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Sunday 30 January 2022

Well, Blow Me Down!

David Hume
Confidence is a funny thing, if you think about it a bit. Most often, we think of something like confidence in quite binary terms; as something you either have or not. However, out there in the wild, I'm sure we'd all recognise it as a many-valued spectrum. What may not be obvious is that confidence is derived from intuition, and that can sometimes be where the wheels fall off, because intuition is a minefield of shoddy thinking if not treated with proper care. Yet we often treat it as something approaching unimpeachable, because it's just common sense, right?

Common sense is something we often appeal to when something is obvious to us, but is it common? Is it sensible? Is it, in fact, obvious?

It is, in fact, an extremely well-understood fallacy of relevance; appeal to intuition. It does not follow that your having a particular intuition about something means that your intuited conclusion is true, and that remains true no matter how many share the intuition. Regulars will have some intuition for what's coming next (SWIDT?), because I have some go-to examples to highlight precisely why it's a fallacy.

Intuition tells us, for example, time runs at the same rate for everybody, no matter what. If this were true, however, our GPS systems would drift out of true at a rate of approximately 10 kilometres per day, which would cause enormous problems for our global communications system. It's obvious something can't be in two places at once. If this were true, the technology to share my thoughts with you would be a pipe dream. Common sense will tell you I can't walk through a wall. If this were true in reality, not only would the microchips in my computer be a fantasy, I would too, as would you and every other organism on Earth.

Intuition in and of itself, then, is generally a terrible guide to reality. This isn't to say intuition is entirely without utility, but it needs to be treated with a modicum of caution, and there are some things not at all obvious about it, as can be gleaned from the way we talk about it. We talk about 'a woman's intuition' or of somebody having 'good' intuition (often within a particular domain). The way we talk about intuition suggests we view it as innate; as a talent. In fact, it's a skill, derived from experience and requiring development, and practice, and perspective. It also requires a little humility. We have to recognise that, on occasion, really well-developed intuitions can steer us up the garden path if we're not careful.

Some years ago, an idea was proposed that did the rounds of all the places where people gather to talk about science. It began as a bit of a brain-teaser, but it turned into hot debate. For my part, my intuition resisted it. It seemed, on cursory inspection, to violate some principles of physics that are pretty much baked into the fabric of how we think about motion, and energy, and force. We've discussed them a fair bit hereabouts; conservation laws. Anyhoo, after a bit of discussion, I satisfied myself that it didn't violate anything, and all was still well in the world of thermodynamics. I pretty much forgot all about it, until I recently came across it again, and was surprised to learn the debate still goes on, despite observational evidence that no physical laws are being violated. It's obvious that some are clinging to their intuitions despite having reasonably well-developed intuitions. So what gives?

It occurred to me what's really happening from a cognitive perspective is that our intuitions are being fooled by the precise way we're thinking about them. In particular, we're excluding an important variable when we're thinking about them, and this is leading us to think there's a paradox. To my mind, all paradoxes are of this nature, and there are two instructive examples that should expose the thinking before we look at the solution to the brainteaser. One of these paradoxes is something we've explored before, but let's have another look.

Xeno's paradox is one of a collection of paradoxes attributed to Xeno of Elea in the 4th century BCE. The best known of them is the paradox of Achilles and the tortoise.

Achilles is in a race with a tortoise. Achilles, being sure of himself, gives the tortoise a head start. When Achilles reaches the point where the tortoise started, the tortoise has moved further on. When Achilles reaches that point, the tortoise has moved on again. Xeno asserts that Achilles can never overtake the tortoise, because each time Achilles gets to where the tortoise was before, it isn't there any more.

Of course, the resolution to this is incredibly simple and straightforward. What's happening here is that, despite time being ostensibly prevalent throughout in the form of 'when', it's actually being excluded throughout, in the form of speed, because the whole problem is cast only in terms of distance, when speed is distance over time \(s=d/t\). In other words, Achilles moves faster than the tortoise. Once we work out how much faster Achilles can run than the tortoise, we can trivially calculate how far the tortoise will get before he's caught. If the tortoise starts 100 metres ahead, and can move at 1 metre per second, while Achilles can run at 10 metres per second, he'll catch the tortoise at a little over 11 seconds. No paradox. The paradox arises only because time has been excluded, showing that this is a problem with intuition.

