The Asteroid Setup That Demands an Explanation
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Ok, so, you're starting from some correct premises (there are no perpetual motion machines) and reaching some correct conclusions (if you extract work from a system that is at thermal equilibrium, you are necessarily destabilizing that system and cannibalizing its structure).
But, there are a number of imprecise, incorrect, and/or sloppy reasoning steps in the middle that are, like in the Planet X thought experiment, causing you to look a lot farther afield for for the details of how and why, when the reality is an explanation that is a lot more mundane.
I'm not going to try to be comprehensive in explaining why, but here are some initial observations:
- The setup is already unstable, whether you extract work or not. The moment you start letting the atmosphere impact the shell, the atoms are imparting momentum, and even if that momentum imparted cancels out on net over time, the very first impact has already pushed some part of the shell closer to, and another farther from, the central asteroid. This creates a net gravitational force that will eventually cause the shell and the asteroid to collide. It also should cause the shell to rotate, but I'm not sure if that matters.
- The discussion of helium atoms and graphene sheets adds unnecessary complexity without changing the conclusions. Make it a classical universe with a perfectly elastic shell and a cloud of tiny, unbreakable rocks, and the rest stays the same, with fewer opportunities to accidentally hide implications from yourself.
- This includes the point that @J Bostock noted where you switch between a thermodynamics model of the helium atoms in some places and an individual-billiard-balls model in others, or between a thermodynamic model of the CMB and a zero temperature model of the shell. If you have sufficient information about the positions and trajectories of individual objects to extract work from them, then you are not operating on a thermodynamic ensemble of atoms at 5K, you're Maxwell's Demon and T=0. If you don't, then 'extract energy from individual atoms as they fall towards the asteroid' is not a meaningful or actionable statement.
- Another is your comment about atoms ending up in lateral motion near the shell. I don't think this is correct, and that would be a lot easier to notice if you replace the mental model you have of a thin gas with a model of a cloud of rocks and dust in orbit around a central asteroid at various altitudes and speeds. You don't get a buildup near the shell. Instead, any object whose initial trajectory would take them beyond the shell ends up in a decaying or highly elliptical orbit that takes it back towards the central asteroid, and then (unless it collides with something) back towards the shell. There's never any kind of pattern of inelastic interactions that slows the objects down to the right orbital velocity to stay near the shell.
- If you want to tighten your thinking up, go back to every single instance of passive voice, and everywhere you use words like "almost" or "sufficiently."
- Replace them with simple declarative sentences. What object has what interaction with what other object?
- What variable approaches what value along what kind of trajectory? Think big-O notation. Does it happen in finite time? If not, how does it approach what asymptotic value? As 1/ln(n)? 1/n? 1/n^2? e^-n?
- The above is important to help understand which small effects dominate and which vanish. You have a lot of effects mentioned that you seem to be considering negligible, when it matters which are more negligible than others. One of the things that I think you'll find approaches zero more quickly than most is the rate at which it is possible to extract work from the system. This is a general property of systems designed to be highly efficient in terms of entropy generation.
Also: Back on the previous topic of simple restatements of the second law, there is always the standard cheeky description of all three laws of thermodynamics:
- You can't win.
- You can't break even.
- You have to play the game.
Thank you Anthony! Truely!
There are many things in what you write that are unquestionably worth looking into.
As I stated earlier, I really think your knowledge as a materials scientist is invaluable for helping me understand this. I do have both questions, and objections. But I do not want to waste your time. If (and as long as) you are interested, I could write down some of my thoughts. Right now, I just fear, I might have presented something that was percieved as ugly, and improper.
If you think a bit of back and forth may be interesting, please let me know. I am quite confident you would find some of my objections and questions at least partly valid. ^^
I add a rather dumb example too (please fill in the obvious blanks):
Suppose you threw tennis balls into the air, several balls each milisecond. Suppose you placed a jagged roof at hight h. Without the roof the tennis balls would travel to height 2*h. Suppose the jagged roof scatter tennis balls in all concievable directions. Would you accept that you get an accumulation of balls in the vicinity of height h compared to what you would get without the jagged wall?
