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The author mentions that having an equation that describes a system accomplishes little if the solutions to that equation are unavailable. He kind of downplays

The author mentions that having an equation that describes a system accomplishes little if the solutions to that equation are unavailable. He kind of downplays quantum mechanics, but I would argue the situation is just as severe as that with turbulence. The Schrödinger equation iħ(∂Ψ/∂t) = HΨ is a very simple equation. Yet the solution to this equation is nearly impossible to solve for all but the simplest systems. A lot of approximations to it have been developed, but to numerically converge on the exact solution is an NP-hard problem in most cases (using QMC). And we all know what that means. There's a famous quote attributed to Max Born: "It would indeed be remarkable if Nature fortified herself against further advances in knowledge behind the analytical difficulties of the many-body problem." In other words, what if we discover all the equations that govern nature, but they are simply infeasible to solve (and worse, we even mathematically prove that they're intractable)? It's an interesting, yet somewhat depressing idea.

avz 12y ago
I'd risk the optimistic expectation that human ingenuity can work with such a state of affairs. Here's one example I can think of: if an interesting measure in a phenomenon follows some equation, we could find some solutions to that equation through experimental approach treating the phenomenon as a program and Nature as the compute engine. Very often you can prove statements about relationships among solutions to an equation (e.g. they may form a linear or affine space). You could then use the original experimental solution to generate further solutions using numerically tractable transformations.
jcr 12y ago
Maybe I'm just a glutton for punishment, but for me, intractable and even undecidable problems are, if you pardon the pun, endless fun. Routing of PCBs and ICs is one of my favorites, and decompliation/disassembly is another. I'm just totally content with knowing that I'll never be "that guy" who gets famous for finding "the" perfect solution (especially if we can already prove a perfect solution doesn't exist). But with enough effort, I might be one of the many people who come up with one of the many increasingly better incomplete solutions. A small step in the right direction, like the approximations of Schrödinger's equation you mentioned, can be very useful as well as an amazingly fun challenge. It's a bit like those video games that keep getting tougher but never end; all the fun is in seeing how far you can get.
drkevorkian 12y ago
Two counterpoints: 1) What you've written, where H is independent of t is reducible to just the Eigenvalue problem for H. Certainly better understood than fluid dynamics. Even with time-dependent hamiltonians we have decent tools for talking about the solutions (Dyson series). 2) Of course for certain values of H, (esp. in continuous space) you can contrive ways of making the eigenvalue problem hard, but you don't have to go as far as quantum mechanics to find difficulty. Just take three bodies under newton's gravitation. Yes, the quantum n-body problem is exponentially harder in n, but that's a fundamentally different type of "hardness" than the hardness of Navier-Stokes.
- Xcelerate 12y ago
  1) is reducible to just the Eigenvalue problem for H. Certainly better understood than fluid dynamics. I feel like that's akin to stating that Fermat's theorem just shows there is no n > 2 such that x^n + y^n = z^n 2) you can contrive ways of making the eigenvalue problem hard You've got this backward. You can contrive ways of making it easy. Almost all physical problems for any system larger than but the simplest of atoms is incredibly hard.

avz 12y ago

I'd risk the optimistic expectation that human ingenuity can work with such a state of affairs. Here's one example I can think of: if an interesting measure in a phenomenon follows some equation, we could find some solutions to that equation through experimental approach treating the phenomenon as a program and Nature as the compute engine. Very often you can prove statements about relationships among solutions to an equation (e.g. they may form a linear or affine space). You could then use the original experimental solution to generate further solutions using numerically tractable transformations.

jcr 12y ago

Maybe I'm just a glutton for punishment, but for me, intractable and even undecidable problems are, if you pardon the pun, endless fun. Routing of PCBs and ICs is one of my favorites, and decompliation/disassembly is another. I'm just totally content with knowing that I'll never be "that guy" who gets famous for finding "the" perfect solution (especially if we can already prove a perfect solution doesn't exist). But with enough effort, I might be one of the many people who come up with one of the many increasingly better incomplete solutions. A small step in the right direction, like the approximations of Schrödinger's equation you mentioned, can be very useful as well as an amazingly fun challenge. It's a bit like those video games that keep getting tougher but never end; all the fun is in seeing how far you can get.

drkevorkian 12y ago

Two counterpoints: 1) What you've written, where H is independent of t is reducible to just the Eigenvalue problem for H. Certainly better understood than fluid dynamics. Even with time-dependent hamiltonians we have decent tools for talking about the solutions (Dyson series). 2) Of course for certain values of H, (esp. in continuous space) you can contrive ways of making the eigenvalue problem hard, but you don't have to go as far as quantum mechanics to find difficulty. Just take three bodies under newton's gravitation. Yes, the quantum n-body problem is exponentially harder in n, but that's a fundamentally different type of "hardness" than the hardness of Navier-Stokes.

Xcelerate 12y ago
1) is reducible to just the Eigenvalue problem for H. Certainly better understood than fluid dynamics. I feel like that's akin to stating that Fermat's theorem just shows there is no n > 2 such that x^n + y^n = z^n 2) you can contrive ways of making the eigenvalue problem hard You've got this backward. You can contrive ways of making it easy. Almost all physical problems for any system larger than but the simplest of atoms is incredibly hard.