills

I’m not sure about the significance of any small random total angular momentum in the particles, but it’s easily fixed, so I’ve done that. I just add $-L/\sum_i r_i$ to each particle’s tangential momentum, where $L$ is the total angular momentum on the first pass.

I’ve also added a choice between random initial conditions that are different for each run, and repeatable initial conditions (i.e. fixed seeds for the pseudo-random number generator). That should make it easier to isolate the effects of various parameters, as opposed to chance differences in initial conditions.

Posted by: Greg Egan on August 7, 2008 1:48 PM | Permalink | Reply to this

### Re: Light Mills

I think that simulating the radiometer in this way is an excellent means to tackle this problem. However, I am not convinced that you have the right physics of the interaction with the vane.

If I understand your model correctly it makes it look like there will obviously be a larger force on the black side because each particle is imparting an impulse that is bigger on the black side. I think that if the physics were so simple then it would not be very contentious. I think you have to model the thermal interaction between particle and vane better to observe the real effect. When the particle strikes a vane it may gain or lose energy depending on the velocity of the particle in the vane it strikes. You seem to have it always gaining energy. ( maybe I missunderstood so just let me know if I’m wrong )

I suspect that when you increase the number of particles in your simulation the vanes accelerate faster (It’s hard to be sure because the simulation slows down) This is not what is observed with a real radiometer and I think that shows that this is not a good enough simulation of the effect.

If you simulate the thermal interactions correctly I think the true radiometer effect may be smaller and you may need a bigger simulation to see it. To fully understand it you would have to plot the temperature and density distribution in the gas and the distribution of forces on the vanes. An interesting question is whether the residual forces are concentrated around the edges of the vanes as claimed in the transpiration theories.

Of course that may require an order of magnitude more work to implement!

P.S. I will not have access to the internet for the next week so wont be posting.

Posted by: PhilG on August 8, 2008 10:20 PM | Permalink | Reply to this

### Re: Light Mills

PhilG wrote:

I think you have to model the thermal interaction between particle and vane better to observe the real effect. When the particle strikes a vane it may gain or lose energy depending on the velocity of the particle in the vane it strikes. You seem to have it always gaining energy. ( maybe I missunderstood so just let me know if I’m wrong )

What’s fixed in the free particle/vane interaction is that the notional particle within the vane is taken to always have a velocity normal to the vane surface (outwardly directed), and a speed that is a deterministic function of the thermal energy of that side of the vane. So what’s missing (along with many other details of the actual thermodynamics of solids) is a distribution of velocities for particles in the vane; what the free particle always encounters is a fixed, average representative.

But you’re wrong when you say that the incident free particle will always gain energy (I assume you mean from the hot side), because that will still depend on its own particular velocity.

What seems to be happening in the cases where the rotor turns persistently is that a stable situation arises where $v_{cold side} \lt v_{particle, RMS} \lt v_{hot side}$. In other words, if the energy flux can keep the cold side sufficiently colder than the gas and the hot side sufficiently hotter, then on average the hot side will get a stronger kick from its interactions with the gas. It’s hard to see why randomising the vane particle’s velocity is going to change that average behaviour, but I’ll try it and see.

Posted by: Greg Egan on August 9, 2008 2:16 AM | Permalink | Reply to this

### Re: Light Mills

Phil, I don’t believe local reasoning about the details of molecule-vane interactions is as important as the vane geometry. Greg kindly implemented the variable vane permitting us to experiment and/or test hypotheses. I’ve noticed the following.

1. At 10-100 (maximum vane, molecules trapped in four regions), there is no effect other than Brownian motion resulting in a random walk. The vane never accumulates any permanent momentum. This behavior is obvious from the facts that (i) total AM = 0, and (ii) the molecules being trapped, the vane and molecules must rotate together and hence always have the same (signed) AM. Solving x+x = 0 gives x = 0. (Greg, this is one reason I wanted the total AM to be zero; otherwise in this trapped situation the vane will rotate forever at a rate determined by the initial total AM, regardless of the temperature of black and white.)

2. At 50-100 (molecules move between the regions by passing near the center) I could believe there is an effect but it is very tiny. In order for the vane to rotate steadily counterclockwise the molecules must balance this with a steady clockwise flow. Because the molecules change regions near the center, for a given velocity the molecules will contribute only small AM, whence the vane can only get small AM.

3. At 10-60 (gap at the outside) the effect is more noticeable. A given flow of molecules contributes (100+60)/(10+50) = 8/3 times the AM of the 50-100 setting. My impression was that the effect was even more than 8/3 times as much so there may be more to it.

4. Regarding your concern about modeling, picture a situation where molecules bounce back and forth between parallel hot and cold surfaces, like ping-pong balls alternating between paddles. Assume that in equilibrium the average normal-to-the-surface component of velocity from hot to cold is U and from cold to hot V. Presumably U > V, significantly so when the temperature difference is significant. Then the average momentum transferred to the hot face by unit mass is V+U, while the average momentum transferred to the cold face is U+V. It follows that the average force on the two faces is the same.

I don’t see how your concerns about inaccurate modeling of collisions invalidates this naive reasoning, which remains sound regardless of how accurately you model collisions. Furthermore this reasoning gives an independent reason why there can be no cumulative rotation in the maximal-vane case: the collisions do in fact alternate (albeit sometimes mediated by one wall or center collision), and the above reasoning almost applies (if the component is measured normal to the bisector between the vanes and the resulting angle compensated for).

One worry is that the wall collisions might affect this. There must be an argument that they don’t, since we know by my earlier argument that AM can’t accumulate, but I don’t currently have a second (geometrical) argument that the walls don’t introduce any asymmetry between black and white.

My talk at BLAST is tomorrow afternoon so I’d better go prepare.

Posted by: Vaughan Pratt on August 9, 2008 5:10 AM | Permalink | Reply to this

### Re: Light Mills

I tried randomising the velocity of the vane particle used in the collisions, and it made the rotor turn faster, so I’ll stick with the original, simpler approach of just using a fixed velocity to represent a given thermal energy.

I’ve now added histograms to the applet which show cumulative sums for the raw number of collisions, the linear momentum transferred, and the angular momentum transferred, for the two sides, binned by radial distance out along the vanes.

Posted by: Greg Egan on August 9, 2008 2:56 PM | Permalink | Reply to this

### Re: Light Mills

PhilG wrote:

I suspect that when you increase the number of particles in your simulation the vanes accelerate faster. (It’s hard to be sure because the simulation slows down.) This is not what is observed with a real radiometer…

Actually it is — up to a certain point, around $10^{-6}$ torr.

With a lot of air in it, the Crookes radiometer doesn’t turn at all. As you decrease the air density it starts turning, but then as you decrease it further it slows down and stops.

According to the Wikipedia article: “The effect begins to be seen at partial vacuum pressures of a few mm of mercury (torr), reaches a peak at around $10^{-2}$ torr and has disappeared by the time the vacuum reaches $10^{-6}$ torr.”

Without air molecules to push the thing, it doesn’t turn.

By the way: I haven’t had time to comment much on this thread, but I’m delighted by it — especially by how Greg has taken the trouble to simulate the Crookes radiometer.

Since it’s much easier to ask for new features in a program than actually do the programming, I can’t help but want a simulated Crookes radiometer where I can put holes in the vanes. The standard explanations say the Crookes radiometer relies on ‘edge effects’ — so maybe it should spin faster if you poke holes in the vanes.

Of course Greg’s radiometer is so small (compared to the average distance between air molecules) that it might already count as ‘all edge’.

Also, edge effects are one place where Greg’s 2d radiometer might differ significantly from a real-world 3d radiometer. I’m not sure.

Posted by: John Baez on August 9, 2008 3:18 PM | Permalink | Reply to this

### Re: Light Mills

I added the ability to put holes in the vanes, but it doesn’t make a significant difference (I can get a small speedup sometimes, but no more than I can get from random changes in the initial conditions).

As you say, the vanes are already very small compared to the particle separation. Given that it’s necessary to have so many parameters different from the real-world values – simply in order to have any effect at all, with the small number of particles – it’s quite hard to translate things. But certainly something about this simulation is such that the extra kick the hot sides of the vanes get is not exactly cancelled by a difference in the particle density between the hot and cold sides. Whether that’s because the vanes are so small, or because the total particle numbers are so low, I’m not sure.

I hope someone can try a real-world experiment with holes.

Posted by: Greg Egan on August 10, 2008 4:59 AM | Permalink | Reply to this

### Re: Light Mills

On the issue of how the rotor speed in the simulation varies with particle number, if all the other parameters are kept at their default values, for particle numbers 50, 100, 200, 300, 400, 500 the net rotation at t=100 is: 0.805, 4.571, 7.786, 5.967, 4.289, 3.196. (Obviously the peak around 200 particles involves a vastly smaller number than in a real radiometer, but then the mass and radius of the particles are vastly larger in the simulation than in reality.)

Posted by: Greg Egan on August 10, 2008 5:45 AM | Permalink | Reply to this

### Re: Light Mills

I’m pleased that my skeptical post got torn apart while I was away. These results look very good.

