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You are the leader of a far-future civilization - human or otherwise - that has endured for trillions of years doing whatever far-future civilizations do. You've surrounded stars with Matrioshka brains and gathered them up with stellar engines, pooled as much mass as you could and did all the computation theoretically possible, but energy is the real problem.

You know you're going to run out of extractable heat soon enough. Black holes radiate too slowly, and while they get stronger in their last moments, they ultimately vanish.

Luckily, your universe contains an anomaly, something totally unique, contrary to the laws of thermodynamics, and, as far as you're aware, totally indestructible and immutable. The anomaly is a single object of a certain size that constantly radiates at a specific temperature, thus producing a constant amount of power.

You wish to maintain your civilization, even if you have to upload your people to computers and shrink them down to a microscopic scale to run everything off the anomaly, for the rest of the infinite span of time past the heat death.

The minimum requirement is the continual, uninterrupted simulation of the equivalent of at least one million human brains for a time that exceeds any finite value given.

How small and cool can the anomaly be?

You have access to:

  • The combined knowledge of a far-future civilization gathered over its lifespan. No technological holds are barred provided they make some physical sense. Super-intelligence is fair game. Assume mind uploading is practical.
  • Several trillion solar masses of initial material of your choice.
  • A few billion active stars and all the initial energy they can provide.
  • One anomaly that is your only thermodynamics-violating object.

You may have to worry about:

  • Proton decay and quantum tunnelling. How do you replace parts of your computing system as they go missing or fuse into each other? Do you synthesize new matter out of energy - and can you do that if your energy source is only at infrared or even radio wavelengths? (Or does it have to be brighter or hotter?)
  • Logistics. Computing systems must be able to simulate everyone and keep track of the physical world with sensors, repairing damage and the like.
  • Sanity. Can a simulation that lasts forever even provide infinite utility to its occupants? Will you resort to memory-wiping or are there other options that a sufficient amount of energy can help provide? (How much will you need?)
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  • $\begingroup$ Assuming 183 kJ/kg/day for a 10 kg brain, gives 1830 kJ/day/brain, times 1 million brains equals 1.83 TJ/day. As long as your anomaly provides more than that, you should be good $\endgroup$ – nzaman Mar 30 at 5:33
  • $\begingroup$ @nzaman Getting really close to the limit set by Landauer's principle (especially with access to a near-zero universe outside) should let you pack emulations of millions of brains into many orders of magnitude less. But is maintaining them still feasible with that little energy? $\endgroup$ – parasoup Mar 30 at 5:57
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    $\begingroup$ Another factor to consider is frame-jacking. If the minds are digital, processing speed can be slowed. Have a trillion trillion minds, but only let each one of them think the thoughts we think in a secound in a million or billion years. Internally perceived time will remain as it is now, but external time will be extremely fast. Yet that doesn't matter because there isn't much going on during the iron star and black hole areas anyways. Obviously this will severely impact energy demand. $\endgroup$ – TheDyingOfLight Mar 30 at 12:26
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    $\begingroup$ A nitpick, but if you can still extract work from anything then the heat death of the universe hasn't occurred by definition. Heat death doesn't have to do with the elimination of energy, but the elimination of energy gradients, since these are necessary to extract work from a system. As to your question, I think the answer heavily depends on the global shape and fate of your universe. For instance, your task is impossible for a static, bounded universe because the anomaly would slowly equilibrate the temperature of the universe with itself, leaving you with less and less energy to extract. $\endgroup$ – el duderino Mar 30 at 15:39
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    $\begingroup$ This does of course assume that the ultimate fate of the universe is "merely" heat death, and that something more destructive like a big rip does not arise. $\endgroup$ – Starfish Prime Mar 31 at 7:11
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An Underclocked Civilization Is Only Constrained By Data Storage

The number and complexity of minds is irrelevant if you adjust the processing speed. Even if the energy trickle only allows 1 computation per trillion years, a trillion trillion stored hyperminds would still be capable of an infinite number of thoughts forever.

So I can think of 4 dangers, in increasingly disturbing order.

A Fading Danger: Gravity Wells and Radiation -- It will be important early on for the civilization to make a run for it from any mass-dense regions of space, because dodging black holes and hawking radiation would be dangerous in low-power mode. Once all undesirable mass has left their Hubble Sphere (i.e. distant enough that the expansion of space exceeds the speed of light), external sources of damage will be removed.

An Uncertainly Fading Danger: Other Hostile Civilizations -- How certain are they that theirs is the only such anomaly? Even if they are incredibly, highly certain, they will probably be wondering about self-defense for a while. But it'll get increasingly safer if FTL travel is impossible, as only civilizations within their Hubble Sphere would matter. But... How certain are they that instantaneous FTL travel is impossible? Even if they are super duper certain of that, they might keep worrying about that, just a little bit...

A Constant Danger: Physical Decay -- This is the big one for your question. Entropy will always be an issue, as well as containment of mass and energy. They must design their computer and redundant data storage to keep maintenance and uncontrollable exhaust under the anomaly's output by a huge margin of error. Mind computation can happily live only within that margin, and nobody but the maintenance janitor would notice. Unfortunately I can't seem to find any source for a theoretical minimum rate of data loss, and the capabilities of future nanotechnology may be unknowable. But Wikipedia's page on Limits of Computation might be a place to start. Anyway, here are some mitigating strategies they might use:

  • In case of catastrophic damage, it would be desirable to safely keep lots of backup energy, possibly stored as mass orbiting in a Dyson Swarm to catch exhaust.

  • An indestructible immutable material would be ideal to help keep particles from whizzing off, so they might even build among the matter of the anomaly itself via quantum tunneling.

  • Then, giving the anomaly a relativistic spin will cause the computer in the faster moving exterior to age slower than the core, pumping up the core's relative rate of energy output. In addition, this will store an incredible amount of energy as angular momentum, but hopefully their computer won't fly apart!

A Growing Danger: Lack of Meaning -- Heaven without its famous, boundlessly deep source of new awesome good stuff is, by definition, hell. Huddling next to a dim glow in endless darkness, having thought and imagined literally everything you can, and knowing there's nothing left, sounds kinda like a bad goal.

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    $\begingroup$ I like how this answer addresses both the early and late stages of the operation. The whole "running out of thoughts" thing occurred to me, but I wasn't confident enough on how long that would take, so I left the goal time unbounded. I guess I was looking for something more numeric, like "if you have X number of atoms running this simulation super slowly, then you need Y energy to maintain them, and the new photons need to be of Z wavelength for these methods to work". I'll see if anyone has any comments on those parameters before marking this one as accepted. $\endgroup$ – parasoup Mar 30 at 18:47
  • $\begingroup$ Yeah, my answer to that is a bit buried in the middle of danger #3. Basically couldn't find anything on such a limit, and I suspect it would be highly dependent on the structure and composition. In fact, the anomaly's immutability may be even more important than its energy output, if they can find a way to harness or build off of that property. $\endgroup$ – BoomChuck Mar 30 at 23:02
  • $\begingroup$ Surprisingly, even though I agree that underclocking wouldn't make any difference, I would feel cheated if I knew I am being underclocked... $\endgroup$ – Aganju Mar 31 at 13:59
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It is infinite.

Orions belt

https://www.youtube.com/watch?v=P7ojSW5pODk

Your thing is not amenable to descriptions of scale. It is infinite, or perhaps has no dimension. Or both - a point opening to another plane. The notion of size does not apply.

Fortunately the notion of "where" does apply so you can find it and use it to make your coffee.

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