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I'm not an expert on particle physics, but the verbiage sounds to me like one of the standard problems in physics: We can write the differential equations. But
by jerf 5y ago
I'm not an expert on particle physics, but the verbiage sounds to me like one of the standard problems in physics: We can write the differential equations. But even simple differential equations can be unsolvable. For example, consider the simple Three Body Problem. The differential equations are simple enough that you can use them as an introduction to the concept of differential equations themselves, but the Three Body Problem is not in general solvable.
The three body problem can generally be acceptably approximated for a reasonable period of time. But that problem only involves inverse squaring of distances. Strong forces decay much faster than that, which makes them more sensitive to errors. Plus you end up with one of the fundamental problems we have trying to understand our universe, which is just how monstrously enormous and monstrously slow we humans are. We operate at "meter" scales and "second" time frames, and particle physics operates at somewhere around 30 and 40 orders of magnitude smaller, respectively. (Not quite all the way down to the Planck sizes, but closer to those than to the macroscopic world.) So when you try to numerically approximate the differential equations, you don't get very far in time or space before your approximations have critically diverged from reality.
It's like we're trying to work out the fundamentals of chemistry and our primary tool is smashing planets together.
- klodolph 5y agoWhen people say that the three-body problem is "unsolvable" they really mean that there's not a way to write out an analytic solution to the problem. In the same way, you can call quintic polynomials "unsolvable", because there's no way to take arbitrary quintic polynomial equations and express the solutions as ordinary algebraic formulas. However, it's very misleading to call quintic equations unsolvable, because we know where the solutions are, and we can use various numeric methods to calculate the solution with arbitrary precision. Any time we can calculate the answer with as much precision as we want, I'd like to say that the problem is "solved" in a very real and meaningful way. The problem is worse with quantum mechanics. With quantum mechanics, not only do we lack analytic solutions to many of the equations used in QM, but we also lack good numeric solutions (using real hardware, at least).
- ethbr0 5y agoFollowing the analogy, as a layperson, what's the nature of the difference with quantum mechanics that doesn't allow it to be solved to arbitrary precision (even if through brute force iteration)? Is it that we haven't discovered the solution generating algorithms? That the state/probabilities of quantum mechanics are fundamentally untenable to similar calculation? Or something else entirely?
- klodolph 5y agoIt's nothing fundamental, as far as I am aware. Depends on what you mean by "fundamental". There is no known way to simulate a quantum computer, using a classical computer, in polynomial time. A quantum computer is just a kind of quantum system, so we know that some quantum systems cannot be efficiently modeled (barring revolutionary advances in simulation algorithms). When your simulations take superpolynomial time, it tends to be easy to find problems which you simply do not have the computational resources to solve, and you may not be able to solve interesting versions of the problem. There are lots of examples of problems like this. However, I don't consider this to be a fundamental difference. For example, satellite navigation systems are just fine calculating directions for driving all the way across the continental US, even though that's a very "large" instance of the problem that they are solving. But if you try to find the fastest route for a delivery driver to make a hundred deliveries within one city, good luck. This is just an analogy, and I'd like to emphasize that "no KNOWN algorithm" efficiently solves these problems, and that we haven't proven whether such an algorithm exists.
- ethbr0 5y agoThanks! By "fundamental," I was curious about the nature of the problem's difficulty, moreso than the progress on solutions. And computational intractability due to system properties makes sense!
- zardo 5y agoAlso a layperson, but as I understand it. We do have the algorithms to simulate quantum systems with n states to arbitrary precision through brute force. But they require that you can put ~n bits into a superposition state. If your computer doesn't have that capability, you can simulate it to arbitrary precision with ~2^n bits.
- crznp 5y agoWell put, though one nitpick: the three body problem is solved. It illustrates another case: even if a solution is known, it might not be practical. https://www.math.uvic.ca/faculty/diacu/diacuNbody.pdf https://www.math.uvic.ca/faculty/diacu/diacuNbody.pdf