I think the most revealing calculation that could be done to understand what was going on here would be to calculate the forces acting on different sections of the vanes to see if the forces are stronger near the edges as predicted. It would be difficult to do that with a real experiment but with the simulation it should be easy enough.

Posted by: PhilG on August 17, 2008 8:44 AM | Permalink | Reply to this

### Re: Light Mills

PhilG wrote:

I think the most revealing calculation that could be done to understand what was going on here would be to calculate the forces acting on different sections of the vanes to see if the forces are stronger near the edges as predicted.

I added histograms at the bottom of the applet that show cumulative sums for the number of collisions, the amount of linear momentum transferred, and the amount of angular momentum transferred, along the vanes, for each side.

These make it pretty clear that the net force isn’t uniform along the vane, but I find it quite hard to make sense of what’s actually going on here.

I’ve just tweaked the applet further, so that instead of specifying the rotor’s mass, initial thermal energy, and thermal energy flux, what you specify is a linear density for those three quantities. This makes it easier to compare rotors with different geometries, on the basis that they are made of the same material and exposed to the same ambient conditions.

It’s quite striking to note that a rotor stretching from 20% to 60% of the container has, with 10 trials, an average net rotation at t=100 of 6.177 (with a standard deviation of 0.367), whereas a rotor one quarter the size, stretching from 40% to 50%, has a net rotation of 28.470 (sd 2.184). If we treat these net rotations as average angular velocities over the time period, and attribute the corresponding gain in angular momentum from rest to a torque arising from a uniform force spread across the length of the vane, it works out that the force per unit length on the smaller rotor is five times larger, i.e. the total force on the smaller rotor is comparable to, and in fact a bit larger than, that on the larger rotor.

I previously said that punching holes in the vanes made no significant difference, but I was using fixed rotor masses and thermal fluxes before; using fixed densities for those quantities instead, punching a single hole (of size 10% of the container) out of the vanes changes the average net rotation at t=100 from 6.177 to 8.686, which is nearly 7 standard deviations.

Posted by: Greg Egan on August 17, 2008 1:53 PM | Permalink | Reply to this

### Re: Light Mills

The histograms show the edge effects beautifully. It is wierd that you actually get more collisions near the edge of the vanes. That seems counterintuitive but it must be what is happening.

Posted by: PhilG on August 18, 2008 7:52 AM | Permalink | Reply to this

### Re: Light Mills

Naively, at least, it makes some sense for the collision count to be high for the white side outer edge, because the vanes are turning into a counter-flow of particles that develops outside the bounding annulus of the vanes.

But of course that explanation only works once we already have the rotor turning and the counter-flow in existence to balance its angular momentum. So inasmuch as the final state of the system seems to explain some of the excess collisions at the edge, the whole causal sequence remains very obscure to me.

Posted by: Greg Egan on August 18, 2008 1:56 PM | Permalink | Reply to this

### Re: Light Mills

I’ve added another histogram at the side of the applet, which plots, against radial distance, the average tangential velocity component of the particles. Like the other histograms, the sums are allowed to accumulate over time, to give a time average as well as an average over the particles’ angular coordinate.

You can see the counter-flow of the particles develop. Even some distance within the bounding annulus of the vane there is a clear bias in the direction of the particles; this tapers off as you get deeper into the vane, but I guess there’s a kind of “Catherine wheel” effect, whereby the hot particles pushed away by the outer edge of the black side feed the counter-flow outside the bounding annulus.

Posted by: Greg Egan on August 19, 2008 1:31 AM | Permalink | Reply to this

### Re: Light Mills

One more tweak to the applet to help visualise what the particles are doing: flow lines!

Or to be precise, I’m now showing average velocities in a grid locked to the rotor.

Posted by: Greg Egan on August 19, 2008 6:50 AM | Permalink | Reply to this

### Re: Light Mills

The flow lines are quite spectacular. The counter flow could be interpreted as transpiration and there is a nice vortex effect between the vanes. But I am still not convinced that the physics is right.

I am trying to understand the forces on the vane but can’t quite make sense of them. There are clearly larger forces acting at the edges of the vanes which supports the transpiration theory, but the total force on the rest of the vane looks like it would be larger and that is not what you would expect.

I notice that the vanes do not continue to accelerate. There is no friction from the vane bearings so the total torque must drop to zero. If the transpiration theory was right I think we would continue to see forces at the edges with opposing air friction forces acting on the rest of the vanes.

However, if I understand the histograms at the bottom correctly then the forces only accumulate at the beginning of the simulation. After it reaches a steady state the forces actng on the vanes are averaging to zero at each pooint. We see that because the histogram bars cease to grow. Have I interpreted that correctly or is there some normalisation that I didn’t take into account?

If the forces do indeed average to zero after some time then that is a bit disapointing from a theoretical point of view. It would mean that all the rotation is coming from the early times during which the temperature distribution is changing. But is that the real physics or is the simulation unrealistic?

I think it would be useful to add a small amount of friction to the vanes so that they will slow down with time. There should also be some friction between the gas and the blub to remove angular momentum from the gas. That would make the simulation more realistic. It should remove any effects due to the intial setup over time.

Posted by: PhilG on August 20, 2008 8:36 AM | Permalink | Reply to this

### Re: Light Mills

PhilG wrote:

Have I interpreted that correctly or is there some normalisation that I didn’t take into account?

The histograms are always normalised so that the highest peak in each one is a fixed size. The fact that they eventually stop changing shape only means that the various counts and forces have settled down to an asymptotic relationship; nothing is going to zero!

The fact that the rotor is no longer accelerating just means a balance has been reached between the net torque and friction with the gas. I can’t see any benefit in adding friction between the gas and the container, or to the rotor’s bearings; the point here isn’t to create the most detailed possible simulation of everything going on in a real light mill, but to understand the one part that’s unclear: how the temperature difference between the two sides of the vanes gives rise to a net torque. We know what friction would do to a system like this, that’s not mysterious at all. If I added it to the program, we’d then just have to mentally subtract it away again to get a clear view of what we’re actually interested in.

It might well be that the many unphysical parameters in this simulation – including the hardest thing to overcome, the small number of particles – mean that it’s not capable of illuminating the reason that real light mills turn. Certainly any rigourous attempt to compare this with the real thing would take a bit of work, to figure out exactly how things should scale, whether 2D vs 3D is important, etc. I think that’s the key to judging the simulation’s relevance, but alas I’m too busy (and too rusty in statistical mechanics) to address this any time soon.

Posted by: Greg Egan on August 20, 2008 1:02 PM | Permalink | Reply to this

### Re: Light Mills

I wrote:

[N]othing is going to zero! … The fact that the rotor is no longer accelerating just means a balance has been reached between the net torque and friction with the gas.

I might have phrased that poorly. To be clear: the histogram for angular momentum records all interactions between the rotor and the gas, and there’s no way to partition those interactions into “friction” and “net torque”. So once the rotor stops accelerating (though there’s always some ongoing Brownian wobble) the sum of all the black bars minus the sum of all the white bars in the angular momentum histogram will just equal the near-constant angular momentum the rotor has attained. And due to the ongoing rescaling – because the quantities portrayed by the individual bars are still growing – if you tried to calculate this by counting pixels on the histogram, you’d soon get exactly zero.

But I don’t think it’s correct to say:

It would mean that all the rotation is coming from the early times during which the temperature distribution is changing.

The thing is, there’s friction between the rotor and the gas, so just because the rotor ceases accelerating doesn’t mean all the physics of interest happened while it was spinning up. The temperature difference between the two sides of the vanes is an ongoing source of torque; if it was not, then friction with the gas would cause the rotor to decelerate. If you let the rotor spin up (say, wait for t=100 with the default settings) then cut the energy flux density down to $10^{-20}$, then by t=200 the rotor is virtually motionless again.

Now, it’d be nice to have a detailed account of what happens while the rotor spins up. But I’d be happy if we could come to grips with the easier part, which is the asymptotic situation where the rotor is neither accelerating nor decelerating, because the “light mill effect” – whatever that is – is balancing the effect of friction with the gas.

Posted by: Greg Egan on August 20, 2008 1:47 PM | Permalink | Reply to this

### Re: Light Mills

Greg Egan wrote: “The histograms are always normalised so that the highest peak in each one is a fixed size”

In that case I am a bit happier with what I see and I agree that the friction idea is unnecessary.

But I am still confused about the histograms. I see just one histogram for Linear momentum, is this the momentum transfer on the black side only, or on the black side plus the white side or the black side minus the white side? The reason I am confused is that if the vanes are not accelerating I expect to see a total force around zero but all the bars have the same sign.

For example it would make more sense to me if the air friction produced a force along the vane on the white side with transpiration forces acting in the other direction ( i.e. on the black side ) near the edges. I realise that the two types of forces can not be separated but the distinction should be clear from the way they act.

