Recently I read about something called photon up-conversion, in which a lower frequency of light is absorbed by a material that then proceeds to emit a higher frequency of light. When the mechanism for up-conversion is two-photon absorption, a molecule absorbs two lower-frequency photons, emits a higher-frequency photon, and then returns to it's initial state.
How would the change in entropy be calculated in a situation like this?
The intuition you're using is that entropy should be linear in particle number, since each particle contributes some amount of entropy on its own, so that reactions that decrease the particle number should decrease the energy.
This isn't quite accurate for a number of reasons, but the most important one here is that the entropy isn't linear in the number of particles, it's sublinear, since you have to divide the partition function by $N!$ because the particles are identical. So the first high-energy photon that comes out of this reaction can contribute a lot more to the entropy than the two low-energy photons did. (An additional factor is that each high-energy photon already contributes more entropy because it has more possible momentum states.)
As more and more high-energy photons are produced, the entropy gain of the reaction decreases. Eventually, when the maximum entropy is reached, the reaction will run backwards at the same rate it runs forwards.
Two people go into your room, each with a cup of water. You pour the two cups together and walk out with a pint. All the water is conserved. Nothing was created from thin air.
Two little photons go in, on big photon comes out is also not inherently wrong, as long as the energies of the two incoming photons adds up to the energy of the exiting photon. This assumes the material is in the same energy state before and after, which appears to be the case you are asking about.
I believe this question suffers from trying to explain why a microscopic process does or doesn’t occur, with reference to the second law. Perhaps it is best to consider entropy as an emergent property that cannot be understood when considering only one (or two/three) photons and a single atom. The same issue would ensue if we followed two molecules of a gas in a mixture that have a collision and “concentrate” at one end of a container. You could argue that they also break the second law. Entropy would only decrease here if there was a bulk concentration that occurred spontaneously. I would ask if there was net energy dispersal that accompanied the”concentration” of the two photon absorption and single emission.
