This came out of a discussion I started yesterday and a related discussion I found.
I'll recap the problem quickly: Consider two blackbodies, with surface areas $A_1$ and $A_2$ and temperatures $T_1$ and $T_2$. Assume $A_1>A_2$. If I use a lens-mirror system to transfer all the radiation emitted by Blackbody 1 onto Blacbody 2, we must have $T_2>T_1$ as a consequence of $A_1>A_2$. This violates the second law of thermodynamics. The resolution was that we cannot transfer all the heat from Blackbody 1 onto Blackbody 2 precisely because $A_1>A_2$.
A statement made in one of the answers in the related discussion by user @Divergence is "Basically, given any source of light radiating from finite surface to half space, you can never concentrate the entire emitted radiation to a smaller area than the original emitting area."
I've tried to read about this and connect this to conservation of Etendue but I'm having a lot of trouble seeing how to prove this statement. Does anyone have an insight into what forbids a lens-mirror system from concentrating all the emitted radiation from one body onto a smaller surface area?