A related issue comes up quite a lot in discussion of climate change. The key slogan is, 'it's the sun, stupid', and the argument goes like this.
CO2 can't be the cause of the rise in average global temperature because, when we compare historic records of carbon dioxide levels being much higher, we also have records that the temperature was much lower, therefore the cause of global warming is simply changes in the output of the sun. 

They even have a graph I first encountered from not-science-advisor-to-Margaret-Thatcher-and-never-published-a-paper-in-a-peer-reviewed-journal Lord Christopher Monckton from two sources and comparing temperature to CO2 over 600 million years. There it is on the left.

See, no correlation. Not even anything like it. But, of course, there's something missing from this picture, and the denier has been sloganising it; it's the sun, stupid. In fact, when we look at the sources, we find that Berner, the source of the data for the CO2 portion of that graph, has shown along with Dana L. Royer et al., using data from GEOCARB III that, once corrected for all other known sources - including the evolution of the sun during main sequence - there's excellent correlation[1].

It's a really easy thing to do, allowing your first intuitions to lead you up the garden path, and entrenchment is never far behind.

So, let's get back to the brainteaser. I'm going to try to construct, one step at a time, a device. I'm very deliberately not going to show or explain the device up front, because I'm confident that all the confusion about this device stems from sticking too rigidly to intuition, so I'm going to construct it in such a way as to sidestep the intuitional sticking point. It will consist of a series of simple thought experiments that we'll combine at the end. If all goes well, you'll see that there's actually nothing you need to get your head around. It will amount to a series of physics tautologies leading inexorably to the conclusion that there's no paradox, and no mystery. That, in fact, paradoxes are nothing other than an indication that we're missing something important.

Fig 1

Let's start with a simple statement. All processes involving energy ultimately boil down to the equalisation of differentials. Energy is differentials, in a very particular sense. To see what that means, let's start with a simple board affixed to the ground. There are no forces acting on this board other than gravity*, so it's currently at its lowest energy state in this environment. That concept is important, because when we talk about the equalisation of differentials, this is what we're talking about. At all times, the board wants to get back to whatever the local analogue is of this state for whatever environment it finds itself in. What that is in any given situation will depend greatly on the attributes we give our board and the environment we construct for it, because we want to look at one idea from a couple of different perspectives, so that we can test our conservation laws properly and ensure that all symmetries remain intact as per Noether's Theorem**.

Anyway, we can imagine standing up against this board and pushing it, creating a differential by expending biothermal energy derived from food. Indeed, the food you eat generates a differential in you, which is why you can exert energy externally in this manner. There's a problem, though. The board is fixed to the ground, so the differential can't equalise until you stop pushing. You can't do any work. Work is the name we have for the process of equalising differentials. If you equalise a differential, even if only partially, you have performed work. This is what work means in physics. Force, as in F=ma, is the distance covered by work. You expend energy by doing this, sending energy down the gradient to higher entropy. This also constitutes work of a sort, but this work is generally excluded from our definition of work, because it's usually lost to the system as dissipation of thermal energy via friction, for example. 

Now we have our crash test dummy, let's add a sustainable force. For diagrammatic convenience, we'll situate our thought experiment in a wind tunnel. This gives us a means of generating masses moving relative to each other. We have two masses; a fast-moving mass - the wind - and a slow-moving mass - the ground - to which our board is affixed. The relative motion of these two masses generates a differential.

Fig 2
So now we have wind blowing on our board, which creates an area of high pressure on the windward side, meaning that the pressure on the leeward side is lower.

A naïve intuition might suppose that the low pressure area would be the same as the right-hand side in Fig 1 above, but the wind rushing past acts as a vacuum and, well, vacates it. In any event, we have a differential, and it wants to equalise. It wants to do work to get to a lower energy state. Unfortunately, the board is fixed, so there's nowhere for it to go, and there's nowhere for the wind to go, so almost all of it gets buffeted out to the side while that area of high pressure is maintained more or less consistently. This - as well as above when you're pushing manually on the board - is what's known as 'static thrust', where the thrust applied is insufficient to overcome all the contributions to drag and do work. Some of the energy expended will be given up as heat, because all energetic processes lose some energy to heat via friction. This is one of those cherished physical laws we mentioned earlier, in this case, our old friend the second law of thermodynamics. 