Posted by: PhilG on August 20, 2008 3:31 PM | Permalink | Reply to this

### Re: Light Mills

PhilG wrote:

The reason I am confused is that if the vanes are not accelerating I expect to see a total force around zero but all the bars have the same sign.

Sorry this is unclear. The white and black bars need to be read as implicitly having opposite signs. The reason I draw it this way is (a) to save vertical space, and (b) to make it easier to compare the heights of black bars with white bars.

Posted by: Greg Egan on August 20, 2008 11:12 PM | Permalink | Reply to this

### Re: Light Mills

Aargh, I just went and checked the applet on a Windows computer, and now I understand why you’re so confused! The histograms are not being drawn properly; on my Mac they’re what they should be (black and white bars on a grey background), whereas under Windows they’re black bars only, upside-down, on a white background!

This must be some arcane Java bug or quirk. I’ll try to find a work-around as soon as possible.

Posted by: Greg Egan on August 20, 2008 11:33 PM | Permalink | Reply to this

### Re: Light Mills

I think I’ve got the histograms working under Windows now. My apologies, Phil, no wonder you found this utterly baffling! If you (or anyone else reading this) don’t see both black and white bars for all 3 histograms at the bottom of the applet, please let me know.

Posted by: Greg Egan on August 21, 2008 12:17 AM | Permalink | Reply to this

### Re: Light Mills

Yes I can see the white bars now thanks!

The result is not as I would like to have seen. The larger forces at the edges act equally in both directions ( I am using an inner radius of 70 and an outer radius of 80). I dont think that is what the transpiration theory would predict.

I then thought about it a bit more and realised that the larger forces at the edges are probably due to a much less sophisticated effect. The particles are large so they have a higher cross-section for hitting the corner of the vanes. If you make them much smaller the apparent edge effect decreases. However the mean free path then goes up as well.

Posted by: PhilG on August 21, 2008 2:50 PM | Permalink | Reply to this

### Re: Light Mills

Greg wrote:

I can’t see any benefit in adding friction between the gas and the container, or to the rotor’s bearings…

Indeed, friction of these kinds are clearly not necessary for understanding why the radiometer spins. They would just muddy the picture.

The crucial kinds of dissipation, as you note, are the gas molecules hitting each other and the vanes of radiometer. These are what keep the radiometer from spinning faster and faster forever. And you’ve got those.

It’s very nice to have the simplest possible model that displays a mysterious effect, and I think you’ve got that. Now the world can puzzle over it.

Posted by: John Baez on August 20, 2008 8:37 PM | Permalink | Reply to this

### Re: Light Mills

You have not convinced me.

The collisions are inelastic so there is no dissipation there (and there shouldn’t be) I think conservation of angular monentum and energy is built in to the simulation so it is not clear that any dissipation is present.

I still cant see where the forces are coming from and I think that the rotation we see could be an effect from the initial dynamics rather than the ongoing physics. If you added a tiny bit of vane friction you could dispell that possibility. to compensate for the energy drain you would have to pump in more energy via the black vane than you take out via the white vane so you might need to vary the flux on each side independently.

Why not add a vane friction parameter I can vary just to humour me?

Posted by: PhilG on August 21, 2008 3:09 PM | Permalink | Reply to this

### Re: Light Mills

Having said that, when you watch it for a while it does look like it is being driven rather than coasting. That suggests that the rotation is not due to the inital dynamics.

There is another reason why I would like some vane friction though. With friction there would have to be a residual torque on the vane from the gas so we would get a chance to look at where it acts.

Posted by: PhilG on August 21, 2008 5:50 PM | Permalink | Reply to this

### Re: Light Mills

PhilG wrote:

The collisions are inelastic so there is no dissipation there (and there shouldn’t be) I think conservation of angular monentum and energy is built in to the simulation so it is not clear that any dissipation is present.

Conservation of angular momentum and energy are built into our universe, too — but we still speak of ‘dissipation’ in situations where energy gets transferred from macroscopic degrees of freedom to microscopic ones. For example, I’d call it ‘dissipation’ when a spinning vane slows down through elastic collisions with air molecules — that’s a form of ‘friction’. I’d also call it ‘dissipation’ when circulating air slows down due to collisions that transfer energy from high-velocity molecules to slow ones — that’s ‘viscosity’. We’re seeing both these effects in Greg’s simulation. Neither requires any indeterminism or failure of conservation laws.

If energy weren’t being pumped into this system, the vanes would slow down and come to a halt. But Greg is keeping the black sides of the vanes hotter than the white ones, simulating the effect of sunlight. So, the system remains far from equilibrium. It’s thus a dissipative system in the usual sense.

I still can’t see where the forces are coming from…

Which forces, exactly? The forces should be described in Greg’s blurb about the applet. In particular, he writes:

When a particle collides with a rotor vane, the event is initially modelled as if the particle has struck a second free particle of equal mass which is moving normally to the vane, at a velocity consistent with the vane’s thermal energy. The incoming free particle is deflected on that basis, and then a balancing change is made to the rotor’s angular velocity and thermal energy, so that total energy and total angular momentum are conserved. This is a crude way of dealing with the interaction, but it allows the free particles to exchange kinetic energy with the vane’s thermal energy.

I’d like to know what ‘at a velocity consistent with the vane’s thermal energy’ means. Is this velocity a deterministic function of the vane’s thermal energy, or is there randomness built into the velocity of the vane particles’ motion?

I’m wondering if this is a deterministic dynamical system or a stochastic one. I’m betting the former is true, since Greg offers repeatable initial conditions, but my eyes aren’t good enough to see if the molecules do the same thing every time when we repeat the experiment with the same initial conditions.

and I think that the rotation we see could be an effect from the initial dynamics rather than the ongoing physics.

You mean the vanes start spinning because… why? The total angular momentum of the particles is initially zero: that’s not going to make the vanes start spinning.

What makes the vanes start spinning is that the black sides are hotter than the white ones. You can see this by starting out the simulation at a very low energy flux density and turning it up later. When you turn this up, the vanes starts spinning.

The question is just: precisely how does this work?

Posted by: John Baez on August 21, 2008 5:31 PM | Permalink | Reply to this

### Re: Light Mills

At the beginning all the particles are quite cold. When they hit the black side they get a large kick and fly off as hot red particles. You can see it happen. There is no surprise that this could give an initial push to the vanes which gets them started.

The problem is that this is exactly the explanation that Maxwell discounted. Once it reaches a steady state the gas is hotter and less dense around the hot side. The simple recoil effect on the vanes is supposed to go away and be replaced by a transpiration effect near the edges. Apparently Einstein came to a similar conclusion in a different way so I am not going to be quick to discount it.

So you see that the rotor is being kick started by the effect that we believe is not there once it enters a steady state. I suppose that effect could be real when you start the radiometer but once it turns there must be other effects keeping it going. We cant see what they are because we have no friction.

I agree that there should be viscosity and drag but we have not done anything to observe it. I dont really think that the motion is all due to the initial effect but without friction I don’t want to discount that yet.

Posted by: PhilG on August 21, 2008 9:18 PM | Permalink | Reply to this

### Re: Light Mills

Phil wrote:

The problem is that this is exactly the explanation that Maxwell discounted. Once it reaches a steady state the gas is hotter and less dense around the hot side. The simple recoil effect on the vanes is supposed to go away and be replaced by a transpiration effect near the edges. Apparently Einstein came to a similar conclusion in a different way so I am not going to be quick to discount it.

Just to be clear, I’m not discounting any of their work or offering any theory of my own. I just don’t think including friction in the vanes will help me understand what’s going on. But if you want it… well, see if you can persuade Greg to include it.

Besides the precise recipe for how particles bounce off the vanes, there’s something else I’m puzzled by. The flow lines on the applet seem to have little arrows on them, and these arrows seem to point in the opposite direction from how the particles are flowing. I typically see the little arrows pointing counterclockwise around the edge of the light mill — the same direction that the vanes are turning. But this seems to contradict both the total angular momentum being zero, and what I actually see when I carefully stare at the particles.

Posted by: John Baez on August 21, 2008 11:25 PM | Permalink | Reply to this

### Re: Light Mills

John Baez wrote:

I’d like to know what ‘at a velocity consistent with the vane’s thermal energy’ means.

As you guessed, I’m making this deterministic, for the sake of simplicity and repeatability. It’s just $v=\sqrt{2E_{side}/M_{side}}$ where $E_{side}$ is the thermal energy allocated to the appropriate side of the vane, and $M_{side}$ is just half the total rotor mass.

Posted by: Greg Egan on August 21, 2008 11:50 PM | Permalink | Reply to this

### Re: Light Mills

John Baez wrote:

The flow lines on the applet seem to have little arrows on them, and these arrows seem to point in the opposite direction from how the particles are flowing.

Actual arrowheads are extra work to draw – and can look quite ugly when very small without even more work to anti-alias them – so I’ve used a reversed convention of marking the “base points” of the vectors, at the centres of the averaging grid. I did mention this at the top of the page:

Each vector points away from the small black point at one end of the line.

