"Einstein and the Daytime Sky"
C. Imitation of opal
Since Einstein was addressing a more general question than the color of the atmosphere, his results have a wider interest.
Smoluchowski had originally considered a more dramatic effect on light, brought on by fluids in a very different state. We've been discussing gases and have mentioned liquids. In our usual experience, a fluid is generally either one or the other, depending on whether its temperature is above or below its boiling point-for example, water is a dense liquid below 100oC and a much less dense gas above. But if we increase the pressure on a fluid, we find that its boiling point rises and the density difference between its liquid and gaseous forms decreases. Once the pressure is high enough, the "boiling point" becomes almost meaningless because the density difference completely disappears. Any light waves passing through the fluid under these conditions, whether the waves' frequencies are high or low, turn out to be strongly scattered. This effect is called critical opalescence. The "critical" refers to the state of the fluid, at or near the exact pressure and temperature that gas and liquid become the same; "opalescence" refers to the scattered light's resemblance to the light scattered by opals.
Smoluchowski showed that near these conditions, a slight expenditure of energy could drastically change the fluid's density. Huge density fluctuations throughout the fluid are therefore practically inevitable. These huge density variations would cause huge variations in the speed of any light passing through them, thus producing the strong light-scattering of critical opalescence. Einstein's effort to calculate the exact effects on the light waves revealed a connection between critical opalescence and the appearance of the daylit sky. These phenomena are different, but both involve the scattering of light due to variations in fluid density.
While Einstein's analysis was adequate for understanding the light from the sky, it was actually limited in its applicability to critical opalescence itself, because it didn't account for the following. If the molecules in one part of a fluid move together into a smaller volume to make the fluid more dense there, they have to come from the space around that volume, thus making the fluid in that space less dense, unless other molecules from elsewhere take their place. In other words, spontaneous compression in one place means spontaneous expansion in another place; density fluctuations in different parts of the fluid are interdependent. Since the molecules ordinarily don't travel very far in a short time, the interdependence will mostly be between small regions of fluid close to one another, and the effect of this interdependence on light scattering is small. But near the fluid's critical state, the density variations are much larger, and the molecules travel much greater distances going from one region to another. A density change in one part of the fluid can thus affect other parts of the fluid much further away, and omitting the effect of this on light makes Einstein's analysis quite inaccurate for critical opalescence itself.
In analyzing the color of the sky, though, Einstein was able to go further than Rayleigh, by using information about fluid density to indirectly account for how fluids' molecules are distributed. A few decades after Einstein, Zimm again considered molecules directly as Rayleigh had done, but this time with new mathematical techniques to account for the likelihoods of different molecular distributions. One difference between these likelihoods and Einstein's likelihoods for density variations is that the newer techniques implicitly account for the way density changes in different locations balance out. Zimm's more general analysis implies Einstein's formula whenever a density change in one region of fluid has little effect beyond that region's immediate neighborhood, but will give quite different results otherwise. (.....continued)