What you're doing by making the roof more jagged is relaxing what you mean by being 'in the vicinity of height h.' You don't have a precise enough definition for that to be a well-formed question. The jaggedness means the roof's height is not really a single number, it's a range. We haven't discussed either the specific roof shape or the distribution of the balls' trajectories (and thus their horizontal momentum and their kinetic energy distributions). On colliding, a ball will either be deflected net-down or net-up, and in the latter case it will soon hit again, and again, until it deflects sufficiently net-downwards or until gravity reduces its vertical speed to zero. So, sure, when the roof's jaggedness increasing its maximum height by some j<h, then on average the balls will stay in the air longer, and the additional time will mostly be spent between height h and height h+j. And because the vertical speed at height h+j will be lower (even for undeflected balls!) than at height h due to gravity, the fall will start out slower than you'd get from a perfectly elastic deflection from a flat roof at height h. If j is tiny, the roof can't be that jagged, and so the effect on ball distribution will also be tiny. If j is large, with such a shape that many balls can actually make it significantly beyond h, then you can't call it a 'roof at height h' anymore.
Suppose h=10', and j=1', and the jaggedness is set up in a way that makes the average roof height 10.5'. Then what you're saying amounts to something like: Before the balls were vertically distributed quadratically, like if you'd had them following the usual gravitational parabolic trajectories but just truncated off all the time they'd have counterfactually spent in the top half of a height-2h room. But now the room is ~5% taller, and the balls spend nonzero time in the new top 5% of it, and we're only truncating the top 47.5% of the parabolic trajectory on average, and we have added more ways for the room to interconvert vertical and horizontal moment.
Obviously I haven't done any simulations or written down any equations to estimate the actual new distribution quantitatively. That would depend on the specific roof shape in ways I can't easily capture in a simple equation (maybe someone else could, but I can't). Even still, rephrased the way I put it above, that's not nearly as surprising as it sounds when you stay vague and handwavy about it.
I suspect you could define a roof shape and a distribution of horizontal momentum vectors such that the balls would on average be deflected down faster than in the case of the flat roof.
Now, if I were to make the roof sticky instead of jagged, then sure, the balls spend more time there right at height h. But then the roof is absorbing the momentum and kinetic energy, producing heat in the process.
Okay. So, I will try to break this down into sections. We have what I believe are agreements, questions and possible disagreements.
To set the stage, let me start with agreements (most of the times followed by a “however”):
Agreements
- The standard cheeky description: I 100 % agree. However, I think speaking of structural degradation may serve as a guide for finding ways of using up structure as a means of converting a fraction of heat used in the breaking of structure into useful energy.
- Big-O: I 100 % agree, and I find your intuition “One of the things that I think you'll find approaches zero more quickly than most is the rate at which it is possible to extract work from the system. This is a general property of systems designed to be highly efficient in terms of entropy generation.” Entirely valid, and plausible. However, this system is complex, and I lack many of the skills you have.
- Model switching: Yes, this is something I do. However, I am interested in what happens at different levels of the setup. If one model can be used in one instance (to best explain certain features), and another model seems better suited for analysing a different section (for certain features), I will do so. I am interested in things like: “Will any atoms evaporate?”, “How will this affect the atmosphere?”, “Will we get a density accumulation close to the shell?”.
- Not adding unnecessary complexity: Yes, that is good advice. However, when I do this, I may be criticized for doing so (e.g. for not adding details about keeping the shell in place).
- What you call the correct premises.
- I agree that the apparatus being used for energy extraction will necessarily have a degradation scaling with the energy extraction. However, this would be equally true for e.g. solar cells. Are you saying solar cell structural degradation are bigger than energy gains? Basically, that energy converted by the solar cell, would not be sufficient to fix solar cell structural degradation (if structure from the outside couldn’t be brought in). If so, I agree that this is an intriguing possibility (from my current knowledge base and vantage point).
Questions
- Would you prefer for me to tighten my language, as best as I can, or shall we deal with specific questions in a back and forth right now? I am not questioning your main point here.
- Follow up: Would you like for there to be a Google Doc, where I try to express things like you suggest, like “What object has what interaction with what other object?” and so on?
- Have I missed something obvious in this reply? Like mischaracterized you, or seemingly missed something important you said?
Possible disagreements
- Perhaps not necessary, but the shell could be seen as “mostly outside the gravity well”, say 31 radii away, making gravity 0.1 % of gravity close to the asteroid. The shell can be attached with wires or pillars, to ensure stability. A symmetrical arrangement of four carbon nanowires ought to be enough in an extremely stable environment with very low level of unbalance and temperature (5 K).
