Why Hopf's last hope''?

In 1942 E. Hopf [EH42] might have (or might have not) formulated what we call Hopf's vision of turbulence.''

The story so far goes like this:

In 1949/50 Robert Kraichnan was assistant to Albert Einstein. His child prodigy work - age 19 - was on quantum theory of (linear) gravity - he showed that a massless spin 2 particle coupled to stress-energy tensor leads to gravity. Einstain told Kraichnan don't do general relativity, do fluid dynamics.'' So Kraichnan did fluid dynamics. Asked, much later, by Ed A. Spiegel, what was harder, he said General realtivity is much, much harder.''

In 1960 E.A. Spiegel was Robert Kraichnan's research associate. Kraichnan told him:

Flow follows a regular solution for a while, then another one, then switches to another one; that's turbulence''.

It was not too clear, but Kraichnan's vision of turbulence moved Spiegel. In 1962 E.A. Spiegel and D. Moore investigated a 3rd order convection equations which seemed to follow one periodic solution, then another, and continued going from periodic solution to periodic solution. Ed told Derek:

This is turbulence!'' and Derek said This is wonderful!''

and was moved. He went to give a lecture at Caltech sometime in 1964 and came back angry as hell. They pilloried him there:

Why is this turbulence?''

they kept asking and he could not answer, so he expunged the word turbulence'' from their 1966 article [MS66] on periodic solutions. In 1970 E.A. Spiegel met Kraichnan and told him:

This vision of turbulence of yours has been very useful to me.''

Kraichnan said: That wasn't my vision, that was Hopf's vision''.

What Hopf  actually said and where he said it remains deeply obscure to this very day. There are papers that lump him together with Landau, as the Landau-Hopf's incorrect theory of turbulence'', but he did not seem to propose incommensurate frequencies as building blocks of turbulence, which is what Landau's guess was.

However, Eberhardt Hopf says in his beautiful 1948 paper [Hopf48]:

It is convenient to visualize the solutions in the phase space $\Omega$ of the problem. A phase or state of the fluid is a vector field $u(x)$ in the fluid space that satisfies [the Navier-Stokes equations] and the boundary conditions. The totality $\Omega$ of these phases is therefore a functional space with infinitely many dimensions. A flow of the fluid represents a point motion in $\Omega$ and the totality of these phase motions forms a stationary flow in the phase space $\Omega$ [...]. , which, of course, is to be distinguished from the fluid flow itself. What is the asymptotic future behavior of the solutions, how does the phase flow behave for $t \to \infty$~? And how does this behavior change as [the viscosity] $\mu$ decreases more and more? How do the solutions which represent the observed turbulent motions fit into the phase picture?...

In early spring of 1976 Mitchell J.~Feigenbaum and Predrag Cvitanovi? sat in a Princeton bar, pondering. What makes cloud a cloud"? Mitchell lamented lack of mathematics to describe patterns. Five years later and feeling so much richer, the two were lounging in the establishment then known as Le Charme Discret de la Bourgeoisie'', now long defunct, still pondering. Mitchell produced a printout from his HP desktop computer, with many periodic points marked over the H\'enon attractor.

Due to viscosity $\mu$, the motion of fluids was assumed by E. Hopf to asymptote on some finite-dimensional manifold $\inertM(\mu)$ in the infinite-dimensional state space. This statement is again dated back to Hopf:

The qualitative mathematical picture which [E.~Hopf] % author conjectures to correspond to the known facts about hydrodynamic flow is this: To the flows observed in the long run after the influence of the initial conditions has died down there correspond certain solutions of the Navier-Stokes equations. These solutions constitute a certain manifold $\inertM = \inertM(\mu)$ invariant under phase flow. Probably owning to viscosity $\inertM$ has a finite number $N= N(\mu)$ of dimensions. This effect of viscosity is most evident in the simplest case of $\mu$ sufficiently large. In this case $\inertM$ is simply a point, $N=0$. Also the complete stability of $\inertM$ is in this simplest case is obviously due to viscosity. On the other hand, for smaller and smaller values of $\mu$, the increasing chance character of the observed flow suggests that $N \to \infty$ monotonically as $\mu \to 0$.''