I guess this can go against your visual intuition if you’re expecting arrowheads, but if you think of the lines as little hairs that the particles are “combing” and tugging out of their follicles … well, maybe that sounds weird, but it works for me.

Posted by: Greg Egan on August 22, 2008 12:06 AM | Permalink | Reply to this

### Re: Light Mills

Greg wrote:

Each vector points away from the small black point at one end of the line.

Oh, okay! I was expecting arrowheads; they’re so tiny and my eyes are so bad that I couldn’t tell they’re just dots.

Thanks for answering my question about the ‘thermal’ behavior of the vanes, too.

Posted by: John Baez on August 22, 2008 2:54 AM | Permalink | Reply to this

### Re: Light Mills

I’ve now added an option for a very simple model of kinetic friction: you get to specify a constant angular deceleration for the rotor, and the lost KE is turned into heat that is distributed equally throughout the rotor. So this model conserves energy, but not angular momentum.

Phil wrote:

The particles are large so they have a higher cross-section for hitting the corner of the vanes. If you make them much smaller the apparent edge effect decreases. However the mean free path then goes up as well.

You can always increase the particle number to bring the mean free path down again. The applet doesn’t grind to a painful halt for, say, $r=10^{-3}$ with $N=1000$. (And even if the applet becomes annoyingly slow, you can always just leave it running and do something else. It will record the net rotation at $t=100$ and leave it visible.)

The problem is that this is exactly the explanation that Maxwell discounted. Once it reaches a steady state the gas is hotter and less dense around the hot side. The simple recoil effect on the vanes is supposed to go away and be replaced by a transpiration effect near the edges.

OK, but don’t expect the relevant edge effects to be sharp, given the small size of the vanes compared to the mean free path. You’ve noted that the excess collisions at the very ends of the rotors are due to the large particle diameter, but when we shrink the particles (and increase their number), the rotor still turns, so something is still driving it.

Obviously in this simulation the drop in density on the hot side fails to perfectly cancel out the difference in pressure, hence the rotor keeps turning despite dissipative effects. Now, to some degree, somewhere the same must be true for real radiometers (even if it’s only near their edges). The strong likelihood that the “non-cancellation” will be spread over the whole vane in the simulation (not exactly uniformly, but not sharply concentrated at the edges either) is not in itself a reason to give up and say the simulation is telling us nothing useful about the reason why real radiometers turn.

Posted by: Greg Egan on August 22, 2008 2:57 AM | Permalink | Reply to this

### Re: Light Mills

Thanks for adding the friction. I find that with friction the vanes move faster at the start and then slow down. With small vanes they stop altogether. This seems to vindicate my criticism.

I was hoping and half expecting to be proved wrong. If anyone can argue that it does not slow down then I am ready to listen.

The angular momentum of the gas is not being dissipated which could be a problem. Perhaps the initial effects are not being fully lost because of that.

I dont think we should give up. My feeling is that the simulation needs more physical realism to show the right physics. It may need dissipation of heat via the bulb surface, an edge thickness to the vanes, smaller particles and much more of them. If that makes it more complicated we could look at simulation of other simpler but related situations such as transpiration through a pore, just to see if the right physics is accessible.

Is it possible to optimise the code so that we can run with more particles? If too much time is spent rendering it could be sped up by showing only a sample of them, or by not rendering every iteration. With many particles the interactions will become the bottleneck. There are ways to optimise the calculation e.g. using a grid. Has that been tried already?

Greg, I realise you must have other projects so your time to spend on this has limits. As it happens I also have some experience of writing Java applets so if you were willing to make it open source I could help out.

Posted by: PhilG on August 22, 2008 10:05 AM | Permalink | Reply to this

### Re: Light Mills

Phil wrote:

I find that with friction the vanes move faster at the start and then slow down. With small vanes they stop altogether. This seems to vindicate my criticism.

In what way? Sorry, but I honestly don’t understand the point you’re trying to make.

Friction slows the rotor down; this isn’t surprising, and I don’t see why you think it implies that there’s something crucial missing from the frictionless simulation. Having opened the door to non-conservation of AM with a crude model of friction at the rotor, sure, it might well take yet more complicated refinements – such as friction between the gas and the container – to make the whole thing act entirely sensibly. But when you say:

My feeling is that the simulation needs more physical realism to show the right physics. It may need dissipation of heat via the bulb surface, an edge thickness to the vanes, smaller particles and much more of them.

… I have no idea what you think is justifying this list. The simulation as it stands rotates (with or without friction; the fact that it will ultimately halt with a crude, partial model of friction is beside the point), and I can’t see that you’ve made any argument, yet, as to why it can’t be rotating for reasons broadly similar to the reasons why a real radiometer rotates. I’m not claiming that it certainly does rotate for the same reason, I just can’t see why you think you’ve shown that it doesn’t.

There are ways to optimise the calculation e.g. using a grid. Has that been tried already?

Yeah, I bin the particles into a grid, set a time step so the fastest particle can only cross one grid cell, and then only check for collisions between particles in the same, or neighbouring, cells.

As it happens I also have some experience of writing Java applets so if you were willing to make it open source I could help out.

It would take a certain amount of work to clean up and document the code to the point where it was shareable … and given the kind of things you’re suggesting you might find it much simpler to start from scratch anyway.

Posted by: Greg Egan on August 22, 2008 11:54 AM | Permalink | Reply to this

### Re: Light Mills

Phil, I’ve been re-reading your comments trying to understand your criticism of the simulation. Is it the case that you still believe that the rotor is only moving because of an initial kick that it gets before the hot gas has had time to expand and exert a lower pressure?

If you do still believe that, can you explain why your hypothesis is not disproved by the following. If you run the simulation with the default settings, at t=100, 200, 300 its net rotation is 6.111, 12.075 and 18.968. If you re-run it, but at t=100 cut the energy flux down to $10^{-20}$, its net rotation at t=100, 200, 300 is 6.111, 8.292, 8.291. Surely this proves that it is the ongoing energy flux that keeps the rotor turning, not some initial kick?

Now, if you run the simulation with all the defaults plus a rotor friction parameter of $10^{-4}$, its net rotation at 100-unit time intervals is 5.426, 10.602, 15.279, 19.806, 22.701. So sure, not only is it slower than with no friction, it’s gradually decelerating. Perhaps that’s just an artifact of the extra negative angular momentum building up in the gas, I don’t know. Maybe I should see what happens if I re-enforce conservation of AM by having the gas lose a balancing amount of AM to the container. But regardless of the can of worms that’s been opened up by rotor bearing friction … the first two trials still stand, and they still prove that the original simulation is not just turning because of an initial kick.

Posted by: Greg Egan on August 22, 2008 2:29 PM | Permalink | Reply to this

### Re: Light Mills

No I am not saying that I think the rotation just comes from an initial kick. I am just saying that we need to make sure that any intial effect is dissipated correctly and I think the best way of doing that is through realistic friction etc.

My other reason for wanting the firction is that it means there will be a net force from the gas on the vanes when it reaches a steady state so we have some chance of looking at where the driving force acts.

If you can solve the problem of understanding the forces without the friction then that would be fine too.

I’ll try and find time to set up my own version of the simulation so I dont have to bother you with my ideas. Meanwhile dont take my “criticisms” in a negative way. I think the simulation is very good and I just want to understand the physics better.

Posted by: PhilG on August 22, 2008 5:34 PM | Permalink | Reply to this

### Re: Light Mills

Greg wrote:

These make it pretty clear that the net force isn’t uniform along the vane, but I find it quite hard to make sense of what’s actually going on here.

Great!

A problem worthy of attack
Proves its worth by fighting back.

While your 2d light mill isn’t the same as the 3d version, it’s a perfectly respectable physics problem in its own right, with the advantage of being mathematically well-defined, fairly simple — yet still challenging!

I still haven’t had the energy to really make sense of the various ‘edge effect’ theories people cook up to explain the light mill. But, I’m getting the feeling that you’re seeing edge effects!

In my ideal world, you and Vaughan would coauthor a review article on the light mill, with him tackling the experimental side and you describing the results of this 2d simulation.

In reality, I’ll at least rewrite the FAQ at some point to reflect the discoveries you two have made, and point out some of the things we don’t know.

But before I do, I should at least read the literature on the subject. It seems like Vaughan has read a lot of the original papers by Reynolds, Maxwell, Einstein and others. That would be fun. Those guys weren’t dopes, and I’m sure we could all learn a lot from trying harder to understand their work on this problem.

I may also write a This Week’s Finds to drum up more interest in this problem — there could people lurking out there who know a lot about it, who don’t visit the $n$-Category Café.

Posted by: John Baez on August 17, 2008 3:31 PM | Permalink | Reply to this

### Re: Light Mills - sign and speed experiments

When I got home from BLAST on Sunday night one of the three radiometers I’d ordered (Bought Now from three \$10 sources on eBay) had already arrived. (The other two came on Monday, the photos had looked different but they turned out to be almost identical.)