- If I have a container in an environment of comparatively high air density, I could just close a lid on a jar in order to trap some air. No information about individual objects needed. If I sat in the International Space Station, I could easily entrap some air. And if the ISS hang above Earth, I could definitely lower my jar towards earth, get some potential energy before earth density became a thing to worry about, open the jar, and release the air, and pull my jar back up again.
- I will try to tackle this “lateral motion” question as well as I can. You say: “There's never any kind of pattern of inelastic interactions that slows the objects down to the right orbital velocity to stay near the shell.” Let us be careful here. I will list a few premises and statements, and you can tell me were I am wrong:
A) Premise: Temperature and gravity well are aligned such that only a tiny minority of atoms will be able to overcome the gravity well.
B) Premise: There are no rotations to take into account (as opposed to here on Earth)
C) Statement: A majority of molecules leaving the atmosphere will move almost radially towards the shell. Leaving radially requires the least amount of speed. Given the speed distribution of molecules at a certain temperature a vas majority of the helium atoms will travel towards the shell along a very radial path.
D) Statement: The speed of atoms hitting the shell are very low for most atoms.
E) Premise: The shell is not microscopically smooth. It is a standard surface, and atoms “bouncing” against it are very unlikely to go out along the same line they came in along.
F) Statement: Temperature in solids can be thought of as random vibrations. As an atom comes in, it will get some kind of push redirecting it. This will cool the shell (since the incoming atom is so slow). Some incoming atoms will get a slow speed (compared to the reference mean speed associated with a temperature of 5 K), some will get a fast speed. Some will go back radially towards the asteroid, most will not.
G) Statement: Most of the atoms traveling out will be pulled back to the asteroid eventually, due to gravity.
H) Statement: There will be an increase in density close to the shell, as opposed to in the space between the shell and the asteroid. I suspect you may have objections here, but I am not entirely sure why.
Agreements
3. Model switching: Having multiple models at different levels of precision and abstraction is useful and switching between them is useful. But, you need to make sure that when you switch, you really make all necessary changes and understand which points you can carry over and which need what kind of reassessment or adjustment. Otherwise you're introducing new and unnoticed errors every time you switch. Doing this well enough to form a useful thought experiment means writing down, as an equation or very precise verbal description, every boundary condition, every initial condition, and every force or law governing the evolution of the system.
4. Complexity: The point is, it is a mistake to consider such details unimportant. You mention keeping the shell in place - in place relative to what? Those "details" mean either some sort of active thrusters that consume work, or some sort of extremely long tethers or pillars that change the set of reference frames with respect to which you're defining the velocity of the particles moving around. They mean your shell and asteroid are not at rest with respect to the reference frame of the CMB, which creates some Doppler shift so the flux is not spatially uniform (turning momentum into a temperature differential, among other things), and also the speeds and frequencies at which the atoms hit the shell and return to the asteroid are not spatially or temporally uniform.
6. It's not about degradation, it's about being able to define such a mechanism at all. You've essentially define a balloon around a thin gas gravitationally bound to an asteroid, such that it has a scale height. If you know where every atom is, then sure, you can intercept the ones falling down and ignore the rest and thereby do work. But then you can't talk about T=5K, because you actually have and are relying on your knowledge of the specific microstate. For you, who somehow has such knowledge, T=0, or at least, T<5K. Otherwise, if you don't have such knowledge, then whatever your try to set up will have to also deal with the atoms moving upwards balancing out the atoms moving downwards, and produce no net work. Solar panels do not have this problem. They have a net flux of high-T sunlight with a known thermal distribution of photon energies coming in to a lower-T environment, and this creates a very predictable theoretically efficiency limit based on the panels' composition. What gets 'degraded' is the sunlight's photon distribution, not the panel's structure.
Questions:
- I think tightening up your language as described will require a lot of tightening up of your thinking, and make it clearer what is going on. So yes, you should try to do that first and then see where that leaves you. But feel free to ask what you want to ask along the way.
- No preference.
- I don't think so, no.
Possible disagreements:
- It really can't. This does not work or help. See above. You're still trying to call things "small" without comparing their effect size to the other effects you're using them to balance or dismiss. To put it another way: Is there some finite limit to shell radius beyond which you think your model doesn't hold up? What happens as you increase the shell radius without bound, or decrease it to be much small? Which effects scale in which ways, which claims break in which order? If there is no clear upper bound you can reason out in this way, then you could just remove the shell entirely (aka place it at infinity) without changing any implications, which I don't think you believe.
