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Comment by leephillips | Hacker News Reader

Since all these phenomena of chaos are inherent in the differential equations, how can they possibly have anything to do with quantum states? The sensitive dependence on initial conditions is a mathematical property of the description of the chaotic systems. It’s a closed mathematical system that makes no reference to any physical reality.

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Here's the quote, it was in some talk he gave:

The chaos can act as a magnifier of quantum fluctuations so that they can produce sizable effects in the world around us. But we know that that can happen often. - Murray Gell-Mann

I guess the idea is that if you have a collection of particles in a system that displays sensitive initial dependence, then any quantum jiggling gets rapidly amplified.

[edit] As you note this doesn't explain the chaos phenomena, because a stable dynamic system would squelch any quantum weirdness rather than amplify it.

yes, chaos theory, including strange attractors, is still deterministic. it acts at a level where the tiny probabalistic effects of a multitude of tiny particles cancel out in ways to miraculously make the larger subjects of interest predictable.

it's the same phenomenon underlying the wisdom of crowds (and the pricing function in fair markets): each individual can be wildly incorrect in an observation, but averaging many (independent) observations together reveals stable insight.

The differential equations which give rise to physical chaotic systems operate on quantum states interacting with each other at the atomic level, so there has to be a link of some kind.

I don’t understand your comment. Differential equations don’t give rise to physical systems.

Some nonlinear DEs exhibit chaos. That’s a purely mathematical property. Whether any particular DE is a useful model of a particular physical system is a matter for the imagination, and either backed up or refuted by experiment.

> Differential equations don’t give rise to physical systems.

Sure, but differential equations describe physical systems, and there is a canonical way to derive a quantum differential equation from a classical one by quantifying the classical Lagrangian using the Path Integral formulation. Giving a fairly natural distinction between the types of equations

> Whether any particular DE is a useful model of a particular physical system is a matter for the imagination, and either backed up or refuted by experiment.

This doesn't make sense to me. The Navier-Stokes equations are known to describe the classical behavior of water and are experimentally confirmed to predict things like trajectory. Their effectiveness has nothing to do with my imagination. If I write x=x' for the position vector of atoms in a fluid that will completely fail to describe anything physical.

The NS equations are a good description of approximately Newtonian fluids like water within certain regimes (nonrelativistic velocities, larger than atomic scales, far above 0°K, etc.). “Imagination” was not the best choice of word on my part. I meant that DEs are mathematical objects; their connection to physical systems is made by the scientist, not inherent in themselves. Whether the scientist guessed right is determined by experiment. The first people to suggest that the NS equations were a good description of Newtonian fluids had a model for fluid behavior in their imaginations. We know it was a good model because of experiment. But even if the NS equations described nothing in nature, their solutions, chaotic and otherwise, would have whatever properties they have.

Note that there is no Largrangian for the NS equations, by the way.

Yes, the quantumness of a differential equation is not a property of the differential equation, but a statement about one possible taxonomy of differential equation. Then, whether quantum-type diff eq's have unique properties pertaining to chaos conditioned on our knowledge of them being labelled 'quantum' is an interesting mathematical question.

> Note that there is no Lagrangian for the NS equations, by the way.

I don't know much about fluid dynamics, but I was under the impression that Bennett derives the Lagrangian form in the book Lagrangian Fluid Dynamics

There is a clash of terminology: the Lagrangian formulation of fluid dynamics follows the path of fluid particles, in contrast to the Eulerian form, which observes the fluid passing by a fixed coordinate system. In general dissipative systems don’t have time-independent Lagrangians.

yep, that sounds right to me. also I see you're the author of the Noether article from Ars Technica. I enjoyed the read.

I’m glad, thanks!

Good discussion, thanks!

The article presumably being https://arstechnica.com/science/2015/05/the-female-mathemati...

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