# Why can't the Navier Stokes equations be derived from first principle physics?

At the 109th UCLA Faculty Research lecture, Seth Putterman gave a talk on Sonoluminescence. During the lecture he emphasized that "The Navier Stokes equations cannot be derived from first principles [of physics]".

In physics there are lots of first principles, and so the first question is what set of first principles would one expect to derive the Navier Stokes equations?

And the second, and main question is why does a derivation fail? Are we missing some yet to be discovered set of first principles in this area of physics?

• I don't know about any first principles in physics other than experiments. I am looking forward to being enlightened though, what other people think physical first principles are supposed to be. – CuriousOne Jun 29 '15 at 3:50
• Are not the NS equations a moment of the BBGKY hierarchy? It's been a while since I looked at this, but I recall it going something like this: mpe.dimacs.rutgers.edu/2013/10/08/… – Declan Jun 29 '15 at 5:27
• @CuriousOne: I'm skeptical that experiments can be first principles. An experiment is formulated with reference to a theory - for example they might be done by humans that have a notion of "particle", "mass", "location" which are theoretical physical concept which are more of less hard to define and not necessary. Maybe it works if "an experiment" is nothing than a sheet of numbers to you. As soon as you imply "this is a sheet with distances, measured in time intervals of five clock ticks", you're deep in theory land, speaking in context of a bunch of notions that people made up. – Nikolaj-K Jun 29 '15 at 7:51
• Technically, empirical disciplines, like Science, do not have first principles as such because they do not have axiomatic assumptions. Rather, this is a term generally applied to logical disciplines such as Mathematics. – RBarryYoung Jun 29 '15 at 15:21
• @docscience: Found it: Around 46:20-49:50 and 51:22-53:30. – Qmechanic Jun 29 '15 at 18:37

None of the interesting equations in physics can be derived from simpler principles, because if they could they wouldn't give any new information. That is, those simpler principles would already fully describe the system. Any new equation, whether it's the Navier-Stokes equations, Einstein's equations, the Schrodinger equation, or whatever, must be consistent with the known simpler principles but it has also to incorporate something new.

In this case you appear to have the impression that an attempt to derive the Navier-Stokes equations runs into some impassable hurdle and therefore fails, but this isn't the case. If you search for derivations of the Navier-Stokes equations you will find dozens of such articles, including (as usual) one on Wikipedia. But these are not derivations in the sense that mathematicians will derive theorems from some initial axioms because they require some extra assumptions, for example that the stress tensor is a linear function of the strain rates. I assume this is what Putterman means.

Later:

Phil H takes me to task in a comment, and he's right to do so. My first paragraph considerably overstates the case as the number of equations that introduce a fundamentally new principle are very small.

My answer was aimed at explaining why Putterman says the Navier-Stokes equations can't be derived but actually they can be, as can most equations. Physics is based on reductionism, and while I hesitate to venture into deep philosophical waters physicists basically mean by this that everything can be explained from a small number of basic principles. This is the reason we (some of us) believe that a theory of everything exists. If such a theory does exist then the Navier-Stokes equations could in principle, though not in practice, be derived from it.

Actually the Navier-Stokes equations could in principle be derived from a statistical mechanics treatment of fluids. They don't require any new principles (e.g. relativity or quantum mechanics) that aren't already included in a the theoretical treatment of ideal fluids. In practice they are not derivable because those derivations based on a continuum approach rather than a truly fundamental treatment.

• Disagree on the first paragraph - we use useful equations covering bulk behaviour which could be derived from simpler principles because it is easier to use (where the derived relations are sufficient). $PV = nRT$ is not fundamental, and it breaks down for anything too far from an ideal gas. But no-one would throw it away when considering gas behaviour for a combustion engine, for example, because starting from first principles would get you to the same, or worse result given an amount of resource effort. – Phil H Jun 29 '15 at 9:30
• The Chapman-Enskog equation is the link between statistical mechanics and the Navier-Stokes equations that you allude to at the end. – tpg2114 Jun 29 '15 at 23:54
• While I appreciate your clarification, I frankly think that your first sentence is completely ridiculous (no offense - I know that you have an extremely deep understanding of physics). Most non-introductory physics textbooks begin by listing all the fundamental postulates in the first chapter, and then the entire rest of the book is dedicated to deriving corollaries from those postulates. Would you say that no equation outside of that first chapter is ever "interesting" or "gives new information"? – tparker Jul 26 '17 at 8:36

They are derivable from classical mechanics using either the continuum or molecular points of view.

Starting with a continuum view, one applies conservation of mass, momentum, and energy to a control volume and the result is the Navier Stokes equations. The Navier Stokes equations, in the usual form, apply to Newtonian fluids, that is fluids whose stress and rate-of-strain are linearly related. One might regard this as an assumption but it can also be viewed as the first term in a power law expansion.

Starting with a microscopic point of view, one can derive the Navier-Stokes equations from taking moments of the Boltzmann equation. In this approach, the linear relation between stress and rate-of-strain appears naturally as the first term in the Chapman-Enskog expansion.

Many undergraduate fluids textbooks include a derivation from the continuum point of view. The derivation from a molecular point of view is done in first-year graduate textbooks such as Introduction to Physical Gas Dynamics by Vincenti and Kruger.

I once asked Putterman after a similar colloquium what he meant by this statement, and his answer was "long time tails". Long time tails are fractional powers that appear in the long time behavior of correlation functions, see, for example, here and here. These fractional powers are seen in molecular dynamics (they are more difficult to see experimentally), but they are not accounted for by the Navier-Stokes (NS) equation, and it is not completely obvious where these effects are hidden in the standard derivations of the NS equation from kinetic theory.

Long time tails are related to fluctuations, and so are ultimately a reflection of the fact that any coarse grained description must depend on a scale, and that the most general theory of non-equilibrium correlation functions at long distances and long times must involve more than a deterministic, continuous partial differential equation such as the Navier-Stokes equation.

The role of noise terms has been studied by a number of people, beginning with Landau and Lifschitz. The basic conclusions are:

1) There is a systematic low energy (long time) theory of correlation functions, which involves a gradient expansion of the conserved currents, and averaging over noise terms fixed by fluctuation-dissipation relations. The Navier-Stokes approximation corresponds to linear derivatives in the stress tensor, and no noise terms. This is a consistent approximation in three dimensions (but not in two).

2) At higher order noise terms have to be included, and kinetic coefficients become scale dependent. The hydrodynamic equations require a cutoff, and the best we can hope for is that low energy (long time) predictions are cutoff independent order by order in the low energy expansion.