- This works because of the jar, and does not work without the jar. Without the jar, you're essentially claiming that there is some height above the air at which you can place some apparatus in 'still' air which will nevertheless produce net work by extracting energy from falling gas atoms in a way that is not balanced by the forces being applied to the apparatus by rising gas atoms. This is why the thermal-vs-individual-objects model switch matters. You can see the jar, and choose to act on information you have about the jar, and the cost of acquiring such information is small relative to the work you can get from the jar. This is not true for individual atoms.
- Ok
- Assuming this is about the starting conditions, sure. Not true if you're saying this stays true over time
- Ditto
- Almost is not entirely. Over time the 'almost' adds up
- True, if you set up the initial distribution carefully, but again, drawing conclusions based on this requires a semi-quantitative understanding of what 'very low' means.
- Ok, so not graphene as described, then :-) I was assuming physisorption on a low-energy atomically-flat surface with very high and uniform emissivity due to it being a zero bandgap semiconductor. It's been a while but I took a whole class in grad school on low temperature vacuum pumps
- Yes, a lot of the thermal energy in solids is phononic. No, your conclusions don't follow, because the real interactions are mediated by specific mechanisms that meaningfully change the result, especially in a thin atmosphere at very low temperature. Classical approximations like heat and temperature are inadequate to predict outcomes here. The phonon density and wavevector distribution, and their effects on momentum and scattering, are quantized and that matters. Example: I once had a conversation with someone who was building quantum computers. They told me about a time they had a piece of metal sitting on a substrate in a very cold vacuum and it just wouldn't cool off, instead staying at the same temperature for days. Turns out the metal wasn't clamped hard enough to the surface it was sitting on. There was essentially a phononic band gap between the metal and the surface that meant the heat just couldn't get out in their setup by conduction without phononic tunneling through a large energy barrier. And it was already too cold for radiative cooling to help, especially since that also would need to account for the phonon momentum distribution since photons have very little momentum. "Very slow atoms bouncing off a very cold surface" is the kind of scenario where you just can't rely on classical approximations.
- This is also an instance of insufficiently careful model switching. Are we talking about a gas at rest on average, with a shell positioned at many times the atmosphere's scale height? Or are we talking about atoms which tend to convert radial into orbital motion over time (with large enough mean free path that they can orbit in opposite directions without too many collisions)? In the former case, sure. In the latter case, no. Your original description assumes both, without relative quantification of how large each effect might be.
- I think my other response about the tennis ball thought experiment should help clarify this. Again, you're stating multiple assumptions that are approximate and that push in opposite directions, or that can be operationalized in ways that have many different implications.
Thank you, Anthony!
I learn some new things here. Mostly I learn about how much I do not know. I appreciate that. However:
Let us back away from what I have written as specific conditions.
Suppose we have a very advanced tech base to play around with. Anything not forbidden, they can build. Suppose we can jump into any time in the universe = any CMB temperature between say 3 to 3000 K. Suppose we can make an asteroid of any shape and any mass. Suppose we can add any type of atmosphere (our starting condition atmosphere).
Suppose we can shape the shell in clever ways, and attach it in clever ways at the most advantageous distance. Suppose we can have shell geometry that funnels incoming atoms, built in order to trap them (in order to build up a small density gradient near the shell kind of analogous to how earth plus atmosphere makes for a geometry that traps heat in a way the moon does not).
Suppose we can have any kind of fancy way of passively entrapping pockets of atoms that exist in a built up density gradient. Maximally clever. Suppose we can have any way we want to send the entrapped atoms through the near vacuum between the shell and the asteroid, where we convert potential energy into work, after which we release the atoms back to the atmosphere again.
Would you say (in your own words) something like:
A) This clearly can't work, no matter how you tweak it!
B) Hmm... With the right amount of tweaking, perhaps ambient heat would be able to create a temperature gradient, kind of analogous to Planet X. Structural degradation would ensure it couldn't keep going forever, but it would be an impressive setup in the meantime. In the meantime we probably could get cycles, where each cycle would generate work in excess of the energy cost for the cycle.
C) Fascinating! I know precicely how I would want to tweak the parameters.
