This question is concerned with a thermodynamic paradox for radiating bodies and radiation in a cavity of a specific shape.

Consider two nested shells that are axisymmetric ellipsoids with the same two foci, A and B, as shown in the figure (line AB is the axis of symmetry). Cut the system along the vertical plane of symmetry and remove the right side of the outer shell, and remove the left half of the inner shell. Then connect the two halves with vertical surface, as shown in the figure, to make it a continuous enclosure. The result is a figure of rotation shown by the thick black line in the figure.

Next, make the inner surface of it a perfect mirror. The property of such a cavity is that each ray emitted from point B comes to point A; but not each ray emitted from point A comes to point B - some rays emitted from A (shown in blue) come back to A.

Now, put two small black bodies (say, two spheres of some small radius) at points A and B. Thermodynamic equilibrium requires that eventually the temperatures of the two spheres equilibrate. However, according to the geometric properties of this cavity, all energy emitted from B comes to A but only a fraction of energy emitted from A comes to B; so the equality of temperatures is not consistent with balance of emitted and absorbed power. How to resolve this paradox?

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I'm sure the answer will come to me, but for now I just want to say I'm very impressed by the cleverness of the setup. –  Nathaniel Jun 29 '13 at 6:58
How to you define your path $A$ to $A$? If you define it to by symmetric relatively to the axis from A to the corner made by the great ellipsoid and the vertical, it is just wrong. –  Trimok Jun 29 '13 at 9:58
@Trimok The rays coming from $A$ and reflected by the vertical surface will seem to come from the other focus, $B$. –  mmc Jun 29 '13 at 17:50
@mmc : The path from $A$ to $A$ (avoiding $B$) is in blue color. –  Trimok Jun 29 '13 at 18:16
@Maxim I think the problem you are having comes from assuming point-like bodies, as that gives you infinite radiance. See this article for more details. –  mmc Jun 29 '13 at 18:17

Paradoxes arise if one mixes two systems of reference. I think there is some confusion here between geometrical optics and thermodynamics. It is true that if one follows rays and one could count them ( they are infinite) the statement is geometrically correct.

But energy is not carried by rays,it is carried by photons which interacting with matter at A and B may even change frequency/energy.

Geometric optics cannot give rise to thermodynamics. Thermodynamical equations arise from statistical mechanics, the statistics of many particles . Particle means an (x,y,z,t) location and a (p_x,p_y,p_z,E). Rays are not an ensemble that can give rise to thermodynamic quantities because they do not have these attributes. Thus in no way an ensemble of rays (infinite in concept) can give a number for a temperature.

Geometrical rays are useful models for electromagnetic radiation but are not particles.

Photons are the particles of light and their behavior is well described by the appropriate statistical mechanics.

In this case it may take a longer time to reach equilibrium in such a set up.

Think of an ideal gas in this setup, molecules bouncing around from the walls completely elastically. They will follow they usual thermodynamic equations and have a definite temperature, the same for both cavities, even if it were different initially and the separation was removed. Equilibrium will be reached.

It is similar with photons, the particles of light. So there is a photon "gas" generated by the intrinsic heat of the balls at A and B,( even if no extra light is shined in), and the temperature of this photon gas will reach thermodynamic equilibrium with the temperature of the balls.

I repeat, paradoxes appear when one confuses the mathematics of two mathematical frameworks describing physical situations. Each mathematical framework has its principles theorems and attributes of the ensemble it describes. One can not use a la cart, unexamined, attributes from one framework into another because paradoxes may appear, as in this case.

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What exactly is the statement here? That geometric optics is in contradiction with thermodynamics? Why is that? What if the cavity was a simple (left-right symmetric) ellipsoid? Wouldn't this produce temperature equilibration, as should be - in spite of geometric optics? –  Maxim Umansky Jun 29 '13 at 14:39
The statement is that geometric optics cannot give rise to thermodynamics. Thermodynamical equations arise from statistical mechanics, the statistics of many particles, en.wikipedia.org/wiki/Statistical_mechanics . Geometrical rays are useful models for electromagnetic radiation but are not particles. Photons are the particles of light and their behavior is well described by the appropriate statistical mechanics. –  anna v Jun 29 '13 at 15:45
It is a very broad statement that geometric optics cannot give rise to thermodynamics. I thought as long as you can use the eikonal approximation the concept of light ray is valid for describing radiation, correct? If so, does the eikonal approximation break down here and that's causing the problem? Or you are suggesting that we are missing the quantum statistics of photons which is the issue? I certainly appreciate the wiki reference on statistical mechanics; however check out that paper by Palmer that mmc pointed to. –  Maxim Umansky Jun 30 '13 at 2:52
Thermodynamics needs particles with energy at least, which are not attributes of rays. Particle means an (x,y,z,t) location and a (p_x,p_y,p_z,E). Rays are not an ensemble that can give rise to thermodynamic quantities because they do not have these attributes. –  anna v Jun 30 '13 at 3:07
Collisions between particles is NOT necessary for establishing a thermal equilibrium, it is enough if particles interact with the "black body". Simple example - photon gas :) –  Maxim Umansky Jun 30 '13 at 5:50

the finite sizes of the black bodies lead to a resolution of the apparent paradox

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