What low-level process drives a frequency-doubling crystal? I was reading about second-harmonic generation (SHG) crystals (or frequency-doubling crystals) used to produce green laser light from IR. 
What low-level process in the crystal is actually driving the combination of two photons into one? I'm thinking of a process at the QED level (or at least motivating what happens at that level). The wikipedia entry on SHG doesn't really address this (it just mentions it's a non-linear phenomena..).
Or put another way - what's the simplest system possible (theoretically) that can show the phenomena even if at a very low efficiency?
Edit: A supplement to the question, is what entropy-related constraints are there on the incoming photons in these processes, in order to avoid for example turning heat into nicer higher-energy photons? 
 A: One thing to note is that it is not just the crystal, but a crystal which is coherently pumped by a strong laser field in which one sees SHG. In this setting, the pump creates a periodic modulation of the index of refraction (thus the requirement for a non liner medium) which effectively acts as a phase grating.
One way of thinking about SHG is that it is the dynamical Casimir effect. Here, instead of a mirror oscillating at optical frequency, the optical path length of the crystal oscillates at this frequency due to the nonlinear response of the crystal, creating photons from the vacuum, at the expense of the photons modulating the crystal.
In terms of more traditional photon-photon processes, you can of course write out Feynman diagrams and calculate the scattering matrices for the annihilation of two photons in one mode and creation of another in a phase matched direction. An example is given (For the inverse process) here. (see page 100 or so).
Also, to address your last point, (which I'm not as familiar with). As I understand it, parametric processes such as SHG do not transfer energy to the medium and so there no reservoir to couple thermal energy to the photons. The process is entirely coherent.
Hope this helps.
