Although I know many arguments against concentrating blackbody radiation to create a spot hotter than the blackbody, I encountered this confusing counter-example.
Consider a thin sphere blackbody of the radius $R$ locked up in a sphere mirror. And there is another blackbody of ball shape with the radius of $r$, at the center of the sphere blackbody.
Now, by doing some calculation based on the Stefan-Boltzmann law, the outgoing radiation power of the ball blackbody is given as
$$r^2 T_{ball}^4$$
ignoring all the irrelevant constants, and the all the inward radiation from the sphere blackbody that is absorbed by the ball blackbody is
$$R^2 T_{sphere}^4 (1-\sqrt{1-(r/R)^2}).$$
(This results from the geometrical calculations from the ratio of solidangles.)
And since $$ \sqrt{1-x} > 1-x $$ for all $0<x<1$, we have $$T_{sphere} > T_{ball}.$$ Now, this result is very confusing since two balckbodies never have the same equilibrium temperature. If you placed a hotter ball blackbody at the center from the beginning, you end up with reversed temparature distribution at the end. You may be able to build a permanent engine from this setting. Obviously there is a critical flaw in this reasoning but I cannot see it. Any kind of help of advice is appreciated.