Let's add our first attribute and give the system a means of equalising that differential and moving toward a lower energy state for the environment.

OK, now it's starting to look like a machine. Anyway, we're obviously all good. Now the wind can push on it and the board has something it can do with the force that's being applied to it. The wind will push against the board, accelerating it. In the process of allowing the differential to equalise, of course, we've done something else. We've found a way to grab hold of the energy differential and do work with it. This is important. Any time you can find an energy differential and find a way to grab onto it, you can do work with it.

Anyhoo, the wind will keep pushing until the differential is equalised which in this case is a maximum somewhere short of the speed of the wind. That's because the wind isn't the only player in the game, as there's another player in the game; drag.

So drag, as the name suggests, is any influence that works to slow the board. All other things being equal, the wind would accelerate right up until the point where the pressure was exactly equal, where the pressure from the wind was equal to the headwind. This would be the local analogue of Fig 1, except headwind isn't the only contributor to drag, because the contact between the wheels and the surface generates rolling resistance, also a source of drag. The full analogue of fig 1, then, is where there are no further differentials in the system to do work. The board is travelling as fast as the wind can push it in the opposition to drag. Now, we're nominally at rest with respect to the wind. We still have two moving masses, but now the wind is the slow-moving mass and the ground is the fast-moving mass. 

Let's reset our apparatus, then, and construct another gedanken. This time, we're going to give our whole board an upgrade with a modern analogue of some ancient technology known as an Archimedes screw, and turn it into a fan on a pillar. Here it is. What we're going to assume, just to keep everything simple, is that the surface area of the fan presented to the wind is the same as that presented by the board. This helps keep track of the budget by not having to worry about it. 

Of course, the fan presents a different subtlety. The blades of the fan are angled along the axis of rotation of the fan. This means, of course, that the wind pushing on the blades will eventually turn the fan. We'll assume the blades are angled such that the when the wind hits the blades it will turn the fan clockwise. At this point, the fan is really a turbine, being driven by the wind. We've found another way to grab hold of the differential. We can now take this differential and do work with it, say, generating electricity, either by attaching a dynamo directly on the axis bearing or by using belts or chains to drive a generator. In fact, this is just converting it into a different type of differential (electrical potential) that we've found lots of interesting ways of doing work with. Still, extracting work from the wind isn't the only option to us here. How about if we attach a power supply to the fan?

OK, so now we can power our fan. Once again, let's see if there's anything we can do to get to the local analogue of Fig 1, the lowest energy state. All we have to do is get it turning anticlockwise, against the direction that the wind wants it to turn in, sufficiently fast to equalise the differential, by dragging air through it at a rate equal to that of the wind. 

Let's replicate the 2nd gen in this form††. We now have a cart, driven forward only by the wind. In terms of force, it behaves functionally like the previous 2nd gen. The surface area of the board is the same, and it's operating just like a sail. Add a power supply, and we can rotate it anticlockwise. Get it rotating sufficiently quickly, and we could even imagine, one day, being able to put this on a machine and have it pull you through the air. I'm being slightly facetious now, of course. Everybody knows how propellers work, and nobody's intuition is anywhere near being violated when we discover that propeller-driven aeroplanes are perfectly capable of taking off with a tail wind. We can do work and both equalise and increase that area of high pressure by getting the details of the propeller and the energy input right.

Without a power supply, it will be accelerated up to approximately the same terminal velocity as the original board. but wait, maybe there's a way we can get something from this situation. Let's get a bit relative. That doesn't mean we have to wheel out Einstein's field equations, of course, because relativity has been around in some form since at least Galileo. So what do we mean by relativity in this context? Well, Galileo identified some symmetries in nature. We've met them before, of course. He noted that, when we do an experiment facing East, it should yield the same results as the same experiment conducted facing West. A quick look at Noether's Theorem tells us that this symmetry corresponds to a conservation law - conservation of angular momentum. He also noted that an experiment conducted at one location - Pisa, say - should yield the same result as the same experiment conducted in London. Noether's Theorem tells us that this symmetry corresponds to conservation of momentum. And finally (for our purposes; Galileo was far from done at this point), an experiment conducted yesterday should yield the same results as the same experiment conducted tomorrow. Noether tells us that this symmetry corresponds to conservation of energy.