The first thing I did with it was to confirm the circumstances under which it runs backwards, namely when placed in the freezer for ten seconds with the door closed, and when taken out of a toaster oven after giving the bulb a chance to warm up. In both cases it spins backwards even in complete darkness, showing that spinning could happen without significant incident radiation.

My next experiment (after letting it equilibrate to room temperature) was to place it in a dark box at around 370 K (a big and very solid iron box that I’d preheated on the stove), where it spun vigorously forwards.

By Wien’s Displacement Law for wavelengths the radiation from 370 K peaks at 2898/370 = 7.8 μm (wavelength peak, the frequency peak is 13.7 μm). Oxygen and nitrogen don’t absorb significantly in that neighborhood. CO2 does but it is implausible that at 384 ppm it could have a significant effect.

Although 370 K is much less than the Sun’s 5780 K, because the emitted radiation arrives at the radiometer from all directions the total power incident on the vanes is close to that of sunlight a few hours from noon (the vertical vanes don’t receive noon sun so well). So if frequency-independent heating of the vanes is the relevant effect the vigorous spinning observed in sunlight can be expected to a similar degree in a hot box, which it was.

These experiments show that my thermosphere-inspired vibrating-molecules mechanism is not necessary for the effect. Hence there must be some other mechanism. I am firmly convinced that it is not an edge effect—-some hours of playing with Greg’s great little applet have only reinforced this conviction; the spinning there seems to depend in an essential way on the fact that the mean free path is on the order of 10% of the bulb radius, large enough for edge effects to kick in. I would expect edge effects to be much less significant at 1%, the regime of a real radiometer.

Having shown that my vibration theory is not necessary, I then attacked the other direction: is it at least sufficient? This would be remarkable if so, because then the radiometer would have two mechanisms, that one plus the as yet unknown other one.

So I set up an infrared lamp and ultraviolet lamp and tabulated a lot of combinations of intensities of both. At first I thought I was seeing some multiplicative interaction between the two, but eventually I realized that the spin of the radiometer was obeying a 3/2 power law as a function of the incident radiation. If you’re not expecting this, the first few samples look a bit like multiplicative interaction, but once the 3/2 power law becomes apparent it becomes clear that the net effect of the two lamps is purely additive: the spin depends only on the total radiation from the two light sources. (I have no idea why 3/2.)

This experiment doesn’t prove conclusively that the vibrating-molecule mechanism can’t drive the radiometer, because the glass blocks so much UV and IR that the radiation that does get through is only good for the other (unknown) mechanism. UV targeted more precisely to oxygen absorption lines might help, as might Blake’s suggestion to use a quartz radiometer (wonder what they go for at Swarowski), though from what I’ve been reading, even quartz has a hard time getting down to oxygen’s best absorption lines at 100 nm.

While I’ve seen various pressures cited for when the radiometer spin peaks and vanishes, I’ve never seen an accompanying light intensity. The 3/2 law shows that this is essential: quadrupling the radiation makes the device spin 8 times faster. Claims as to the pressure at which it stops spinning should therefore be taken with a grain of salt. Whether the location of its peak spin depends on intensity is a nice question; if it did one might be able to use it as a poor man’s radiometer over a greater range. The pressure for peak spin that I’ve seen in more than one place is 60 mTorr.

I can certainly vouch for the efficacy of shaking the device to reduce stiction, which at low spin rates can make all the difference between 5 rpm and stopped.

I’m now working on a theory of the radiometer that doesn’t violate the ideal gas laws (unlike the thermosphere-inspired vibration theory). Mean free path is key, as is area (as opposed to perimeter) of the vanes. Holes in the vanes only dilute the efficacy of the mechanism I have in mind. More later if the relevant differential equations pan out. The idea is much simpler than either of Reynolds’s or Maxwell’s explanations, don’t know about Einstein’s because I haven’t grokked his 1924 German paper yet. Supposedly Einstein also focused on the edges, in which case my mechanism is different from his too.

Posted by: Vaughan Pratt on August 13, 2008 6:32 AM | Permalink | Reply to this

### Re: Light Mills - sign and speed experiments

I was starting to think that I was on some Ministry of Information Retrieval watchlist of potential terrorists, as it took an awfully long time for “scientific equipment” to be delivered to my home. But, I have now confirmed Vaughan’s observation with regard to the freezer, and I may be able to do something useful with my housemate’s blacklight LEDs.

Posted by: Blake Stacey on August 16, 2008 7:52 AM | Permalink | Reply to this

### Re: Light Mills - sign and speed experiments

Am I right in thinking that the reverse motion in the freezer makes sense simply in terms of the black side, being a better emitter, cooling down faster than the white side? So any explanation for what drives the mill one way when the black side is warmer than the white side will still apply, reversed?

Presumably the reverse spin in the freezer doesn’t persist for long, as the temperature difference drops.

Posted by: Greg Egan on August 16, 2008 8:50 AM | Permalink | Reply to this

### Re: Light Mills

We’re having an interesting dispute regarding Greg Egan’s simulation of the Crookes radiometer. Phil Gibbs wants to make the simulation more realistic and therefore more complicated. Greg doubts the need to add these extra complications.

The discussion is getting ridiculously nested, since Phil isn’t doing what it takes to prevent this — you need to click ‘Reply to this’, not on the post you’re actually replying to, but to the one it replied to! Sneaky, huh? So, I thought I’d try to move this discussion down here.

Now, there’s a real factual issue in dispute here: does the rotor spin just because of the initial ‘kick’ that it gets before it reaches a steady state, or not? I think Greg’s experiment shows that it’s not just an initial kick effect. But we can continue to argue about this…

On the other hand, there’s also a methodological issue involved. Are we trying to simulate a Crookes radiometer well enough so we feel sure we know how that gadget works? Or should we be happy if we understand how Greg’s simulation does what it does?

I’ve already argued for the latter. Perhaps it’s because I’m a mathematical physicist, but when I see a mysterious effect, I like to start by studying the simplest situation in which this mysterious effect occurs. After all, if we can’t understand a simplified model, how can we possibly understand something more complicated? Understanding of the simplified model is no guarantee that we’ll understand the full reality with all its complexities, but it’s hard to see how it could hurt.

(And, being a mathematical physicist, I generally let someone else tackle the more complicated aspects of a problem after I’ve done the easy stuff.)

Since Greg’s simulation already displays a counterintuitive effect, I’m eager to try to understand that, regardless of how much his model resembles a full-fledged 3d Crookes radiometer with all its complexities.

In fact, I would lean further towards simplification. If I were to seriously study this problem, I would probably want to tackle a radiometer with two vanes instead of four. I would also reduce the diameter of the central ‘spoke’ to zero.

Along these lines, I also enjoy taking Greg’s applet and setting the rotor’s outer radius to 100% and the inner radius to 10% — currently the minimum value, since the ‘spoke’ has nonzero diameter. This divides the radiometer into four chambers: particles can’t get from one chamber to another. In this situation, it doesn’t spin: it just wiggles back and forth randomly. It would be nice to prove this must happen.

It’s also interesting to ponder the flow lines in this situation. Should there be some organized ‘circulation’ of particles in this situation, or not?

To get the radiometer to spin, we need to shrink the rotor’s outer radius. Then we get a counterflow of particles around the outside, balancing the angular momentum of the spinning vanes.

That much is reasonably obvious. The real question, of course, is why the steady state situation has the radiometer spinning instead of at rest.

There’s nothing inherently paradoxical about this, since we’ve got a dissipative system. If you put a pan of water on a very slightly warm burner, the water won’t circulate — heat transfer occurs only by conduction, not convection. But if you turn up the heat, a phase transition occurs at a certain point, and the water starts to circulate, forming so-called Bérnard cells. So, depending on the parameters, dissipative systems can either display ‘circulation’ or not. The question is just how this works for the radiometer.

I haven’t said anything very substantial here, but at least it was fun saying it. I actually do think the last century’s work on nonequilibrium thermodynamics and dissipative ystems is one advantage we have over the greats like Maxwell and Einstein when it comes to trying to explain this gadget. (The ability to simulate things is another.)

For example, we could plot the steady-state angular momentum of the rotor as function of its outer radius, and see if it drops off smoothly as this radius approaches 100%, or whether it drops off sharply at some point, which might indicate a kind of phase transition.

Posted by: John Baez on August 22, 2008 6:00 PM | Permalink | Reply to this

### Re: Light Mills

Phil wrote:

My other reason for wanting the friction is that it means there will be a net force from the gas on the vanes when it reaches a steady state so we have some chance of looking at where the driving force acts.

If things had been different, maybe this would have helped, but what happens is this: In the case without rotor friction, in the steady state, the torques due to particle collisions in opposing directions at individual locations on the vane don’t cancel locally; only the sum of all these torques is zero. And when you add rotor friction, in the steady state the individual torques still display essentially the same pattern, but now there’s a slight readjustment so that the sum is non-zero, in opposition to the frictional torque. So I don’t see how the second case tells us anything new.

By the way, I’ve added an option for friction between particles and the container walls, and when a small amount of this is switched on, it does seem to lessen the deceleration I noted previously when rotor friction is switched on. But the interaction between these two effects is complicated, and it seems much harder to find a steady state than with the original model that guarantees perfect conservation of angular momentum.

