I'm returning to this discussion since if I don't I'm sure Robin will never let me forget about it. :-?
On 17 Sep 1999, Robin wrote:
> Consider the function P(m) which describes the most P(ower) you can extract
> from a star given a certain amount of available M(etal). Since you can
> choose not to use metal, P must be an increasing function.
> If you have two stars, and are trying to extract the most power from them,
> and you have a certain amount M of metal available, your problem is
max P(m1) + P(m2) such that m1 + m2 = M
m1,m2 and m1 >=0, m2 >=0
> You are suggesting that an optimum is m1 = 0 and m2 = M, putting all the
> metal at one star so as to get every last photon and reradiate at near 3K,
> while completely leaving the other star alone. This requires that
> P''(m) > 0 on average, with *increasing*, not diminishing, returns.
The metal may have 2 uses: computation, which presumably requires energy, or memory, which requires very little energy (occasional rewriting to offset cosmic ray damage in excess of ECC limits seems unavoidable).
The computational throughput of a parallel processing "computer" system will be highest value when the material resources are utilized in such a fashion that you get the greatest amount of computation done with the minimal amount of interprocessor communication delays and energy costs.
So the best situation to setup is m1 at the thermodyanmic limits and m2 with the leftovers. For example, this could be m1 = 0.999(m1+m2) and m2 = 0.001(m1+m2). You can't use any more metal at m1 because you are using all the power of your star *and* are radiating at the lowest background temperature possible. You may develop m2 but it is hardly worth the trouble, since the cost of shipping information to m2 (in terms of energy and months to years of transit time) makes it hardly worth the effort of any additional computation you got back (after more months to years).
> Instead I expect any plausible model of metal-limited Dyson sphere to show
> decreasing returns: The first few tons gives lots of power while the last
> few tons gives a lot lot less. This implies that to get the most power one
> should spread the metal evenly across the two stars:
> m1 = m2 = M/2 .
I agree that metal-limited Dyson spheres *do* show decreasing returns. The further out from the star you go, the cooler the layers get and the less energy you have to do useful work. If you want a "global" maxima, your conclusion is correct. But *unless* the AS-SI has agreed to work for me full time for the next 10 billion years (for free of course), it makes little sense to split m1 & m2 evenly. It makes much more sense to make the RB-SI a Matrioshka Brain (hiding the star, operating at the thermodynamic limits) and make the AS-SI a Jupiter Brain (harvesting a small fraction of the stellar output, waiting for more metal to be produced).
In a subsequent message, I wrote
>
>> No. I'm suggesting that there are always decreasing *local* optima.
>> If there were *no* costs to transfering material (or information) from
>> 1 to 2, then doing that would make sense. But if the costs of the
>> material/information transfer *exceed* the cost of local manufacture
>> then it makes sense to reject remote information/material (if you
>> have to pay for it).
To which Robin replied:
> But mass transport costs make the puzzle worse! Let n1 and n2 be
> the initial metal amounts at the two stars. The problem then is: max
> P(m1) + P(m2) such that m1 + m2 = n1 + n2
> m1,m2 - T*abs(m1-n1) and m1 >=0, m2 >=0
>
> (Note that abs(m1-n1) = abs(m2-n2).) Assume without loss of generality
> that n1 > n2. If P(m) has diminishing returns (P''(m) < 0), then when
> transport costs are very high, m1 = n1 and m2 = n2, so nothing moves.
I agree, if transport costs are high, you don't go harvest metals and ship them home. You either breed them locally or make do with what is available.
> And when transport costs are zero, m1 = m2 = (n1 + n2)/2. And for
> intermediate transport costs n2 is in the range [n2,(n1+n2)/2].
In the long run, you will orbit to places where transport costs are lower (in theory, if you can predict the motions of all of the stars in the Galaxy, you can apply small delta-V's that over long time scales (millions of years) bring you in very close proximity to locations (stars or gas clouds) where quantities of metals are available and transport costs are very low.
Do cockroaches question whether to reproduce? No. They do it because they are programmed for it. Do humans question whether to reproduce? Yes. Because they consciously question whether the benefits exceed the costs. If we assume the evolution of intelligence goes hand-in-hand with the evolution of rational-thought, then you have to answer the question of what justifies the production of something that is only capable of thinking things that you thought ages and ages ago?
The point of reproduction (IMO), is to hopefully get something new and better. If you are up against thermodynamic limits, that probably becomes increasingly difficult as the universe ages.
If you believe sacrificing your potential for that of another entity then equi-division of metals might make sense. Once you have decided to harvest everything that provides benefit in excess of the investment, then the question is whether there is any benefit to leaving behind "handicapped" JBrains. That sounds like a pretty sadistic thing to do. As a JBrain it is going to be pretty clear that you got somebody's leftovers.
Sure, a few of the textropians can go off and colonize nearby stars, but they are going to be *years* behind us on the SI evolutionary curve. You could set your sights on a bigger star (more energy), but that has a shorter lifetime and suffers from larger inter-SI node propagation delays (bigger stars with more energy means the radii of the "hottest" computronium layer must be larger to prevent melting). You could shoot for a metal rich star that would have a resource base for faster post-SI evolution, but as soon as you got a few million moon-sized telescopes up and running you might discover the sky filled with individuals who have been pursuing your strategy for much, much longer.
I certainly agree that the math outlined above makes sense. If we were striving for some "global galactic optima" your argument might sense. But in striving for any galactic optima, you have to solve a planning problem that involves information this is ~100,000 years out of date (the light-speed distance across the galaxy). On SI timescales a *lot* can happen in that length of time.
If we could assume that drives & justifications for colonization in humans would apply in SIs, then staking out metal claims might make sense as well. But if *you* are going to claim them, the claiming entity must be an "owned", non-self modifying subsidiary. Questions:
I'm optimistic that *if* SIs exist already, we will see them before we have the ability for interstellar colonization. So framing the exploration/colonization question within that framework is useful.
If SIs do not exist, then survival scenarios (enclaves, escaping bio-nano-goo, etc.), interstellar colonization and galactic optimization discussions are useful. If this is the case, then we also have another interesting problem to discuss -- why is the development or survival of intelligent life so darn difficult?
Also, some of my claims regarding communications costs have raised some eyebrows. A very brief & rough summary of my thoughts in this area, and the references (what people really want) is now up at: http://www.aeiveos.com/~bradbury/MatrioshkaBrains/LogP.html
I'll do my best to answer questions but it may take another 2 months.
Robert