The following image displays specific design characteristics and
scales for thinking about Matrioshka Brains.
The upper row of the graph is intended to show three things (a) the distance
from the sun at which specific computronium materials may be functional;
(b) the variety of different materials from which the computronium may
be constructed; (c) the large increase in radiator size required to radiate
heat at increasingly lower temperatures as distance from the sun increases.
The materials outlined are those for which "computronium" either currently
exists or may be envisioned. For example we already have silicon
and to a lesser extent GaAs based computronium. Advanced semiconductor
laboratories are working on GaN, SiC and diamond based semiconductor circuits.
There are also laboratories working on high temperature superconductor
(HTSC) and low temperature superconductor (LTSC) circuits. It is
not difficult to envision mechanical computers of titanium carbide or sapphire
capable of operation at extremely high operating temperatures based on
principles similar to the mechanical diamondoid computers specified in
(Dre92).
The size of the circles is representative of the radiator size required for computers consuming a standard amount of power (e.g. 105 W/cm3 - based cooling limits specified in Dre92, sec. 11.5.3). [Drexler documents that computational processes that are to be relatively efficient in "real time" must generate heat even when using reversible computing elements.] Heat radiator surface area must increase at lower temperatures due to the black body radiation law proposed in 1879 by Austrian physicist Stephan Josef Stefan. It showed that heat radiation capability scales with temperature to the 4th power (i.e. hot radiators are very efficient, cold radiators are much less so). So as the temperature of the computronium decreases the area of the radiators must significantly increase and there is a decrease in the overall computational capacity [due to speed-of-light communications delays between nodes which must orbit further from each other to enable increased radiator size]). This explains why Matrioshka Brains become less efficient as they grow in size.
The lower level row of the graph details the current dimensions of our solar system (i.e. how close and far from the sun our planets are). It is intended to provide a perspective of how much closer and how much further from an energy source such as the sun the computational nodes of a Matrioshka Brain might operate based on the assumption that each layer of a Matrioshka Brain must radiate the ~1026 W of power generated by the sun.
The impact of the graph may best be understood if one compares the upper
row and lower row in conjunction with the orbit radius scale (i.e. the
distance from the sun at which the computronium elements are likely to
operate).