Today $\inertM(\mu)$ is known as the inertial manifold'' and widely studied. Its finite dimensionality $N(\mu)$ for non-vanishing $\mu$ has been, in certain settings, rigorously established by Foias and his collaborators\rf{FNSTks88}. In hindsight of more recent development in the theory of ergodic systems, Hopf's intuition about the monotonicity'' with smooth changes in system parameters such as $\mu$ has turned out to be wrong - due to the lack of structural stability a family of dynamical systems parametrized by $\mu$ can explore infinity of coexisting attractors, including collapsing to a 1-dimensional periodic attractor, infinitely often\rf{FSTks86}. Nevertheless, the intuition is qualitatively correct.

What E.~Hopf writes next is prescient:

The geometrical picture of the phase flow is, however, not the most important problem of the theory of turbulence. Of greater importance is the determination of the probability distributions associated with the phase flow [...], particularly of their asymptotic limiting forms for small $\mu$.''

The inspiration for the recurrent patterns program'' comes from the way we perceive turbulence. Turbulent systems never settle down, but we can identify a snapshot as a cloud'', and an experimentalist can tell what the values of physical parameters in a turbulence experiment were after a glance at the digitized image of its output. How do we do it?

within what we shall here refer to as the Hopf's vision'' or recurrent patterns program''.

In Hopf's vision turbulence explores a repertoire of distinguishable patterns; as we watch a turbulent'' system evolve, every so often we catch a glimpse of a familiar pattern:

\centerline{ \raisebox{-4.0ex}[5.5ex][4.5ex] { \includegraphics[height=12ex]{figs/Hopf-a.eps} } ~~~ $\Longrightarrow$ ~~ {other swirls} ~~ $\Longrightarrow$ ~~~ \raisebox{-4.0ex}[5.5ex][4.5ex] { \includegraphics[height=12ex]{figs/Hopf-b.eps} } }

For any finite spatial resolution, the system follows approximately for a finite time a pattern belonging to a finite alphabet of admissible patterns, and the long term dynamics can be thought of as a walk through the space of such patterns, just as chaotic dynamics with a low dimensional attractor can be thought of as a succession of nearly periodic (but unstable) motions.

So, if one is to develop a semiclassical field theory of systems that are classically chaotic or turbulent,'' the first problem one faces is how to determine and classify the classical solutions of nonlinear field theories. Given how difficult the implementation of the periodic orbit theory can be already in finite dimensions, and the presence of features in the quantum field theory not captured by semiclassical expansions, its implementation in a full-fledged quantum field theory has for a long time seemed an overly ambitious project.

The recurrent patterns program does not address issues such as the Kolmogorov's 1941 homogeneous turbulence, with no coherent structures fixing the length scale; here all the action is in specific coherent structures. It does not seek universal scaling laws; spatio-temporally periodic solutions are specific to the particular set of equations and boundary conditions. And it is {not} probabilistic; everything is fixed by the deterministic dynamics, with no probabilistic assumptions on the velocity distributions or external stochastic forcing.

What new insights into spatio-temporal chaos or turbulence can the periodic-orbit theory provide? The theory yields accurate prediction for measurable time-averaged observables for a given geometry and Reynolds number, such as the mean frictional drag for the plane Couette flow (see \reffig{Fig:KKdrag}), mean velocity profiles within the flow, or the mean pressure drop per unit length in a pipe flow. It offers a detailed qualitative and quantitative understanding of the geometry of turbulent vs. laminar basins of attraction, with applications to non-local control of such flows.

Suppose that the above program is successfully carried out for classical solutions of some field theory. What are we to make of this information if we are interested in the quantum behavior of the system?

Sadly, searching for periodic orbits will never become as popular as a week on C\^{o}te d'Azur, or publishing yet another log-log plot in Phys. Rev. Letters. Writing a code to find even one unstable periodic orbit can take months for a novice, and there is no way to prepare a canned routine for all seasons - each dynamical system arrives with its own peculiarities and quirks.

References

[EH42] E. Hopf, Abzweigung einer periodischen L\"osung'', Bereich. S\"achs. Acad. Wiss. Leipzig, Math. Phys. Kl.  94, 19 (1942)

[MS66] Moore D W and Spiegel E A , A thermally excited nonlinear oscillator'', Astrophys. J.  143, 871  (1966)

[BMS71] Baker N H, Moore D W and Spiegel E A, Quatr. J. Mech. and Appl. Math.  24,  391 (1971)

[Hopf48] E. Hopf, A mathematical example displaying features of turbulence'', Comm. Appl. Math. 1, 303 (1948)