Posted by: Greg Egan on August 23, 2008 4:05 AM | Permalink | Reply to this

### Re: Light Mills

I’ll repeat and clarify some of my points so they are not lost further up the page.

When you want to use maths to study a problem analytically it helps to keep the problem as simple as possible otherwise the equations can become insoluable. When you do a numerical simulation there are still reasons to keep things simple but they are different reasons. For example we would not want to do a full 3d simulation because of the increased computation required and the increased difficulty of visualising the results.

However, with this kind of problem you cannot avoid the complexity introduced by the fluid flow and other effects. We have seen vortices behind the vanes and we know that these flows create pressure changes due to the Bernoulli effect which then causes drag. Perhaps there are other forces in a moving rarefied gas that I dont understand yet. We are allowing the vanes to accelerate until the drag forces balance the forces that drive the vanes but we cant separate the forces and the effect we are looking for cannot be seen.

The average velocity of an oxygen molecule at room tewmperature is about 400m/s, that’s faster than sound. If we allow the vanes to turn at speeds comparable to the speed of the molecules we would bring in effects that are negligable in real life. I can see that with some parameters the vanes do turn slowly but in all cases without friction the fluid forces are allowed to balance the forces we want to see so we can only make the unwanted forces small by also making the wanted forces small.

To control all the different effects and separate them out we need to have more parameters to vary. We have to increase the complexity. I still think that friction from the vane spindle and friction between the gas and the bulb are necessary elements to slow them down and control the physics so that we can see the real effect.

In a real radiometer I undersatnd that the temperature and density distribution equalises quickly because of the large mean free path and the high velocity of the molecules provided the vanes and the fluid dont turn too fast. According to Maxwell they reach a steady state so that the pressure is constant except in regions were the temperature changes over a distance which is small enough compared to the mean free path. If the mean free path is small compared with the size of the vanes then this is only going to happen near the edges of the vanes where the temperature changes between the hot and cold side. This is the effect we want to observe if we hope to confirm the theory.

In the simulation we start with a state where the temperature and density is everywhere equal in the gas but the black side of the vane is hot. As the vane heats the nearby gas it gives an initial kick. The force acts evenly across the surface of the vane so it is unlike the effect we are trying to observe. Once the gas distribution is in a steady state this force should go away, but the momentum it has given the vanes and the gas could still be present. Greg and John have argued for that this initial momentum is dissipated. However, if total angular momentum is conserved I find it diffcult to eliminate the possibility that some part of that inital effect remains in play. I still think we need friction on both the vanes and between the gas and the bulb to be sure we dissipated it.

Notice that I am giving two separate reasons why I want friction included in the problem, If anyone wants to argue against using it they need to eliminate both arguments.

Greg has added in some new friction parameters which I want to try out, but I have also started working on my own version of the simulation so that I can try other variations out myself without bothering Greg. I’ll try to complete that first.

Posted by: PhilG on August 23, 2008 10:50 AM | Permalink | Reply to this

### Re: Light Mills

I’ve extended the options for rotor and particle/wall friction in the Light Mill applet to include “infinite” values. In a sense these might actually be simpler choices than zero – but they could also help reveal any localisation of the torque on the vanes, as Phil suggested.

By “infinite” rotor friction, I mean locking the rotor, and converting all the energy that would have gone into its rotational kinetic energy into thermal energy.

By “infinite” particle/wall friction, I mean that every particle that collides with the walls has its tangential velocity set to zero (while its radial velocity is increased to conserve kinetic energy; this is a crude approach, but I’m still resisting giving the walls thermal energy). Having particles bounce radially off the walls is a bit surreal, but it guarantees that the particles’ angular momentum doesn’t build up (which would otherwise be a consequence of locking the rotor), and if the mean free path is small enough the particles’ directions will be rapidly randomised by collisions anyway.

I’m trying a run with 10,000 particles of radius $10^{-3}$ (mean free path around .05), to see how the forces on the vane are distributed after the gas has had time to find a steady state with the rotor motionless.

Posted by: Greg Egan on August 24, 2008 9:28 AM | Permalink | Reply to this

### Re: Light Mills

I have made an alternative applet page to try out a few different ideas. My version does not have all the features of Greg’s and it has simpler ways of doing the interactions.

I have fixed the vanes which makes the demo a bit boring but for the purposes of measuring the distribution of forces on the vanes it is optimal.

I fix the temperature of the surfaces and adjust the speed of a particle after impact with a surface to match the temperature. This may be simpler than Greg’s approach but I hope it is good enough. The black sides are hot while the bulb and white sides are cold.

My biggest simplification is the particle interactions. Particles only interact when they overlap at a time step, then they undergo an elastic bounce. I think that has made the simulation run much faster so I have been able to run with 20000 particles.

I start with a cold gas so that you can see the effect of the heat at the beginning. Run it up to about t=4.0000 to see the residual forces on the vanes. The “force differences” are the net forces on the black and white side at the same radial distance. Inner radius is to the left.

Posted by: PhilG on August 24, 2008 5:49 PM | Permalink | Reply to this

### Re: Light Mills

Here’s a snapshot showing what the forces look like when averaged over a long enough time

The overall pressure around the vane equalises until almost constant, but there is a small excess force which is near the edges on the black side. In the central half of the vanes the forces balance almost perfectly between the black and white sides. The excess force acts on the regions that are within a distance from the edge comparable to the mean free path. This is consistent with the theory of the radiometer as explained in the Physics FAQ.

There may be some effects from the gas flow which complicate the force profile. Perhaps that is the reason why the excess force drops a little at the outer most edge where the gas is likely to be flowing fastest past the vane. But the fixed vanes and the gas friction with the bulb wall seems to have reduced these flow effects to a point where they do not obscure the overall picture we wanted to see.

Posted by: PhilG on August 24, 2008 10:45 PM | Permalink | Reply to this

### Re: Light Mills

I think these simulations are fantastic. As I marveled, a thought crossed my mind, “Just imagine what Einstein (or your favorite giant) could have done if he had Java at his disposal!” My initial reaction was that science would have progressed much faster if simulations like this were available from the beginning. Then the contrarian sitting on my other shoulder piped in thinking that Java probably would have hampered the development of science.

I don’t know which shoulder is right, but thought it was something fun to ponder so decided to share it.

Hypothetically speaking: Would science have progressed better or worse if numerical simulations were available from the beginning?

Posted by: Eric on August 25, 2008 12:15 AM | Permalink | Reply to this

### Re: Light Mills

There’s a different interesting question: if you assume that things like Babbage’s difference engine wouldn’t even with the best of circumstances have been capable to have scaled up to even something like Greg’s simulation, then numerical simulation becomes available with general purpose, easily programmable (I wish :-) ) computation. So there’s the possibility of using even simple machine learning techniques along with experiments. The only example I’m aware of in physics of trying to detect “interesting” things in physics using any kind of computational novelty detector is Wolfram looking at cellular automata. Along with the recent trend in “experimental mathematics” which, as I understand it, still relies on computers just for producing results which are still eye-balled for interesting phenomena. I wonder if automatic novelty detection will enter physics sometime in the next couple of decades (something like spotting the surprise 1-loop cancellations in N=8 supergravity being spotted.

(Automatic novelty detection is something I think a lot about in the context of my more engineering based problems.)

Posted by: bane on August 25, 2008 4:24 PM | Permalink | Reply to this

### Re: Light Mills

Eric wrote:

Hypothetically speaking: Would science have progressed better or worse if numerical simulations were available from the beginning?

Maybe some people would have taken the Wolfram view that analytically treatable problems are rare, unrepresentative and unimportant … but I don’t think that view would have triumphed, any more than it has today. As I see it, numerical simulations are just cheap, easy experiments carried out in parallel universes whose physics is much simpler than our own. I don’t think cheap, easy experiments are a threat to theoretical understanding!

Posted by: Greg Egan on August 25, 2008 4:55 AM | Permalink | Reply to this

### Re: Light Mills

That’s nice work, Phil! It’s going to take my own applet a matter of days to reach a steady state with 10,000 particles, but I’ll post the results when I get them. The run I’m doing has the inner edge sealed against the central spoke, so I don’t expect the pattern to be quite identical, but it’s looking as if it will yield very similar results.

Posted by: Greg Egan on August 24, 2008 11:33 PM | Permalink | Reply to this

### Re: Light Mills

Here’s a snapshot of what I got when I locked the rotor and ran the simulation with 10,000 particles up to about t=80. Similarly to what Phil reported, there’s near-cancellation of the forces away from the edges of the vanes, with the bulk of the net torque arising near the edges.

Posted by: Greg Egan on August 28, 2008 5:38 AM | Permalink | Reply to this

### Re: Light Mills

John wrote:

For example, we could plot the steady-state angular momentum of the rotor as function of its outer radius, and see if it drops off smoothly as this radius approaches 100%, or whether it drops off sharply at some point, which might indicate a kind of phase transition.

