Whether a fluid can be made arbitrarily thin

Every essay before this one treats viscosity as a property a substance happens to have. Pitch has a lot of it and helium has almost none, the range is enormous, and nothing in any of the arguments suggests where it stops.

Asking where it stops requires asking what it would be compared against. A viscosity is a pascal-second, and a pascal-second has no privileged value: there is no combination of the fundamental constants that produces one. So “a small viscosity” is not a statement until it is made dimensionless.

There is one division that makes it one. Entropy density — how much entropy a cubic metre of the substance holds — has units of joules per cubic metre per kelvin, and dividing a viscosity by it leaves kelvin-seconds, which is exactly the units of $\hbar/k_B$. So $\eta/s$ is a quantity that can be compared with a combination of Planck’s constant and Boltzmann’s and nothing else.

The conjecture, made in 2005 by Kovtun, Son and Starinets, is that

$$\frac{\eta}{s} \ge \frac{\hbar}{4\pi k_B}$$

for any substance whatever. The number on the right is $6.08 \times 10^{-13}$ kelvin-seconds, and the figure above is drawn in units of it.

What the ratio means before any bound is attached to it #

It is worth understanding the quantity on its own, because its interpretation is what makes a bound plausible.

Viscosity is momentum transport: momentum going sideways, carried by whatever carries it. In a gas the carriers are molecules and the transport goes as the mean free path; in a liquid it goes as the rate at which molecules change places. In both cases the transport is done by something that moves and carries momentum, and the rate depends on how far it gets.

Entropy density counts how many things there are. Entropy is a count of the arrangements available, and per cubic metre it is roughly the number of particles times a number of order ten.

So $\eta/s$ is momentum transport per carrier — how effective one degree of freedom is at moving momentum about. A fluid with a small ratio is one whose constituents are individually bad at carrying momentum, which is to say one in which they do not get far before being interrupted.

That reading is what makes a floor sound possible rather than arbitrary. A carrier cannot be interrupted more often than every time it moves — which is a statement about a length, and the only length a carrier privately owns is the one uncertainty gives it.

Heating a liquid takes it toward the floor and a gas away from it #

Before asking about the floor, it is worth asking which way ordinary substances move.

That picture has a consequence for where a minimum can be. A substance taken along a path that starts as a liquid and ends as a gas has its ratio falling at one end and rising at the other, so it must pass through a minimum in between — and “in between” is where the distinction between liquid and gas stops being sharp, which is the region around the critical point.

That is exactly where every measured minimum is. Helium at three megapascals, nitrogen at ten, water at a hundred: each has its smallest $\eta/s$ near its own critical point, and the values there are eight or nine, twelve and twenty-five times the floor. Water at ordinary conditions is four hundred times the floor and at its own minimum is twenty-five — and pressure alone moves it the other way, since squeezing a liquid thickens it exponentially while doing much less to its entropy — so most of the distance an ordinary substance can travel toward the bound is travelled by taking it to its critical point.

That comparison is the reason the subject is taken seriously. A gas of lithium atoms at a hundred nanokelvin and a plasma of quarks at two trillion kelvin have nothing in common except that in both the interaction is as strong as it can be — in the cold gas because the scattering length has been tuned to infinity with a magnetic field, in the plasma because the coupling is strong at those energies. Both arrive within a small factor of the same number. Nothing about the constituents survives; what survives is the strength of the coupling.

Where a bound of that size comes from without any string theory #

The conjecture was derived by a route that does not resemble fluid mechanics: for a class of strongly coupled quantum field theories, the correspondence between such a theory and a gravitational theory in one more dimension turns the calculation of a viscosity into the calculation of a black hole’s absorption cross-section, and the ratio comes out as $\hbar/4\pi k_B$ exactly, for every theory in the class.

That derivation is not available for water. But an estimate of the same size is, and it is worth doing because it says what the floor is about.

The two lines meet, and that is the whole argument.

For a dilute gas, $\eta \approx \tfrac13 n m \bar{v}\ell$ and $s \approx n k_B \hat{s}$, where $\hat{s}$ is the entropy per particle in units of $k_B$ and is around twenty for an ordinary gas. The number densities cancel and

$$\frac{\eta}{s} \approx \frac{m\bar{v}\ell}{3\hat{s}k_B} = \frac{\hbar}{3\hat{s}k_B}\cdot\frac{\ell}{\lambda},$$

where $\lambda = \hbar/m\bar{v}$ is the particle’s de Broglie wavelength. So the ratio is the mean free path measured in de Broglie wavelengths, divided by a number of order fifty.

The gas is made thinner in this sense by compressing it — the viscosity itself does not change, since the carriers become more numerous exactly as fast as their paths shorten, while the entropy per cubic metre rises. So the ratio falls, in proportion to $\ell$, and it reaches the floor when $\ell/\lambda$ is a few tens.

