# Thermalization of perfect incompressible fluid

Suppose we have an perfect incompressible fluid (no viscosity) in a box. Lets assume we shook the box and let the fluid sit for a long time. Since there is no dissipation, the energy is conserved. Is the liquid going to come to a stable "thermal" state, or the energy is going to cascade down to smaller and smaller vortices? If the thermal state exist, will the statistical distribution of velocities at a particular point satisfy Maxwell's distribution? How different are the results going to be in 2D vs 3D? Has anyone studied statistical physics of ideal fluid?

I am really curious about the questions but couldn't find any answers easily. Any references would be great.

This is a very interesting question. Although I am not able to come up with a full answer, nor do I know if research or established results about this topic are available, here is what I am thinking.

The incompressible inviscid flow is described by the Euler equation instead of the Navier-Stokes equation. According to the Helmholtz theorems, vortices cannot be created or destroyed in an inviscid fluid. That is, if you have thought of vortices somehow splitting into smaller and smaller ones, this does not seem to be possible, as far as I can see. This does not preclude, however, that even a single vortex line could be entangled more and more with itself (without creating knots, probably) during the evolution of the system. If it would finally become so entangled that it spreads all over the whole box, say in a fractal or Peano curve like fashion, that would be your "thermal state", I guess.

This brings us to the ergodic hypothesis. A statistical-mechanical ensemble in 6N-dimensional phase space behaves much like the incompressible inviscid flow of an ordinary fluid in 3 dimensions. The ergodic hypothesis states, that any system in the ensemble will come arbitrarily close to any other system if you wait long enough. Allegedly, and in colloquial terms, the fluid of the statistical ensemble gets naturally stirred so thoroughly that eventually it is perfectly mixed.

According to Hamilton's equations and Liouville's theorem, the ensemble represents an incompressible flow (divergence of phase space velocity is zero, hence, ensemble density is constant along the flow lines, i.e. material derivative of density is zero, and yet, the continuity equation holds). Similarly to the inviscid fluid, energy is conserved due to Hamilton. Since neighboring sample systems in the ensemble are statistically independent of each other by definition, they cannot influence each other, and so I think this is the analogy to the lack of viscosity.

Having said that, I suspect that it is difficult or impossible to prove thermalization of the ordinary incompressible, inviscid fluid, much in the same way that it has not been possible yet to prove the ergodic hypothesis of statistical mechanics (as far as I know). Moreover, there is Poincare's recurrence theorem, that any purely mechanical system under certain reasonable assumptions is almost periodical, which would preclude ergodicity.

However, I don't know if Poincare's theorem is also valid for continuum systems, particularly the incompressible inviscid fluid of your question. My intuition about stirring paint tells me that there is no almost-periodicity, but then again, this intuition has built around viscous flow, so might not be general enough.

• Nice answer. Working in Poincare's recurrence theorem too.+1 Apr 3, 2021 at 23:17
• Thank you for a detailed answer! I would like to point though, that Helmholtz theorems do not seem to preclude a vortex from self-intersecting and generating a loop (or do they?). I am also not sure how to apply Liouvile's theorem to the motion of vortices, since the equations for the vortex motion are of the first order (velocity of a vortex is defined by the position of all other vortices). Apr 3, 2021 at 23:58
• As to self-intersection of vortices: probably true. But my feeling is that vortices might not cross, although I can't prove that. As to Liouville: that was just an analogy to justify the difficulty of proving thermalization. I don't know if you can strictly apply Liouville to vortices. Apr 4, 2021 at 0:02
• Oliver, a small correction: Poincare's recurrence theorem does not preclude ergodicity, it's valid after all even for chaotic systems, for instance the cat map. Apr 6, 2021 at 9:21

It appears the answer to my question is that ideal fluid does not thermalize, but velocity field cascades down into smaller and smaller scales. A good reference for this is the Onsager's paper, or just wiki's part on Kolmogorov's theory. The velocity at a point is non-Gaussian with heavy exponential tails (see a review here, or a very specific paper here).

These are all results for 3D turbulence. 2D turbulence is very different, since there is no energy cascade mechanism. Point vertices though would thermalize, since they obey an effective Hamiltonian dynamics (see the same Onsager's paper)