I’ve plotted that here; it’s pretty smooth, so I don’t think there’s a phase transition. Maybe that will show up if I look at varying the thermal energy flux instead.

Posted by: Greg Egan on August 28, 2008 2:13 PM | Permalink | Reply to this

### Re: Light Mills

It’s good to see that Greg got similar results to me. Having looked at the paper by Scandurra (physics/0402011) I realise that the forces we observed are the ones predicted by Einstein on the faces of the vane near the edges. Scandurra says that Maxwell’s transpiration force acts only tangentially on the edge surfaces which we do not include in the simulation. I am not sure this interpretation of their work is fully justified though. It would require some further investigation of the theories to sort it out.

In that case the FAQ needs a few changes e.g. where the hard vacuum case is mentioned (Did anyone work out carefully what would happen if one side is silver and the other side black as I described it in the FAQ?). And we need to say something about Einstein’s work.

Of course we could improve the FAQ further with some diagrams and equations, or even by using the applets to illustrate the effects, but that would mean a bit more work.

If someone was really keen they could use the simulations to test whether holes increase the force. I fear we would need to work with smaller mean free path and more particles to attack that question.

Posted by: PhilG on August 28, 2008 7:36 PM | Permalink | Reply to this

### Re: Light Mills

“Did anyone work out carefully what would happen if one side is silver and the other side black as I described it in the FAQ?”

I found that for a 4-vane radiometer either the black or silver sides can have the higher force depending on the angle of rotation. There is a stable angle where the vanes are turned by about 20 degrees.

Posted by: PhilG on August 29, 2008 10:06 AM | Permalink | Reply to this

### Re: Light Mills

Phil wrote:

I found that for a 4-vane radiometer either the black or silver sides can have the higher force depending on the angle of rotation. There is a stable angle where the vanes are turned by about 20 degrees.

I’m probably doing something wrong, or assuming something different, but I don’t get the same answer.

Suppose there’s incident radiation of unit intensity parallel to the x-axis, moving in the -ve x direction. I’m neglecting shadowing. A vane inclined at an angle $\theta$ to the x-axis will see the intensity reduced by a factor of $\sin(\theta)$. For a vane silvered on the clockwise side (i.e. exposed to this radiation if $0 \lt \theta \lt \pi$), twice the normal component of the momentum of each photon acts to produce a torque, and the normal component has a factor of $\sin(\theta)$, so overall the torque gets a factor of $2\, \sin(\theta)^2$.

Now, for a Lambertian reflector, the intensity of the re-emitted radiation picks up a factor of $\cos(\alpha)$ where $\alpha$ is the angle from the normal. The integral over a hemisphere of $\cos(\alpha)\,d\Omega$ or $\int_0^{2\pi}\int_0^{\pi/2}\cos(\alpha)\, \sin(\alpha)\,d\alpha\,d\phi$ is $\pi$, while the integral of the normal component gets another factor of $\cos(\alpha)$, i.e. $\int_0^{2\pi}\int_0^{\pi/2}\cos(\alpha)^2 \sin(\alpha)\,d\alpha\,d\phi$, which is $2\pi/3$. So when a matte surface absorbs a photon we get a factor of $\sin(\theta)$, but when it re-emits it the factor is a constant $2/3$.

Two vanes joined so the silver side of one vane at angle $\theta$ faces the radiation, while the matte side of the other vane (at an angle $\theta+\pi$) faces the radiation, will thus have a total torque with a factor of $2\,\sin(\theta)^2 - \sin(\theta)(\sin(\theta)+2/3)$, or $sin(\theta)^2 - 2\,\sin(\theta)/3$.

If we add the torque for a second pair of vanes at an angle of $\theta+\pi/2$, the total simplifies to $1-(2/3)(\cos(\theta)+\sin(\theta))$, or $1-(2\sqrt{2}/3)\cos(\pi/4-\theta)$. Since $2\sqrt{2}/3 \approx 0.942$, this function is always positive. So that would suggest that a 4-vaned rotor would always turn with the silver side trailing.

Posted by: Greg Egan on August 29, 2008 2:55 PM | Permalink | Reply to this

### Re: Light Mills

You are correct. I got so used to our 2D simulations that I calculated the factor for a 2D reflector instead of a 3D one. I got pi/4 where you got 2/3. For factors below sqrt(1/2) the silver side always gives the stronger push. Thanks for checking it.

I can console myself with the fact that this makes the conclusion in the FAQ right even if the radiation from the black side was neglected.

If the light source was inicident at an angle to the plane of rotation then the black side would get more advantage. I would guess that the vanes stop above a certain angle and might turn the other way at higher angles. That could be wrong. If I get time I’ll check it.

Posted by: PhilG on August 29, 2008 4:03 PM | Permalink | Reply to this

### Re: Light Mills

For four vanes with silver on one side and black on the other, I now find that if the angle of the light source to the plane of roation is less than 19.5 degrees the vane turns with silver side trailing. For angles between 19.5 and 48.2 degrees it will stop. Then for angles more than 48.2 degrees it turns with the black side trailing. I hope I got my sums right this time.

I wish radiometers could really be made to work like this as it would be nice to see the radiometer stop and then change direction as the sun rose.

Posted by: PhilG on August 29, 2008 4:47 PM | Permalink | Reply to this

### Re: Light Mills

Phil wrote:

Did anyone work out carefully what would happen if one side is silver and the other side black as I described it in the FAQ?

Did you try the paper I mentioned in the original blog entry?

• M. Goldman, The radiometer revisited, Phys. Educ. 13 (1978), 427–429. (Available here to those with magic powers.)

It’s got a pretty detailed analysis of a radiometer silvered on one side, black on the other, in a hard vacuum.

Of course we could improve the FAQ further with some diagrams and equations, or even by using the applets to illustrate the effects, but that would mean a bit more work.

Surely links to Greg’s applet and yours would be a very good idea, along with an improved bibliography.

Posted by: John Baez on August 29, 2008 7:59 PM | Permalink | Reply to this

### Re: Light Mills

I dont have the magic powers necessary to read Goldman’s paper but what he said in the abstract just confirms what Greg and I already worked out.

The book by Loeb on the kinetic theory of gases looks like a good reference both for formulae and historical facts. I can see about half of it in Google Books. Apparently it includes what is virtually an English translation of Einstein’s contribution. Loeb says Einstein’s work was the most complete and clear account of the effect. I’ll try to get a copy of it.

Posted by: PhilG on August 29, 2008 8:23 PM | Permalink | Reply to this

### Re: Light Mills

I’ve written another simulation, here, which models a single plate free to move only horizontally in a square container with the left and right edges identified. This simpler geometry might make it easier to come to grips with the underlying physics.

Posted by: Greg Egan on September 6, 2008 12:39 PM | Permalink | Reply to this

### Re: Light Mills

I’ve posted some rudimentary results for the simulation of a hot/cold plate free to slide in the normal direction.

These initial plots give the displacement at $t=100$ vs number of particles, $N$, for three different particle radii: $r=10^{-2}, 10^{-3}, 10^{-4}$. Curiously, the displacement reaches a peak around $N=250$ for both $r=10^{-2}$ and $r=10^{-3}$, despite the mean free path being very different. (For $r=10^{-4}$ the data is too noisy to be meaningful.)

Posted by: Greg Egan on September 13, 2008 6:47 AM | Permalink | Reply to this

### Re: Light Mills

Some more results for a single hot/cold plate: plots of average net force on a friction-locked plate versus number of particles are much noisier and flatter than plots of displacement when the plate is allowed to move. The very clear peaks seen in displacement just aren’t there in the force.

Posted by: Greg Egan on September 16, 2008 12:03 AM | Permalink | Reply to this

### Re: Light Mills

Some more plots in the hot plate simulation show something interesting, if obvious in retrospect: friction with the walls enhances the force on the plate. What’s plotted here (the 2nd and 4th force plots) is an extreme case, but the principle is that if the wall applies a frictional force to the particles, ultimately they transfer it to the plate – and the sense of that force is such that it would make the plate move faster.

Posted by: Greg Egan on September 24, 2008 11:56 PM | Permalink | Reply to this

### Re: Light Mills

That’s a nice result to see. If you have zero friction with the walls and fix the plate then the force from the plate should cause the gas flow to accelerate until drag forces round the plate bring it to a steady speed. The net average force on the plate at that point would be zero due to momentum conservation in the direction parallel to the walls. It is only because of friction with the walls that you can get a residual force on the plate in the steady state.

With the original bulb setup the same was true due to angular momentum conservation. Of course the same principle also applies if you allow the plate (or vanes) to move freely and keep the wall friction. To get a residual force over time you need both friction with the walls and friction in the movement of the plate.

Posted by: PhilG on September 25, 2008 8:24 AM | Permalink | Reply to this

### Re: Light Mills, then on to spirals

Sorry I disappeared there. I’m taking off for the UK in a couple of days and have been busy working on related stuff.