D) I don't know. It just got too complex, with too many unknowable hypotheticals.
I would say that if you find a place in the universe where there exists any kind of free energy gradient - any differential in pressure, temperature, composition, or other 'structure' as you've been calling it - then with Sufficiently Advanced (TM) technology you can extract work from it. If you're Sufficiently Smart and patient you may be able to build the equipment in a way that allows you to later reversibly extract the work that went into its construction.
What you can't do is start from a lack of such gradients, and create a system that causes them to passively form. As described, your system can't work, because it's trying to claim it can have passive diffusion that is net in one direction, and work extraction from controlled flow in the other direction that consumes the gradient produced. You're trying to make this work by taking in thermal radiation from the CMB, but this also doesn't work, because the CMB is uniform and there's no process cooling the shell's outer surface below the CMB temperature and no process that passively could even in principle.
All that adds up to (A). You cannot define the system in a way that is both self-consistent and functional.
Let's compare with processes that could work:
- You find an asteroid with a liquid ocean and no atmosphere. The liquid evaporates, and you run turbines off the escaping gas. This is Planet X again.
- You find an asteroid hurtling through space at you. You place some device in its way, and extract work from the energy of collision.
- You find a star, and surround it with a Dyson Sphere. The star consumes its own mass to form a hot plasma emitting lots of photons, which you capture and convert to electricity. No problem! You're working off the stored potential of hydrogen to undergo fusion. You consume some of the work to keep the Dyson sphere in place.
- Your Dyson Sphere enclosed star goes supernova and leaves behind a black hole. You hurl damaged pieces of your Dyson Sphere into it, and the (somehow surviving or repaired) remaining pieces extract work from the gamma rays and other radiation given off as matter falls towards the event horizon. No problem - you're consuming the structure and mass-energy of the sphere's matter.
- When you run out of spare Dyson Sphere parts, you've got to use more of that cleverness. The black hole will be colder than the CMB, and will begin gaining mass by absorbing the CMB, and become even colder in the process. I cannot think of a way to use that to do (a very small amount per unit time of) work, but a Sufficiently Advanced Alien might. This works until, eventually, the universe expands and the CMB cools to below the black hole's temperature, after which the black hole's evaporation by Hawking radiation outpaces CMB absorption. Even then you might be able to do work by turning your equipment around and harnessing the Hawking radiation and using the CMB as the heat sink, right up until it finishes evaporating.
So overall: Yes, I can imagine there could be a system that couples some astronomical object to the CMB, absorbing its heat and doing work. No, from within the universe, "we" cannot set up such a system except by finding a pre-existing gradient of structure to extract from, or by doing more work to create such a gradient than we can extract by consuming it.
I really enjoy imagining your last point, by the way ^^. I do not know if you meant to, but you paint a beautiful picture.
I can't take much credit, they're ideas generally in the zeitgeist at the boundary of physics, sci-fi, and speculative engineering.
If you like sci-fi, and haven't read these already, you may want to check out Asimov's short story The Last Question, William Olaf Stapledon's short novel Star Maker, and Clarke's trilogy A Time Odyssey. All have elements of "What would it take and look like for a civilization to actually survive into the utmost future, long after all the stars have burned out?" They don't talk about these specific mechanisms (the first two were from before we knew about the CMB!) but I find them really interesting and thought provoking.
I like Asimov, and I love Clarke's storytelling. Reading his books, it amazes me how he seemingly predicted some of the technology we take for granted today. I can't help wondering if he may not in part have manefested his predictions, by inspiring the actual inventors. I have never read William Olaf Stapledon. A recomendation, I take it?
I wonder, are you planning to answer these two questions. You have no obligation to do so, obviously. Only if it feels constructive to do so.
Yeah. Stapledon is older - Star Maker was written in 1937, and it builds on the themes of Last and First Men, a book he wrote in 1930. They don't really have much plot to speak of, they're more purely exploratory and written as a kind of future history/scifi cosmogony/speculative evolutionary engineering/secular eschatology. But they're quick reads and I think they're interesting worldbuilding thought experiments.
I do think there's some inspiration of that type that goes on, yes. But also, it is often possible for a field to know early on what some of the theoretical limits are for what can be achieved through it, even if it takes decades or more to even start seeing it happen. The great scifi authors are the ones that ask what it will mean when they do.
Ah! I'm glad I asked. So I had two guesses.