And at $\ell/\lambda = 1$ the calculation has destroyed itself. A mean free path is the distance a particle travels between collisions, and a particle is not smaller than its own wavelength, so a path shorter than a wavelength is not a distance anything travels. There are no quasiparticles left to carry momentum; there is only a medium.

The kinetic calculation reaches the bound at approximately the density at which it stops applying. That is not a proof of anything, and it is the physical statement the bound is making: a fluid cannot be thinner than one in which the carriers of momentum are interrupted as often as they can possibly be, and that is once per wavelength.

How a viscosity is measured in something that lives for 10⁻²³ seconds #

The heavy-ion number is the one most likely to be taken on trust, and the measurement behind it is worth stating because it is an argument from a shape rather than from an instrument.

Two gold nuclei collide, not head-on but offset, so the region where they overlap is not round. It is a lens — longer across the collision than along it — and it is very hot. That region then expands into the vacuum.

If it expanded as a gas of independent particles, the initial shape would leave no trace: each particle would fly off in whatever direction it happened to have, and the debris would be distributed evenly in angle whatever the overlap looked like.

If it expands as a fluid, the shape matters enormously. A pressure gradient is steepest across the narrow direction of the lens, so the fluid accelerates hardest that way, and the debris comes out preferentially in the plane of the offset. The asymmetry is quoted as a coefficient called $v_2$ — the amplitude of the second harmonic in the angular distribution — and it is measured to be large, around six per cent.

Viscosity enters because it erases the asymmetry. Shear stress opposes the differential expansion, so a viscous fluid converts less of its initial shape into a final momentum asymmetry than an ideal one does. Running the hydrodynamic equations forward with $\eta/s$ as a parameter and comparing the predicted $v_2$ with the measured one is how the number is extracted.

Two features of that procedure decide how much it can be trusted. The sensitivity is real: doubling $\eta/s$ from the best-fit value visibly reduces the predicted asymmetry, so the measurement is not merely consistent with a small value but excludes a large one. And the extraction is model-dependent in a way an instrument reading is not — it requires an initial shape, which is itself computed, and a prescription for when hydrodynamics starts and when it stops. The published uncertainty, a factor of about two, is dominated by those choices rather than by the counting statistics, which are excellent.

The same logic, in a completely different setting, produces the cold-atom number. A trapped gas of lithium atoms is released from a trap that is longer in one direction than another; it expands; and a gas of weakly interacting atoms would expand isotropically while a fluid overshoots in the narrow direction and ends up longer the other way. Watching the aspect ratio invert, and how much of the inversion survives, measures the same ratio by the same argument — at a hundred nanokelvin instead of two trillion degrees.

That two experiments this far apart use the same observable is not a coincidence of technique. The asymmetry of an expansion is the most direct thing a fluid does that a gas does not, and it is available whenever the sample is small and free.

Why Planck’s constant turns up in a classical quantity #

Nothing about water at room temperature is quantum mechanical in any way that matters, and the bound it is being compared against has $\hbar$ in it. That deserves an account rather than an eyebrow.

The reason is the one the kinetic estimate makes explicit. The ratio is a mean free path divided by a de Broglie wavelength, and a de Broglie wavelength is where $\hbar$ enters: it is the constant that converts a momentum into a length. The ratio itself is a pure number — how many of its own wavelengths a carrier travels between interruptions — and it is only in restoring the units that $\hbar$ reappears.

So $\hbar$ is not in the physics of water’s viscosity. It is in the comparison, because the comparison is between two lengths and one of them is quantum mechanical. Water’s ratio is four hundred times the floor precisely because a water molecule at room temperature travels a few hundred of its own de Broglie wavelengths between rearrangements, and that number is large for the same reason an ordinary object’s wavelength is too small to have consequences.

The same structure is why the bound cannot be stated for viscosity alone. Pitch and the quark–gluon plasma differ in viscosity by a hundred million, and they differ in this ratio by fourteen decades in the same direction — the plasma’s enormous viscosity is carried by an enormously larger number of degrees of freedom, and dividing by the entropy is what removes the count and leaves the per-carrier statement.

What a small ratio buys, and where it is being looked for now #

A ratio near the floor is not a curiosity about materials; it is a statement about when a fluid description works at all.

Hydrodynamics is the description of a system by a few local quantities — density, velocity, temperature — and it is valid when those quantities vary slowly compared with the mean free path. A small $\eta/s$ means a short path per carrier, which means hydrodynamics applies down to shorter distances and shorter times than one would otherwise expect.

That is the resolution of a genuine puzzle in the heavy-ion experiments. The matter produced when two nuclei collide is about ten femtometres across and lives for a few times $10^{-23}$ seconds, and it is described successfully by hydrodynamics — a continuum theory — over that size and that duration. It can be, because the interactions are strong enough that a local equilibrium is established within a fraction of the lifetime. A weakly coupled plasma of the same constituents would have a mean free path larger than the whole droplet and no fluid description at all.

