How is the 2nd law of thermodynamics obeyed in this system?

[Christofer Br]

Imagine there is an radiation concentrator (winston cone) surrounded with extremely many layers of foil for radiation insulation, except at the smaller opening. Every part of the setup is initially in thermal equilibrium with the surroundings. The amount of thermal radiation flowing through the smaller opening in both directions through the hole would be the same in both directions, except since on one side we attached a winstone cone, there will be more radiation coming out into the surroundings from the inside of the setup since some of the radiation that otherwise would miss the opening is being focused on it by the winstone cone. We assume that the energy losses through the insulation are vastly smaller than the energy of radiation focused on the opening by the winston cone.

It seems now that since there is more radiation coming out of the setup than coming in [through the opening], the temperature inside would spontaneously lower until the lower temperature radiation coming out has the same energy flux as the radiation coming in.
This of course would be at odds with the second law of thermodynamics. How is the entropy decrease prevented in this case?
In case you start wondering about the emissivity of the winstone cone and the foil at its larger opening, note that the 'surroundings' can substitued for a box with an opening shared with the winstone cone, with sides lined with metal (the same material as winston cone), if we consider an isolated system

 

[Dale]

It is not only the area, but also the solid angle or view factor which is important.

Suppose that you have a small flat source radiating thermal energy. It does not send its energy in a sharp beam normal to the surface, but it sends it out over all $2\pi$ steradians. If a receiving surface only covers $1\pi$ steradian then the “view factor” is 0.5 and only half of the energy emitted by the first surface is received by the second surface.

Due to geometry you are guaranteed that any increase in area is associated with a decrease in view factor. So although you can make a large area exchange energy with a small area, only a small portion of the large area’s energy can exchange while a large portion of the small area’s energy will exchange. The product of the area and the view factor are the same on both sides.

The net result is that the exchange is geometrically guaranteed equal at thermal equilibrium.