There are some subtleties to this when we put it all together, and they have interesting consequences. For example, combining all these differentiable symmetries together, we come to the conclusion that, if you were inside a closed box on the cart, there would be no experiment you could conduct that could tell whether you were being pushed along at windspeed in one direction or on a treadmill running at windspeed in the opposite direction. They're functionally identical, and it's all because of these symmetries and the quantities being conserved. In fact, it's not just because of these symmetries, it IS these symmetries. It's what it means for these quantities to be conserved. 

Let's pop our 2nd gen fan-sailed cart on a treadmill and we can see how that manifests in the real world. This allows us to do some interesting experimental revisions that tease out some details. First, we could connect our propeller to the wheel. We've already seen how this works above. With the wind turning the fan, the fan acts as a turbine, and the wheels will turn the treadmill. In practice, you'd have to fix the cart in some way, but there's nothing about using a wind-driven turbine transferring angular momentum to another rotatable entity that rubs anybody's intuition the wrong way, because it's the same principles we've already explored above. A dynamo is mechanically no different from an axle. With the right care, we can easily imagine the turbine slowly being spun up by the wind and driving the treadmill until it reaches the environmental analogue of Fig 1.

Now imagine reversing the process. Switch the wind off, reverse the gearing so that the wheel is driving the propeller, and run the treadmill. Again, because of those conservation laws, this is experimentally equivalent to the situation we just had with the wind on and the treadmill off. We're taking advantage of exactly the same energy differential whether we're tapping into wind energy or the energy of the road rolling by underneath. Noether's Theorem tells us these situations are identical. We can trivially see that the treadmill can run the propeller, because it's mechanically identical to the analogous situations we've already seen. Now, imagine designing your gear ratio and your propeller in just the right way, with just the right angular plane to optimise thrust-to-mass, and you could even see your propeller driving you forward on the treadmill under only the propulsion provided by the treadmill. Any net thrust provided in this manner is, because of all the equivalence and symmetry principles we've discussed above, provided by the propeller, which is getting its motive force from the wheels. The propeller allows it to take the force provided at the wheels and convert it into additional motive energy. Except it isn't additional, it's right there, rotating away for the entire time the wheel is in motion. 

Now for the final stage. Switch the treadmill off and switch the wind back on. Now, we have the same setup as the last experiment, with the wheels providing the rotational force to the propeller, and the wind providing the forward propulsion to the sail. We already know, from our experiments above, that the wind will push on the sail up to the point of equilibrium. We also know that this is exactly equivalent to standing still on a treadmill with the road rolling past at windspeed. We also know, from our last experiment, that the rotational force imparted to the wheel by the road can be transferred by a suitably ratioed gear to a propeller capable of providing a net thrust based only on that rotational force, so we have a question: 

Can we construct a device that is powered solely by the wind that can outrun the wind?

We've touched some important lessons along the way here, albeit very, very rudimentary. Nothing about any of the above is anything other than extremely basic physics at this point, and there should be nothing remotely unintuitive about it. I hope I've shown with extremely simple physics that it must be possible to do this and that now laws are being violated. We know when something like wind is moving past there are ways we can grab onto it. We've generalised that to any differential you can grab onto and extract energy from and turn it into work. So we have our cart with a big fan on the back, moving at approximately windspeed. Is there a differential nearby we might be able to grab onto? Why yes, yes there is. There's something rushing past you rather quickly that you can grab onto. In fact, you've been grabbing onto it all along, and it's been imparting some of the wind energy into your cart ever since it started moving.

The differential is there, and we have a way to grab it, so this becomes an in-practice question rather than an in-principle one. It becomes whether sufficient energy can be extracted from the wheel to drive a fan fast enough. Of course, with good gearing, we can control the ratios pretty tightly so that we can imagine a large surface turning reasonably easily with minimal input, and with the angles optimised for surface area against the wind, taking lessons from high-speed sailing and other areas... 