Phil, you really hit the spot with your mention of the book by Loeb. Section 84 is an encyclopedically complete history of the previous three-quarters of a century of work on the radiometric effect. A number of experiments support Reynold’s, Maxwell’s, and Einstein’s theoretical calculations that it has to be an edge effect. My intuition was wrong that edge effects wouldn’t be strong enough and that the explanation must lie elsewhere.

I think my intuition let me down when I tried to generalize my experience with 1 atmosphere down to 10-4 atmospheres. There is so little resistance to motion down there that far less force is needed to spin the vanes around at high speed than would be required at normal air pressure, helped by the glass bearing seated on a very sharp needle point, a cheap but effective low-friction bearing.

Changing the subject (should I be changing the thread here? I don’t know the convention), here’s another question, not much further removed from n-categories than the radiometer one. If from a point Z in the plane you draw an expanding equiangular spiral of angle α (defined so that α=0 gives a circle) about a point O in one direction, and its tangent at Z in the other, they eventually meet, say at A. Let φ be the angle ZOA. Using the sine rule for triangles it’s not hard to show that φ is related to α by

cos(α) = cos(φ−α)*exp((2π−φ)tan(α))

For the physics application I have in mind, actually just the geometry of moving slices of pie around within a pie plate, it is natural to ask how α depends on φ.

Clearly α tends to zero with φ, and almost as clearly tends to π/2 as φ tends to π (I described the spiral starting from Z rather than A to make this clearer).

Rewriting the above relationship as

α = atan(ln(cos(α)/cos(φ-α))/(2π-φ))

doesn’t give a closed form for α, since α also appears on the right. However if α is initialized to φ2/4π this formula when iterated converges to the answer.

Some numerical experimentation with bc with scale=200 and φ set to various powers of 1/10 reveals that

α = φ2/4π + φ4/24π - φ5/24π2 + .003872729996832391029424035567…φ6 + O(φ7).

Question: what is .003872729996832391029424035567…?

Trying to match this up to some formula involving powers of π and rationals has me completely stumped.

Posted by: Vaughan Pratt on September 14, 2008 8:21 PM | Permalink | Reply to this

### Re: Light Mills, then on to spirals

I think your problem can be recast as:

$e^{-(2 \pi -\phi ) t(\phi )}-\sin (\phi ) t(\phi ) -\cos (\phi )=0$

where $t(\phi ) = \tan (\alpha(\phi ))$. Taking the derivatives of this equation wrt $\phi$ at $\phi=0$ gives a series of linear equations for the derivatives of $t (\phi )$; solving these gives the series:

$t (\phi ) = \frac{\phi ^2}{4 \pi } +\frac{\phi ^4}{24 \pi } -\frac{\phi ^5}{24 \pi ^2} +\frac{\left(45+32 \pi ^2\right) \phi ^6}{2880 \pi ^3} -\frac{17 \phi ^7}{720 \pi ^2} +\frac{\left(455+68 \pi ^2\right) \phi ^8}{20160 \pi ^3} +O(\phi ^9)$

But if you want a series for $\alpha(\phi )$ itself rather than $t(\phi )$, the same general method and a bit more work gives:

$\alpha(\phi ) = \frac{\phi ^2}{4 \pi } +\frac{\phi ^4}{24 \pi } -\frac{\phi ^5}{24 \pi ^2} +\frac{\left(15+16 \pi ^2\right) \phi ^6}{1440 \pi ^3} -\frac{17 \phi ^7}{720 \pi ^2} +\frac{\left(805+136 \pi ^2\right) \phi ^8}{40320 \pi ^3} +O(\phi ^9)$

Posted by: Greg Egan on September 15, 2008 2:16 AM | Permalink | Reply to this

### Re: Light Mills, then on to spirals

Oh, of course. Why didn’t I think to expand cos(φ-α) as cos(φ)cos(α)+sin(φ)sin(α)? In fact I’d done so a couple of times but didn’t notice how much easier it made the differentiation and so pruned that branch as not helping. (I made so many mistakes differentiating my original formula by hand, which should give the same answer but with a lot more work, that I moved the differentiation approach down my priority list.) Bloody good job, thanks! Apparently I shouldn’t have left Perth (left there after eight years).

A computer search for an expression in rationals and powers of π for the constant in my question (which agrees with (15+16π2)/1440π3 in all but the last place, 7 instead of 8) would have been a lost cause: the number of pairs of cubics in π (for top and bottom) with coefficents between -1500 and 1500 is equal to the number of molecules in 244 m3 of air. On the other hand if the strategy were to enumerate the sparse ones first it wouldn’t take nearly so long.

Posted by: Vaughan Pratt on September 15, 2008 6:18 AM | Permalink | Reply to this

### Re: Light Mills

There is now a (still puny) entry on light mills at the $n$Lab.

Posted by: Urs Schreiber on December 1, 2008 10:31 PM | Permalink | Reply to this

### Re: Light Mills

In the Pioneer anomaly the acceleration is changed from calculated 7.8e-10 m/ss but adjusted to 8.74m/ss to account for radio transmission of 5 W on a vehicle mass of 374kg (IIRC). This happened in a vacuum and a 10% change is critically important. This discussion inre light mills raises the question as to whether this radiation force can be taken for granted. The answer appears to be yes and no.

Posted by: John Polasek on July 8, 2009 7:56 PM | Permalink | Reply to this

### Re: Light Mills

I hesitate to jump into this fascinating discussion among you professionals with a layman’s confused ideas, but I can’t resist. I would like to carry to the extreme John Baez’s suggestion that we simplify the problem. Let’s make the light mill one-dimensional with only one particle and one vane, with the motion of the particle and vane restricted to a circular track. I can think of two ways to define the particle-vane interaction that will give a working mill.

First, assume that the vane is totally absorbing on the cold side, so that a collision of the particle with the vane’s cold side is completely inelastic (all the particle’s kinetic energy is converted into the vane’s thermal energy). Now imagine that we have a pump that will move both heat and particles from the cold side to the hot side (does this sound a wee bit like transpiration at the edges?). Initially, imagine the lone particle as stuck to the motionless hot side. In due course the particle will gain enough internal energy from the hot side’s thermal vibrations to exceed its binding energy with the vane. At that point it will pop off the surface and give a slight recoil kick to the the vane. The particle wizes around the track and the vane moves slowly in the reverse direction. They collide, the particle sticks and all macro-motion stops (all kinetic energy being converted into thermal energy). The pump moves the particle and the thermal energy it left behind over to the hot side, and the cycle repeats. Notice that averaged over time we have steady-state motion with no initial transient.

For the second scenario, assume the vane is a one-way mirror, totally transparent when seen from the “cold” side and totally opaque when seen from the “hot” side. As before, start with the particle stuck to the hot side (actually, both sides can be at the same temperature in this scenario). As before, the particle acquires energy and pop off with a recoil kick. This leaves the vane moving slowly around one direction while the particle wizes quickly around in the reverse direction. There is no furhter interaction between the particle and the vane because of the transparency. In this scenario the initial kick is the whole story.

Perhaps I have simplified the problem so far as to beg the question; but, I would like to hear what others think.

When I was a kid I had a Crookes radiometer. As I recall, it would initially spin rapidly under a strong light; however, after a minute or so it would slow down, but not stop. So, I wouldn’t be surprised if the motion has both transient and steady state components. Then again, it has been over 50 years since I played with the thing.

John Baez - I love your blog. What amazing moxie you have to even imagine that you can usefully convey ideas as subtle as n-categories to an interested lay audience.

srwenner

Posted by: Steve Wenner on September 5, 2009 1:56 AM | Permalink | Reply to this

### Re: Light Mills

Thanks for taking a crack at this problem in a simplified form, Steve! I’m too distracted to think about this now, but someday I will write a This Week’s Finds on the light mill problem, and then I’ll try to get my thoughts in order. I hope that attracts the attention of some more people… someday this problem will be solved.

Thanks for crediting me with ‘moxie’. I think math and science are way too cool to let a few experts be the only ones who get to have the fun. The trick is figuring out how to explain the essence of an idea without getting bogged down in technical details. It’s hard. But it’s really worth it. For one thing, it’s a great way to understand the idea better!

Posted by: John Baez on September 7, 2009 7:23 PM | Permalink | Reply to this

### Re: Light Mills

Change the shape of the vacuum and you’ll work this out. The sphere’s circular equator compliments the blade rotation along itself, in either direction. All else being equal (including the vacuum’s imperfection), replace the spherical container with perhaps a cube or an irregular volume, and notice the blades dramatically slow down because photons and/or rarefied gas bouncing off the white side will now achieve only incoherent momentum. But here’s the real mystery: Which volume shape(s) will cause a total halting of the blades?

Posted by: John Titor on March 3, 2011 3:56 AM | Permalink | Reply to this

### Re: Light Mills

Bailing into this way above my paygrade discussion with my diffeq math and physics 181 background I have a question for the assembled geniuses here:
Do we have a predictive equation for the radiometer? And if so what is it?

Posted by: James Altman on July 15, 2012 1:46 PM | Permalink | Reply to this

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