A Model of Flow with Zero Viscosity

Superfluidity

At 4 kelvin, helium condenses into a liquid with properties similar to water or any other liquid. At 2 kelvin, helium becomes a superfluid with properties unlike any other fluid we can create.

One of the strangest properties of superfluid helium is that it has zero viscosity. A flowing liquid experiences viscosity that causes it to slow down; for instance, stirred coffee eventually stops spinning. Superfluid helium has zero viscosity, and it spontaneously creates vortices that spin without resistance.

In Knot Physics, superfluidity results from the properties of the branched spacetime manifold.

Branched Spacetime

We will describe a few aspects of quantum mechanics in Knot Physics in order to understand superfluidity. (For a full description of quantum mechanics in Knot Physics, see the Knot Physics website. For a quick overview of quantum mechanics, see the 3-minute video here.)

In Knot Physics, spacetime is a branched manifold. The branched spacetime manifold allows a quantum superposition of multiple states, where each branch is a quantum state.

The branches of spacetime split and recombine randomly. 

Branches with zero and one recombination
On the left, we show the branched spacetime manifold which consists of a large—but finite—number of branches. On the right, we see the branched spacetime manifold with one example of branch recombination.
Spacetime branches
For simplicity, we will represent the branches of spacetime with 1-dimensional lines. On the left, we show a simplified version of the branches of spacetime. On the right, we show one example of branch recombination.

Elementary fermions—like electrons and quarks—are knots in the spacetime manifold. A fermion consists of one knot on each branch of spacetime. All the knots collectively are in a superposition, which is the quantum wave function for the fermion.

In order for two branches to recombine, the knots on each of the recombining branches must match.

Spacetime branches with one fermion
We show an elementary fermion, which has one knot on every branch of spacetime. On the left, we see a pair of matching knots. On the right, two branches recombine such that the pair of matching knots recombines into a single knot.

Atoms are made of fermions, so an atom has one “instance” on each branch of spacetime. We refer to each instance of an atom as an atom-instance. In order for two branches to recombine, the atom-instances on each of the recombining branches must match.

Spacetime branches with one atom
This diagram represents a helium atom, with one helium atom-instance on each branch. On the left, we see a pair of matching atom-instances. On the right, two branches recombine such that the pair of matching atom-instances recombine into a single atom-instance.

If there are N atoms, then each branch has N atom-instances.

Spacetime branches with three atoms
We show three helium atoms. Every helium atom has one atom-instance on each branch. On the right, two branches recombine such that a pair of matching atom-instances recombine into a single atom-instance.

In Knot Physics, the branches of spacetime must frequently recombine. This is because branch recombination has entropy, and the entropy of branch recombination is greatest when the branches recombine frequently.1 For branches to recombine, the atom-instances must match; specifically, the positions and momenta of the atom-instances on the recombining branches must match.

Superfluidity results from a particular type of flow of atom-instances on the branches of spacetime. In constructing this flow type, we will use the fact that positions and momenta of the atom-instances on the recombining branches must frequently match.

Superfluidity results from a particular type of flow of atom-instances on the branches of spacetime.

Superfluid Flow

Atoms consist of atom-instances on the branches of spacetime. If the atoms are a fluid, then the atom-instances can flow, and the way in which those atom-instances flow is their flow type. We construct a superfluid flow type using three basic flow types. Each basic flow type describes the flow of atom-instances on a single branch.

In the first basic flow type, the atom-instances are at rest with respect to their container.

Atom-instances at rest

In the second basic flow type, the atom-instances move with constant velocity relative to their container.

In the third basic flow type, the atom-instances move with a variable velocity. This variable velocity could be described as constant motion plus a longitudinal wave.

We recall that the branches of spacetime must frequently recombine. For branches to recombine, the positions and momenta of the atom-instances on the recombining branches must match.

Clearly the first two basic flow types cannot recombine with each other. The momenta of the atom-instances on those basic flow types never match.

Some atom-instances at rest and some moving with constant velocity

The first basic flow type can, however, recombine with the third. This recombination includes translation such that the third flow type translates relative to the first flow type before the branches recombine.

Single atom-instance recombination with variable velocity
We show the first flow type (top) recombining with the third flow type (bottom). The vertical motion is not literal; it only indicates splitting and recombination of the branches.
Some atom-instances at rest and some moving with variable velocity
We again show branch recombination between the first and third flow types, but in this animation, we show recombination over entire branches. For simplicity, we will use recombination of entire branches in the following animations, but we remember that recombination can also occur in just a small region, as in the previous animation.