There will be limits, of course. First, if you try to grab onto that rotating axle too hard, you're going to increase drag, which requires more force to combat. The limits are ultimately a matter of how much you can extract from a given amount of wind from the friction imparted to the wheels for a given amount of drag. It's not a trivial machine to construct, certainly, but nor does it involve anything beyond fairly basic engineering and aerodynamics. It requires technology that's been around since Archimedes and principles of physics we've had in our locker since since Galileo. 

Meet Rick.


Oops, sorry. That's not Rick. This is Rick.
Rick is the man who posed this brainteaser. And yes, that's him, in the vehicle - Blackbird - and yes, of course it works, for all the reasons we've seen above. This photo was taken while setting a landsailing world record, and is actually running upwind at approximately twice windspeed by reversing the gear-train so that the fan is operating in turbine configuration and driving the wheels. Downwind, the record is 2.8 times windspeed, all entirely powered by the wind via tapping into this entropy sink and extracting work from it. This was the lightbulb moment for me when I examined this a decade ago. As soon as I saw where the differential was, I was satisfied it would work. 

Here it is again, working, upwind. 

So, what have we learned? As I pondered why this might have run up against people's intuition, I cast back to my own experience with it more than a decade ago. The problem lies in the thing that's being excluded. Once we've defined the experimental setup in our minds, we do exactly the equivalent of what we do in the other two problems above, and treat a critical factor as if it isn't even there. In Xeno, we excluded time, in the climate denier argument, they excluded the sun. Here, we're excluding the ground. 

What's really happened is that we defined the ground as the backdrop, rather than what it actually is, namely a dynamic participant in the experiment in all its iterations. Once we've defined the experiment, we forget that the ground is even there, and only think about the fast-moving mass - the wind in this case, when the role of wind and ground is dynamic in this case, and they swap places from the cart's perspective as velocity increases or decreases. Because of this dynamic relationship, and because of the relationship you have at the interface of these moving masses, you can rub one against the other and get some static, as it were. You can extract work from them by grabbing onto this differential.

It is, in fact, very much the same failure of intuition we encounter when we hear about Schrödinger's Cat. We think of the experiment as going on inside the box when in fact cat, box, and observer are all critical components and may not be separated in this way (caveats on 'observer'. Regulars will recognise that misunderstandings of the observer effect are among my top pet peeves).

Anyhoo, I hope this has been fun, and has suitably armed you to be wary of when even well-developed intuitions can fail. 

[1] CO2 as a primary driver of Phanerozoic climate - Royer et al - 2004

 Further reading: 

On The Shoulders Of... Emmy Noether and the unification of conservation laws.

Order, Order! Entropy and thermodynamics.

All Downhill From Here. Entropy, energy gradients and lowest-energy states.

The Dark is Already There Gravity, freefall and equivalence.

Paradox! A Game for all the Family  Paradoxes and intuition

Rick Cavallaro's Youtube channel for real world experiments showing all of the above, including more videos of Blackbird in action.

* Counter-intuitively, gravity is acting to stop the board from free falling. We think of gravity as something that holds you down, but that would mean you were subject to it in free-fall. As Wesley Crusher taught us in The Dark is Already There, free-fall is the freedom from all forces, meaning gravity doesn't keep you down, it stops you from falling.

** In fact, Noether constructed her theorem from first principles with no prior exposure to thermodynamics. She'd been asked by world-renowned mathematicians David Hilbert and Felix Klein to come and explain to them how energy was conserved in this theory by this new fella Einstein. She went back to first principles on conservation so that she could build up sufficient understanding to address the question, and unified all our conservation laws in one go. She should be as famous as Einstein, or Curie. For more on Emmy Noether and Noether's Theorem, see On The Shoulders Of... linked in further reading.

† The way I've phrased this has the potential to mislead. We're not, of course, at rest with respect to the wind, hence 'nominally'. In fact, because of drag, we're still moving at less than windspeed.

†† In fact, even this is slightly misleading, because the fan on a pole is more directly analogous to the board on wheels. In the rest frame of the ground, the fan is a way to grab hold of the wind. In the rest frame of the wind, the wheel is a way to grab hold of the ground. Each is performing precisely the same function from their own perspectives. That observation is a slight complication for our purposes at this point.

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