1) What if you could use the Cosmological Degradation as your entropy sink? What if you could tweak the asteroid sufficiently cleverly to make this work. The law about "entropy always increasing" would not be broken.
2) What if the structure of your setup was your gradient? What if the asteroid and its gravity well, the shell, the atmosphere and the energy conversion equipment would degrade in an edgecase like this in a way that could never be repaired from the energy converted. The law about "entropy always increasing" would not be broken.
Do you see anyting absolutely forbidding those two possibilities? And for 2, I would be interested in your intuition both for the asteroid setup, and in general.
Hey, sorry, I thought I'd responded to this one and apparently hadn't.
I think my black hole discussion is essentially my answer to (1). I don't think I could think of a way to make it work with an asteroid or similar setup. I am not entirely sure your discussion of cosmological degradation is well-defined enough to answer more precisely than that.
For (2), my other comments about you can of course do work to create a gradient you then consume, and get some of the work back. But as written, no, that doesn't mean the setup as described can work.
Actually I think this fails for ordinary reasons. Key question: how are you getting energy out of lowering the helium?
- If you mean the helium is chemically bound to the sheets (through adsorption) then you'll need to use energy to release it
- If you mean the helium is trapped in balloons, then it will be neutrally buoyant in the ambient helium atmosphere unless you expend energy to compress it.
Big thank you! I made an edit to the post, clarifying this point (the one in my previous reply). Do you think I need to address the point of enclosing the gas directly?
Ah! Good question. Perhaps I should have given more details.
I think enclosing the helium (without compression) is the way to go. And the density will spike close to the shell (gas comming from the asteroid will accumulate there). You will this have:
*The atmosphere, with the greatest density.
*The vicinity of the shell wall, with a spike in density.
*Space in between, very close to a vacuum.
You will be able to lower the enclosed helium through a lot of space without any bouyancy, turning potential energy into work.
2Why do you think gas will accumulate close to the shell? This is not how gases work, the gas will form an equilibrium density gradient with zero free energy to exploit.
Here’s why I believe a slight density increase near the shell is not only possible but statistically inevetable:
- The shell can be placed sufficiently far from the asteroid such that escaping helium atoms travel nearly radially from the surface. Most will have very low kinetic energy, having just barely escaped the gravitational potential well.
- Sufficiently far from the asteroid, these atoms feel almost no gravity. They’re essentially coasting in near-inertial trajectories.
- If the shell were absent, these atoms would simply escape into space. But with the shell in place, their outward motion is halted. They bounce off the shell instead of escaping.
- Since their approach is slow and nearly radial, many will strike the shell with low momentum. After bouncing, some will scatter at non-radial angles and may linger in the vicinity. Given enough atoms and time, this creates a diffuse accumulation zone close to the shell. A mild, geometry-induced density spike.
- This isn't a violation of thermodynamic equilibrium. It’s a boundary condition effect arising from system geometry and the kinematics of slow-moving atoms arriving at a barrier. The system as a whole trends toward equilibrium, but that equilibrium includes local features shaped by containment. Also, we do not have to wait for equilibrium before collecting atoms. It is enough that we calculate that at some point in time there will be a density spike close to the shell.
So yes, under these assumptions, some degree of helium accumulation near the shell is to be expected.
No, when you carry through the calculations you will find that in equilibrium the density is monotonic with distance from the asteroid.
One easy way to see this: if there were increased density near the shell without any counterbalancing force attracting them to the shell, then there would be a net flow of particles away from the shell reducing the density. So this cannot be an equilibrium.
There may be transient microscopic density variations, but no macroscopic ones (absent some sort of Maxwell's Demon).
It is also an incorrect assumption that the motion is nearly radial. At all heights the direction distribution is still uniformly random.
Here’s why I believe a slight density increase near the shell is not only possible but statistically inevetable:
This sort of issue is what people invented numbers and equations for.
OK, so the issue here is that you've switched from a thermodynamic model of the gas atoms near the asteroid to one which ignores temperature at the shell. I'm not going to spend any more time on this because while it is fun, it's not a good use of time.
One of the properties of the second law is that if you can't find a single step in your mechanism which violates it, then the mechanism overall cannot violate it. Since you claim that every step in the process obeys the second law, the entire process must obey the second law. Even if I can't find the error I can say with near-certainty that there is one.
