# Navier-Stokes from Quantum Mechanics?

by Q_Goest
Tags: mechanics, navierstokes, quantum
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In Victor Stenger's book, "Quantum Gods" he states:
 ... the Navier-Stokes equation is a classical physics equation and is easily derivable from Newtonian mechanics. I have done it in class when I taught undergraduate fluid mechanics. Second, everything in classical mechanics follows as a limit of quantum mechanics and so one can acurately argue that Navier-Stokes is deducible from quantum mechanics.
Do you agree with the second part of this (part that is in bold)? Why or why not? If yes, how would you suggest deriving those equations? If not, what makes the NS equations underivable.

Edit: If you have references that back up your responce, please provide.

Thank you!
 P: 438 If you come up with the proper limits, which usually involve taking some statistical averages, then yes I think that's correct. If we really believe quantum mechanics is correct, and we have a set of classical equations which work beautifully at a certain scale, then I think it's quite intuitive that one should be a limiting case of the other - with some additional assumptions in place.
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 Second, everything in classical mechanics follows as a limit of quantum mechanics
Agreed, for classical mechanics of point particles See http://lcbcpc21.epfl.ch/Course/MDMC_1_2013.pdf (or read the book referenced there).

 and so one can acurately argue that Navier-Stokes is deducible from quantum mechanics.
Yes ... but there is another step that needs to be justified, namely that the NS equations (and many other things in classical physics) approximate a collection of point particles as a continuum. it seems reasonable (to a mathematician) that some parts of the baby may get lost in the bathwater when making that approximation - but I'm no expert.

 Sci Advisor HW Helper PF Gold P: 2,905 Navier-Stokes from Quantum Mechanics? Hi AlephZero. Thanks for the responce. Looking at the reference, it seems to be geared toward predicting molecular properties from quantum mechanics as opposed to NS equations. I think you're right in that fluids can be approximated as a continuum from point particles for a gas at least. As I recall, that is done using van Der Waals forces and adding in conservation of energy and momentum. For a liquid, I would think one needs to consider molecules with a finite volume. I would think the derivation must have been done many times. I really don't doubt the derivation is possible, I just wonder what the best reference might be. Someone must have done this many years ago. In fact, doing a Google search seems to bring up lots of hits, but I'd be curious to see where it started and how it's best explained. Consider that if the statement is true, there should be a reference to the appropriate individual's work on this topic and not just a statement provided as a suggestion.
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 Quote by Q_Goest In Victor Stenger's book, "Quantum Gods" he states: Do you agree with the second part of this (part that is in bold)? Why or why not? If yes, how would you suggest deriving those equations? If not, what makes the NS equations underivable. Edit: If you have references that back up your responce, please provide. Thank you!
I disagree with Stenger's claim- AFAIK, there is no complete quantum theory of dissipative processes, which makes viscosity not derivable. Also, 4 of the N-S equations (momentum and energy balance) involve constitutive relations (stress and strain in the momentum equations, heat and temperature in the energy equation), which are in general not derivable from first principles.

To be fair, there is some work applying the fluctuation-dissipation theorem to hydrodynamics- a good summary is in Chapter 8 of Chaikin and Lubensky's "Principles of condensed matter physics". Dattagupta and Puri "Dissipative phenomena in condensed matter" also has a good summary of current research in quantum dissipative systems.
 Sci Advisor HW Helper PF Gold P: 12,016 Victor Stenger makes a completely nonsensical claim. As AndyResnick says, constitutive relations are critical features in the modelling leading up to N-S, and generally none of them has been justified "ab initio". They are, essentially, (at most!) empirically justified, which is quite a different beast, showing Stenger doesn't know what he talks about. But, even the empirical justification is not really that well founded; rather, the standard N-S equations are the SIMPLEST modelling, in a mathematical sense to work with.
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Andy and arildno, thank you for your explanations. I’d like to understand this a bit better though. Perhaps you can shed a bit more light on this?

I was watching a YouTube video the other day with Steven Weinberg who suggested that the “laws of thermodynamics are entailed by elementary particle physics”. By extension, Steven seemed to infer that classical laws such as described by the NS equations were entailed by elementary particle physics therefore, I assume, they are entailed by quantum mechanics. Arlindo has pointed out that “proving compatibility of NS with QM is NOT the same as deriving NS from QM.” Perhaps Victor’s statement that NS are deducible from QM steps over the line, making the statement false?

Just to clear up one lingering doubt about what is meant here, I take it that constitutive relations are (in general) not derivable from quantum mechanics? That surprises me. I would have thought simple things like stress/strain in a solid for example, simply follow from forces between atoms and how those forces vary depending on distance. Similarly, I would have thought that forces between fluid molecules (or atoms) could be described using QM equations such that in principal at least, one might in some way derive NS equations from QM equations.

The section of Victor Stenger’s book that discusses this regards “bottoms-up” emergence. At the beginning of this section he says:
 If emergence simply means that particle physicists are incapable of deriving the principles developed by workers in most areas of great complexity, where that complexity is measured in terms of the large number of particles involved and their detailed interactions, then there is little more to say. If the conclusion is that the reduction of all material systems to elementary particles is thereby refuted and some additional laws of nature are required, then there is a lot to say. As I mentioned above, no laws of physics are violated in complex systems.
I’d like to stick to the subject regarding NS being derivable from QM and I understand that is being taken to mean that the NS equations are derivable from QM equations and apparently that is incorrect. I understand that dissipative processes and constitutive relations can’t be derived. I’d like a better understanding of why they can’t be derived. Is there an ‘in principal’ reason for this or is it a problem with the mathematics? If in principal, then what more specifically is the issue and can that issue be given a descriptive explanation or is there only a mathematical explanation of the issue?

Thanks again.
 P: 438 I don't see why what he says is so wrong - starting from the quantum mechanics formalism, and then applying the tools of statistical physics to deal with many particles, you should be able to arrive at the N-S equations just fine... Things like dissipation and entropy will come when you make use of the statistical formalism. Unless, you're arguing that you should be able to derive things like the N-S equations purely from the Schrödinger equation, without introducing any other assumptions or tool.