Can you think of examples where a macroscopic law coexists with a fully known mi

Question

Can you think of examples where a macroscopic law coexists with a fully known microscopic law, but the former hasn't been derived from the latter (yet)? Or maybe a rule of thumb, which works but hasn't been understood yet. Or some emergence which is hard to see from the individual interacting parts. The microscopic arrangement should be simple. So plain complexity like in biology is something different. Most thermodynamic results and entropy-related results you can derive from microscopic assumptions.

I heard apparently Fourier's law isn't that clear http://www.citebase.org/abstract?id=oai%3AarXiv.org%3Amath-ph%2F0002052

Maybe there are more examples from other physical areas (quantum mechanics, chemistry etc.)? Basically something where there are fully understood microscopic laws and also a macroscopically observed law which theoretically should be derivable.

Explanation / Answer

There are thousands of such examples, it is basically all situations in condensed matter physics. You see a lot of regularities that have no explanation.

Here's one of the most annoying ones for me: Moseley's law--- you can knock out one of the two electrons most tightly bound to a heavy atom (in the K-shell). This leaves a hole orbiting the nucleus. The energy of this hole can be calculated from the Bohr model, except that you need to use a nuclear charge reduced by exactly 1 unit. This is due to electron screening.

But why is this exactly one unit? Measurements in heavy atoms show that the K-shell Moseley screening is one electron charge. But the other K-shell electron is orbiting at the same r, and the far-away electrons contribute different amounts, and yet somehow when you sum up all their screening contributions, no matter what the atom, you end up reducing the nuclear charge by one unit. This is not understood. I will ask it as a question.

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