The third basic flow type can also recombine with the second.

Some atom-instances moving with variable velocity and some moving with constant velocity
We show the third basic flow type (top) recombining with the second basic flow type (bottom).

The combination of these three basic flow types is viable because it allows frequent branch recombination. We call this combination the superfluid flow type.

Superfluid flow of atom-instances
We show the superfluid flow type. The third basic flow type (center) recombines with both the first basic flow type (top) and the second basic flow type (bottom).

The superfluid flow type has atom-instances that are at rest, atom-instances that are in constant motion, and atom-instances that alternate between those velocities. The superfluid flow type is a quantum superposition that includes a fluid state that is at rest and a fluid state that is in motion.

The superfluid flow type is a quantum superposition that includes a fluid state that is at rest and a fluid state that is in motion.

We now show why the superfluid flow type can only occur at very low temperatures.

Temperature

The superfluid flow type depends on translation of the third flow type relative to the first. By considering the impact of temperature on branch recombination, we can see how translation depends on temperature.

When branches recombine, the positions and momenta of the atom-instances must match. The temperature of atom-instances affects their momenta, but branches can recombine even if the atom-instances are hot. After a branch separates, the momentum of an atom-instance matches the momentum of the atom-instance from which it just separated, allowing branch recombination.

Cold atom-instances can also, of course, recombine in this way.

Recombination of cold atom-instances

Cold atom-instances have another mode of recombination that includes translation.

Some atom-instances at rest and some moving with variable velocity

Translation is not possible for hot atom-instances because the momenta of the atom-instances do not match after the translation.

Hot atom-instances cannot translate
Momenta of hot atom-instances do not match
For hot atoms, the non-matching momenta of atom-instances prevents branch recombination after translation.

We recall that the superfluid flow type requires translation. Because heat prevents branch recombination of translated branches, it is only possible to construct a superfluid flow type at very low temperatures. 

Because heat prevents branch recombination of translated branches, it is only possible to construct a superfluid flow type at very low temperatures. 

We now show why the superfluid flow type must have zero viscosity.

Viscosity

Viscosity occurs when the kinetic energy of fluid motion dissipates into the kinetic energy of heat. We can show that the superfluid flow type prevents this kind of energy dissipation.

First, we recall that branches can only recombine if the atom-instances have matching momenta. This means that hot atom-instances cannot recombine with cold atom-instances.

Hot and cold atom-instances cannot recombine
Here we show an impossible pair of branches. They have atom-instances with different temperatures. For that reason, the branches cannot recombine. Branches that cannot frequently recombine are prohibited by entropy of branch recombination.

We now consider the three basic flow types that comprise the superfluid flow type: The first flow type has atom-instances at rest, the second has atom-instances with constant velocity, and the third has atom-instances with variable velocity. The first flow type cannot dissipate kinetic energy into heat because the atom-instances are not moving and do not have kinetic energy. Suppose atom-instances in the second or third flow type could dissipate kinetic energy and become hot. The hot atom-instances would no longer be able to recombine with the cold atom-instances of the first flow type. As such, the three branches could no longer frequently recombine with each other.

This never occurs. Entropy of branch recombination requires that branches frequently recombine. As such, the atom-instances of all three basic flow types stay cold in order to frequently recombine with each other. None of the atom-instances can dissipate their kinetic energy of motion into heat, and this means that the superfluid flow type has zero viscosity.

None of the atom-instances can dissipate their kinetic energy of motion into heat, and this means that the superfluid flow type has zero viscosity.


In Knot Physics, superfluidity results from the interaction of different flow types on the branches of the spacetime manifold. By understanding those interactions, we see that the superfluid flow type allows flow but does not allow viscosity. For this reason, a superfluid flows without resistance.

Superfluid flow of atom-instances

Notes

1. To learn more about the effects of branch recombination entropy, see the post “Schrödinger’s Cat”.


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Cliff Ellgen

Cliff Ellgen graduated from Caltech in 1999 with a degree in mathematics. He has been contemplating Knot Physics, a geometric unification theory, for more than fifteen years. Since August of 2014, he has been collaborating with Garrett Biehle, who received his Ph.D. from Caltech in astrophysics under advisors Kip Thorne and Roger Blandford.

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