For anyone reading: Please note that I do not claim this is a perpetual motion machine. I do not claim the setup breaks the second law. In fact, I claim the opposite, and I even think I have found a mechanism where entropy does increase with the required scaling. I think the answer to why the second law doesn't break may be important and interesting.
There is an error in thinking that the high-altitude helium will be at a lower temperature than the low altitude helium. If the helium is not being continually stirred (which would take energy input), then the equilibrium state has the density decreasing with height, but the temperature is uniform. The high-altitude atoms are just as energetic as the low-altitude atoms. This is a basic fact about thermal equilibria: each of the degrees of freedom has the same time-averaged energy. Temperature is just energy per degree of freedom.
If the initial setup is not in that equilibrium state, then you can extract work from it, but only a finite amount as it approaches equilibrium.
The continual stirring of the Earth's atmosphere is a substantial contributor to the decrease of temperature with height.
You raise a point about thermal equilibrium, and you're absolutely right that in a static equilibrium, temperature would be uniform throughout the gravitational field.
However, the system I'm describing isn't in thermal equilibrium as we "start" the setup (e.g. insert the atmosphere). It's in a state with continuous evaporative cooling. When the fastest atoms escape the gravity well, they remove more than the average kinetic energy from the atmosphere (that's why they can escape).
This is exactly analogous to evaporative cooling of water, just with gravity instead of intermolecular forces. The key is that the CMB at 5K provides continuous heat input to compensate for this cooling. So we have:
1. Fast atoms escape → top of atmosphere cools below 5K
2. CMB radiation heats the atmosphere back toward 5K
3. This maintains a steady temperature gradient
4. If we waited enough, eventually an equal amount of atoms would be falling back to the asteroid as is evaporating
5. Eventually we would reach an equilibrium
6. However, we do not let that happen, since as soon as density accumulates near the shell, we enclose the atoms (metaphrically like just screwing a lid on a jar "trapping" some air), and send them down in a way where we can extract some of the potential energy before releasing them.
7. This requires no knowledge of individual atoms. One can calculate statistically when there will be a higher density near the shell (or have a measuring device).
When you release the lowered atoms back at the surface, do you have to fight against the atmospheric pressure?
Good question!
Short answer is no. Here on Earth space "starts" at around 100 km. Above this we kind of have a vacuum. The same would be true for the asteroid (above a certain point the pressure is very low). As long as we release the lowered atoms above this point there would be no atmospheric pressure to fight against.
You're sorta talking here about extracting work from an initial pressure differential by converting it to a temperature differential, just like in the Planet X example.
That would be fine, but it contradicts your post, where you specifically state that everything starts in thermal equilibrium at 5K. The CMB is still irrelevant and unneeded, and does not provide the kind of T gradient you're claiming. (6) does not work, and (7) is not true.
It seems to me that this setup is equivalent to "skim air from the top of Earth's atmosphere, drop it back to Earth, extract gravitational energy", with some more details that don't change much. This fails for density reasons, unless I'm missing something.
You're right that it superficially resembles the Earth atmosphere example, but there's a crucial difference that makes it more interesting.
In the Earth case, you're fighting against atmospheric buoyancy. The air you're trying to drop is surrounded by denser air, so no net energy gain.
In my setup, the helium that accumulates near the shell has escaped the asteroid's atmosphere entirely and is drifting in near-vacuum. When you capture it at the shell (e.g. 31 radii out, gravity ~0.1%), you can lower it through essentially empty space (no buoyancy to fight against).
The key insight: atoms that barely escape arrive at the shell with near-zero kinetic energy, creating a density enhancement. Most will arrive radially and scatter in all directions as they hit a microscopically jagged 5 K surface. Some will bounce multiple times at different parts of the shell before returning to the asteroid.
You will get an ever so slight density increase close to the shell compared to the near-vaccum between the atmosphere and the shell. You're not "skimming atmosphere" - you're collecting atoms that have already paid their full gravitational escape cost.
I've laid out a bit more of the mechanism in my response to AnthonyC if you're interested in more details. Happy to address specific objections!
The interior surface of the shell is larger than the surface of the asteroid, reducing the density. I don't know if this completely compensates for that effect or if there's also something else involved, but you didn't even consider it. (And if you try to fix this by making the asteroid so big that it's more like a flat sheet, the flat sheet's escape velocity, at the scale where it behaves like a flat sheet, is infinite.)
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