is trivially true, and this is why I introduced solving the many-body S.E. as approximately equivalent to the n-body problem.
Alright, I will take your word for it. I had never seen anyone say that the classical Newtonian-mechanical sort of n-body problem was almost identical to a quantum intra-atomic version, though.
Exactly why I think you’ve made a good point. I need to look at the approximation and see if it’s possible. If it has 10^24 derivatives to get chemical accuracy, and scales poorly with respect to n, then it’s probably not useful in practice, but the argument you make here explicitly is exactly the argument I understood implicitly from your previous post.
If nothing else, it’s an interesting example of a data/computation tradeoff.
(To expand for people not following: in the OP, he claims that an algorithm/AI which wants to design effective MNT must deal with problems equivalent to the n-body problem; however, since there is no solution to the n-body problem, it must use approximations; but by the nature of approximations, it’s hard to know whether one has made a mistake, one wants experimental data confirming the accuracy of the approximation in the areas one wants to use it; hence an AI must engage in possibly a great deal of experimentation before it could hope to even design MNT. I pointed out that there is a proven exact solution to the n-body problem contrary to popular belief; however, this solution is itself extremely inefficient and one would never design using it; but since this solution is perfect, it does mean that a few chosen calculations of it can replace the experimental data one is using to test approximations. This means that in theory, with enough computing power, an AI could come up with efficient approximations for the n-body problem and get on with all the other tasks involved in designing MNT without ever running experiments. Of course, whether any of this matters in practice depends on how much experimenting or how much computing power you think is available in realistic scenarios and how wedded you are to a particular hard-takeoff-using-MNT scenario; if you’re willing to allow years for takeoff, obviously both experimentation and computing power are much more abundant.)
Alright, I will take your word for it. I had never seen anyone say that the classical Newtonian-mechanical sort of n-body problem was almost identical to a quantum intra-atomic version, though.
There are differences and complications because of things like Uncertainty, magnetism, and the Pauli exclusion principle, but to first order the dominant effect on an individual atomic particle is the Coulomb force and the form of that is identical to the Gravitational force. The symmetry in the force laws may be more obvious than the Hamiltonian formulation I gave before.
F_G=frac{Gm_1m_2}{r2}:and:F_C=frac{k_eq_1q_2}{r2}
The particularly interesting point is that even without doing any quantum mechanics at all, even if atomic bonding were only a consequence of classical electrostatic forces, we still wouldn’t be able to solve the problem. The difficulty generated by the n-body problem is in many ways much greater than the difficulty generated by quantum mechanics.
This is sort of true. The fact that it turns into the n-body problem prevents us from being able to do quantum mechanics analytically. Once we’re stuck doing it numerically, then all the issues of sampling density of the wave function et al. crop up, and they make it very difficult to solve numerically.
Thanks for pointing this out. These numerical difficulties are also a big part of the problem, albeit less accessible to people who aren’t comfortable with the concept of high-dimensional Hilbert spaces. A friend of mine had a really nice write-up in his thesis on this difficulty. I’ll see if I can dig it up.
Why do we have to solve it? In his latest book, he states that he calculates you can get the thermal noise down to 1⁄10 the diameter of a carbon atom or less if you use stiff enough components.
Furthermore, you can solve it empirically. Just build a piece of machinery that tries to accomplish a given task, and measure it’s success rate. Systematically tweak the design and measure the performance of each variant. Eventually, you find a design that meets spec. That’s how chemists do it today, actually.
Alright, I will take your word for it. I had never seen anyone say that the classical Newtonian-mechanical sort of n-body problem was almost identical to a quantum intra-atomic version, though.
If nothing else, it’s an interesting example of a data/computation tradeoff.
(To expand for people not following: in the OP, he claims that an algorithm/AI which wants to design effective MNT must deal with problems equivalent to the n-body problem; however, since there is no solution to the n-body problem, it must use approximations; but by the nature of approximations, it’s hard to know whether one has made a mistake, one wants experimental data confirming the accuracy of the approximation in the areas one wants to use it; hence an AI must engage in possibly a great deal of experimentation before it could hope to even design MNT. I pointed out that there is a proven exact solution to the n-body problem contrary to popular belief; however, this solution is itself extremely inefficient and one would never design using it; but since this solution is perfect, it does mean that a few chosen calculations of it can replace the experimental data one is using to test approximations. This means that in theory, with enough computing power, an AI could come up with efficient approximations for the n-body problem and get on with all the other tasks involved in designing MNT without ever running experiments. Of course, whether any of this matters in practice depends on how much experimenting or how much computing power you think is available in realistic scenarios and how wedded you are to a particular hard-takeoff-using-MNT scenario; if you’re willing to allow years for takeoff, obviously both experimentation and computing power are much more abundant.)
There are differences and complications because of things like Uncertainty, magnetism, and the Pauli exclusion principle, but to first order the dominant effect on an individual atomic particle is the Coulomb force and the form of that is identical to the Gravitational force. The symmetry in the force laws may be more obvious than the Hamiltonian formulation I gave before.
F_G=frac{Gm_1m_2}{r2}:and:F_C=frac{k_eq_1q_2}{r2}
The particularly interesting point is that even without doing any quantum mechanics at all, even if atomic bonding were only a consequence of classical electrostatic forces, we still wouldn’t be able to solve the problem. The difficulty generated by the n-body problem is in many ways much greater than the difficulty generated by quantum mechanics.
Also, nice summary.
I am not a physicist, but this stack exchange answer seems to disagree with your assessment: What are the primary obstacles to solve the many-body problem in quantum mechanics?
This is sort of true. The fact that it turns into the n-body problem prevents us from being able to do quantum mechanics analytically. Once we’re stuck doing it numerically, then all the issues of sampling density of the wave function et al. crop up, and they make it very difficult to solve numerically.
Thanks for pointing this out. These numerical difficulties are also a big part of the problem, albeit less accessible to people who aren’t comfortable with the concept of high-dimensional Hilbert spaces. A friend of mine had a really nice write-up in his thesis on this difficulty. I’ll see if I can dig it up.
Why do we have to solve it? In his latest book, he states that he calculates you can get the thermal noise down to 1⁄10 the diameter of a carbon atom or less if you use stiff enough components.
Furthermore, you can solve it empirically. Just build a piece of machinery that tries to accomplish a given task, and measure it’s success rate. Systematically tweak the design and measure the performance of each variant. Eventually, you find a design that meets spec. That’s how chemists do it today, actually.
Edit : to the −1, here’s a link where a certain chemist that many know is doing exactly this : http://pipeline.corante.com/archives/2013/06/27/sealed_up_and_ready_to_go.php