>Imagine trying to understand how a modern computer running a video game works, but all you understand is the basics of logic gates. Sure, in principal you could understand what’s going on in terms of bit flips, but it’s hopeless in practice.

See "Could a Neuroscientist Understand a Microprocessor?":

https://journals.plos.org/ploscompbiol/article?id=10.1371/jo...

(which is actually looking at transistor-level data, not gate-level, but the games examined are simple ones)

Yes, I was thinking of that article when I wrote the above comment. Thanks for the link!

Thank you. This is encouraging me to believe that huge materials science progress can be made in the next 50 years driven partially by computational advances.

That depends on what you mean by computational advances. If you mean advances in computer hardware, I wouldn't hold your breath on that helping much since these are generally exponentially scaling problems meaning that tiny incremental improvements in simulation fidelity or system size require doubling the computational resources.

Quantum computers, if large enough ones ever materialize could give us a revolution in computational many-body physics because they could actually circumvent this exponential scaling.

The most promising straightforward avenue though is actually just developing better algorithms that make more insightful, appropriate approximations to correctly capture the physics we care about. That may or may not be what you meant by computational advances.

> The trouble, is not the fundamental building blocks, but how you take a mathematical description of electrons bumping into ions and then generalize that to 10^23 electrons and ions.

Has anyone tried to generalize it to, say, 10^2 electrons and ions? What is the smallest system that empirically exhibits superconductivity?

Most Quantum Monte Carlo work on Fermionic systems deals with a number of particles on the order of 10^2. The numerical work suggest even such a small number can superconduct and we have seen experimentally that there are superconducting nano-particles that can be very small indeed. I don’t recall off the top of my head how many atoms are in the smallest ones we’ve seen though.

As far as pen-and-paper theoretical work goes though, something like 10^2 atoms is actually more difficult than 10^23 of them because 10^23 is basically infinite for our purposes so there are all sorts of useful limits one can take. When you consider something like 10^2 atoms, many useful approximations go out the window. You can no longer ignore the physics at the edge of the material since everything is very close to the edge, you can’t make assumptions about homogeneity, statistical arguments about the aggregate behaviour of electrons become unreliable, etc.

We known that machine learning gets attention of physicists these days, and although some of these applications look unrealistic or far-fetched, it sees to me from your description that condensed matter physics can be attacked from this direction, and it might be feasible.

You start with 1D problems, move to small 3D volumes, and build from there. ML is good at "interpolating" or guessing what the answer must be from a given input. Maybe, one will need many nested levels of ML models to realistically simulate a solid-state material, but this is not entirely impossible, I think.

If such an ML simulation can be done reasonably efficiently, then likely there could be a theory that can be formulated in terms of equations, approximating the "theory of everything" with sufficient detail.

A lot of people are trying to explore modern ML methods in physics, but they’ve so far had limited success at least in terms of interesting theoretical results. One glaring problem is that ML methods tend to be black boxes, so even if someone solved all of condensed matter physics with some super neural network, the question would remain “how and why was this able to work?” “What is going on physically?” which is what physicists tend to actually care about more than being able to make a prediction.

Believe me though, people are trying.

On the other hand, there is plenty of work on rule extraction from trained neural networks being done. Wouldn't it be amazing if we extracted actual laws of physics from a network trained by observations?

Yes, I know that.

What I'm saying is that if an ML model can approximate quickly the solution, then there could be a simple theory expressed in terms of equations, approximating the full problem. I.e. if there is an efficiently computable procedure, then it's a sign that there might be a good simple approximating theory.