Viscosity II: String Theory

Non-experts might want to read Viscosity I first.

String Theorist and blogger Joseph Djugashvili has a post on what string theory has to say about viscosity, and in particular, about the ratio of viscosity to entropy density. My first reaction was “Say what?” Why should string theory have anything to say about viscosity? It’s a long post, with several interesting references and a number of the standard boring rants, but I won’t attempt to summarize – go read it yourself.

Getting back to the question, recall that viscosity is all about transverse transfer of momentum. At low energies, transverse forces have a long time to act, so it’s not implausible that viscosities could become arbitrarily large, and they do. Conversely, it’s also not really surprising that transverse momentum transfers should saturate at high energies.

Consider the case of a bullet smashing into a steel plate. If you have an ordinary bullet, and a reasonably thick plate, as the bullet hits it transfers momentum to the atoms of steel directly in front of it, but the forces holding the plate together transfer momentum transversely allowing the bullet to be slowed to a stop. This process has time to act because ordinary bullets travel slowly compared to the speed of sound in steel – the speed with which the momentum can be distributed throughout the larger plate. When a hypervelocity bullet – one travelling fast compared to the speed of sound in the plate – strikes, the atoms of steel in front of it have no time to call on their neighbors for help, and they just get pushed ahead of the pellet. Thus a small hypervelocity pellet can penetrate inches of steel armor.

Transverse momentum transfer saturation has been an important theme in high energy physics, if what I dimly recall is not too far off. Naïve question: Is it possible that such momentum transfer saturation has to occur if we want to avoid non-renormalizable forces?

Back to string theory:
The key paper cited by Lumo is Viscosity in Strongly Interacting Quantum Field Theories from Black Hole Physics by P.Kovtun, D.T.Son, A.O.Starinets. The paper has a key result, and a conjecture:

The ratio of shear viscosity to volume density of entropy can be used to characterize how close a given fluid is to being perfect. Using string theory methods, we show that this ratio is equal to a universal value of $\hbar/4\pi kB$ for a large class of strongly interacting quantum field theories whose dual description involves black holes in anti--de Sitter space. We provide evidence that this value may serve as a lower bound for a wide class of systems, thus suggesting that black hole horizons are dual to the most ideal fluids.

I can’t be trusted to adequately summarize their string theoretic argument, but I will try anyway. (Experts should read the paper themselves, and the curious should consult somebody who understands ST). My understanding is as follows: they consider a strongly interacting quantum field theory which has a holographic dual black brane (in the AdS/CFT sense). They use this duality to compute the transverse components of the stress energy tensor in terms of the graviton absorption cross section on the brane. The entropy density can be computed from the temperature of the brane, and the transverse components give the viscosity.

Their argument applies to a class of quantum field theories. The conjecture is that the viscosity/entropy ratio computed above serves as a lower bound for all (or at least “a wide class”) of fluids. They present an uncertainty principle based argument to support it. Again, my summary can hardly do justice to it, but I think that the most crucial idea is that as the entropy decreases, the ability of transverse modes to transfer momentum does as well.

Finally, they have this discussion argument, also used by Lumo, which looks bogus to me.

Discussion.—It is important to avoid some common misconceptions which at first sight seem to invalidate the viscosity bound. Somewhat counterintuitively, a near-ideal gas has a very large viscosity due to the large mean free path. Likewise, superfluids have finite and measurable shear viscosity associated with the normal component, according to Landau’s two-component theory.

The superfluid component really does have zero viscosity. Of course it also has zero entropy density as well, so the bound isn’t really defined. So let’s not clutter up a perfectly good argument with bogus nonsense.

So did string theory really tell us something important here? Well, at a minimum, it seems to have suggested an important idea, but the fact that they were able to deduce a version of their bound from the uncertainty principle alone hints that the bound may not depend on the field theory – black hole duality. Maybe relativity and quantum mechanics alone sufficiently constrain transverse momentum transfer.