The place the same question is now being asked is not hot at all. Electrons in a very clean conductor normally scatter off impurities and lattice vibrations far more often than off each other, which is why a current obeys Ohm’s law rather than the fourth-power law a viscous fluid in a pipe obeys. In graphene near its neutrality point, with the impurities reduced far enough and the temperature in a particular window, the electron–electron scattering is the fastest process — and then the electron gas is a fluid in the hydrodynamic sense, with a viscosity of its own.

What it does is what a fluid does. The current profile across a channel is parabolic rather than flat; the resistance falls as the channel is narrowed over part of its range, because a viscous flow’s resistance depends on the channel differently from a diffusive one; and vortices appear, with the current running backwards beside an injection point. All three have been measured since 2016. The estimated $\eta/s$ for that electron fluid is within an order of magnitude of the floor, which puts a two-dimensional sheet of carbon in the same conversation as a gold-on-gold collision.

A conjecture with counterexamples, and entropies read off tables #

It is a conjecture and it has counterexamples. Within theories that have a gravity dual, adding higher-derivative corrections to the gravitational side lowers the ratio below $\hbar/4\pi k_B$, and in a theory with a large number $N$ of species of field the ratio can be made to fall as $1/N$ — because the entropy counts all the species and the momentum transport does not. Both are constructions rather than substances. No experimentalist has a bottle of either, and no measured system is below the bound.

Every value on the scale for an ordinary substance is a computation from two tabulated numbers, and the entropy is the shakier of the two. An absolute entropy density requires an absolute entropy, which is got by integrating a heat capacity from zero and depends on the reference state; different tabulations differ by a few per cent, and for pitch — which is not a single substance — the figure is an estimate. The ratios are good to a factor of two at worst and the argument is about orders of magnitude.

The cold-atom and heavy-ion values are extractions rather than measurements. Neither system can have a viscometer put in it. In the heavy-ion case the number comes from fitting the angular distribution of the debris with hydrodynamic simulations, and it is the simulation’s transport coefficient that is quoted; in the cold-atom case it comes from watching how fast a deformed cloud relaxes. Both are model-dependent, both have been revised, and the uncertainty on the heavy-ion value is at least a factor of two.

And $\eta/s$ is not the only dimensionless measure of thinness. The kinematic viscosity has its own proposed lower bound, of a different form — set by $\hbar$ divided by the geometric mean of the electron mass and the molecular mass — and it makes a different statement about a different question — a kinematic viscosity is the diffusivity of momentum, which is what sets how far an alternating shear reaches rather than how much stress a flow carries. Which of the two is the right thing to bound depends on what the viscosity is being asked to do, and the two are not equivalent.

A viscosity of ten to the eleventh, drawn as the smallest bar #

They cannot show that the quark–gluon plasma’s viscosity is enormous. Its $\eta$ is of order $10^{11}$ pascal-seconds, which is a hundred million times that of pitch, and the figures here show only the ratio. Every popular description of that material as “the most perfect liquid” is a statement about the ratio and reads as a statement about the viscosity, and the two are as far apart as it is possible for two statements about one substance to be.

Nor can they show a path through the phase diagram. The minima quoted are minima along particular isobars, and the surface $\eta/s(T,p)$ has structure the scale flattens into one number each. Where a substance’s true global minimum is, and whether it is at the critical point or somewhere on the supercritical crossover beyond it, is a question about a surface that a single bar cannot ask.

And they cannot show the uncertainty. Three of the values on the scale are extractions with error bars comparable to their own size, and they are drawn as bars of definite length beside values computed to three figures. The figure treats them alike because the axis is fourteen decades long and on that axis a factor of two is invisible — which is the right decision for the comparison being made and the wrong one for any use of the individual numbers.

Still open: whether anything can be pushed below #

The two systems closest to the floor were reached by making the interaction as strong as it can be, and in both cases “as strong as it can be” has a precise meaning: the scattering length is infinite in the cold gas, and the coupling is at its strongest available value in the plasma. Neither can be pushed further by the means that got it there.

What would settle the question is either a measured violation or a proof. A violation would most plausibly come from a system with many species of low-energy excitation, since that is where the theoretical counterexamples live, and some proposals point at materials with many nearly degenerate bands. A proof for ordinary matter does not appear to be close: the arguments that exist are either restricted to theories with gravity duals or are kinetic estimates of the kind above, which give the right order and cannot give a bound.

Meanwhile the ratio has become a standard way of describing strongly interacting matter, quoted for everything from neutron-star interiors to electrons in graphene, and the interesting result is no longer whether the floor is exact. It is that two systems twelve orders of magnitude apart in temperature, made of different things, land within a factor of five of the same number — which says that what decides how a fluid transports momentum is the strength of its coupling and not what it is made of.

The habit worth carrying away is about making a quantity dimensionless before asking whether it is small. A quantity with no natural unit has no small values; a quantity measured in $\hbar/k_B$ does. The question “how thin can a fluid be” has no answer and “how thin can a fluid be per unit of entropy” has a candidate one, and the whole of the difference is a division that changes the units.