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.

I really enjoy imagining your last point, by the way ^^. I do not know if you meant to, but you paint a beautiful picture.

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.

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.

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 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).

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!

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?

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

1. 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.
   
    
2. 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.
   
    
3. 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?”.
   
    
4. 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). 
   
    
5. What you call the correct premises.
   
    
6. 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

1. 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.
   
    
2. 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?
   
    
3. Have I missed something obvious in this reply? Like mischaracterized you, or seemingly missed something important you said?
   
    

Possible disagreements

1. 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). 
   
    
2. 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.
   
    
3. 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.

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. ^^