The osmotic pressure of excitations within a fluid
The helium-4 fluid forms a "background space" for excitations. Let us have the following setup.
x = excitation
|________|
|________| | x x |
| | | x |
|________========________|
capillary
helium-4 at 0 K helium-4 at 2 K
The capillary is so narrow that the excitations cannot go through it. Superfluid helium-4 flows easily through the capillary.
Entropy wants to grow: the excitations cause a negative "osmotic pressure" in the fluid on the right side and pull fluid from the left vessel to the right. The pressure is quite large and produces the fountain effect of superfluid helium-4.
Erik Verlinde's paper about entropic gravity (2010) contains a nice illustration of what exactly might be the force which pulls fluid from the left vessel to the right vessel. The polymer filament in the picture is hit by quanta in the heat bath. The quanta tend to pull the filament in. In our diagram, the filament represents the superfluid in the capillary.
The second sound
x = warm location
-------------------------------
| x x x ---> |
-------------------------------
() tube of superfluid
| |
| |
periodic
heating
Let us have a tube full of superfluid helium. We heat one end of the tube periodically at very short intervals.
What happens after the first pulse of heating? The osmotic pressure pulls cool fluid to the heated end and pushes warm fluid to the right. The inertia of the fluid keeps it flowing for some time, even after there no longer is pressure.
The inertia pushes the x in the diagram to the right and imports cool fluid to the left end. Here we need to assume: cool fluid can flow to the left end without getting completely mixed with the outflowing warm fluid.
A wave phenomenon requires a force and inertia. The force is the osmotic pressure and the inertia comes from the mass of the fluid.
The wave is called the second sound.
Let us look once again at the two vessels of the previous section. Let us assume that the temperatures are initially equal at 0 K. Then we suddenly heat the right vessel. The osmotic pressure pulls fluid from the left vessel. Thanks to the inertia of the pulled fluid, it keeps flowing for some time even if the temperature on the right drops below the left vessel. We have a temperature oscillator here.
excitation x hits the wall
--------- ---------------
| | | x ----> |
| -------- |
---------------------------------
body of superfluid
We may regard the body of superfluid helium-4 as a "space" whose volume stays constant. Excitations are particles which bounce from the botders of the body and try to stretch the body at the places where they bounce. They cause pressure which tries to deform the body.
In this section we did not really use the assumption that helium-4 is a superfluid. Any fluid with a very low viscosity can exhibit these phenomena.
A body of superfluid hits a solid wall: the momentum is not absorbed by excitations
Our blog post on March 26, 2022 studied if the dispersion relation of Landau's "phonons" and "rotons" can justify superfluidity. We concluded that it only can explain superfluidity if we assume that the body of the fluid is not able to absorb any of the momentum p' in a collision and the excitations get all the momentum. Why would that be so?
body of superfluid solid wall
----------------------------------
o o o o o -----> |
----------------------------------
capillary
If a body of a superfluid in a capillary hits a solid wall, the fluid will certainly stop flowing. The body absorbs most of the momentum transferred in the collision. That momentum does not go to excitations.
Thus, the claim that the dispersion relation would be able to explain superfluidity is clearly wrong.
In his 1955 paper about liquid helium Richard Feynman writes that low-energy phonon or roton states in the superfluid are "scarce", and that explains frictionless flow. His claim is strange, since the quanta of sound waves, phonons, are bosons, and many bosons can exist in the same state. In our example of a body of a superfluid hitting a wall, there will be macroscopic sound waves, each one containing a huge number of quanta.
In their 1947 paper, Max Born and Herbert S. Green claim that the phonon-roton theory of Lev Landau does not explain superfluidity. The authors do not believe that there is a Bose-Einstein condensate in a superfluid, either.
Order can remove viscosity
Let us study how we in classical mechanics can avoid some loss of momentum and energy when spheres move past obstacles.
o o ----> ●
o o o ----> ●
o ---->
●
spheres obstacles
If we have a random configuration of spheres flying to the right, the spheres will certainly lose some of their momentum to the right direction when they bump into obstacles. If collisions are elastic then the spheres do not lose kinetic energy, though.
chain of spheres
o - o - o - o ●
\ ●
o - o - o ---->
●
obstacles
The obvious way to avoid collisions is to bind the spheres into a chain which dodges the obstacles. Randomness makes "friction" inevitable and order can remove the friction.
Summing a large number of waves
chain of spheres
o o o o o o o o o o ---->
● ● ● ● ● ●
evenly spaced obstacles
Let us then have a chain of spheres moving past an ordered structure of obstacles. When a sphere moves past an obstacle, there probably is some acceleration and deceleration of the sphere. The sphere will emit a sound wave, and may emit an electromagnetic wave, too.
1. Evenly spaced obstacles. We assume that the spacing of the spheres is different from the obstacles. The chain of spheres slides over the obstacles. Each obstacle creates a sound wave or an electromagnetic wave. The phases of the individual waves are
n α,
where n is an integer and 0 < α < 2 π.
Let us assume that the phase difference α between consecutive obstacles is larger than
s / v * 2 π f,
where s is the spacing of the obstacles and v is the speed of sound or an electromagnetic wave, and f is the frequency of the wave. Then the phase difference differs from 0 when observed from any direction. The assumption may be broken if the speed of the chain of spheres is close to the speed of sound. This would explain the critical velocity in Landau's superfluid model.
Another reason for Landau's critical velocity might be that if the hit to an obstacle happens faster than the speed of sound, it creates a shock wave instead of a normal sound wave. Our analysis of destructive interference may not work for shock waves.
Let us look at the sum wave from very far away, at a point x. The amplitudes of the individual waves are approximately the same since we are very far away. The phases of the waves at the point x are approximately
n β,
where n is an integer and 0 < β < 2 π is a constant
Let N be the number of the individual waves that we sum.
According to the link, the amplitude of the sum stays less than some constant C as N goes to infinity. We conclude that destructive interference wipes out almost all of the created wave.
Bloch's theorem is a similar result to ours. It states that in a periodic potential, the solution to the Shrödinger equation is a plane wave modulated by a periodic function.
chain of spheres
o o o o o o o o o o ---->
● ● ● ● ● ● ● ● ●
"amorphous" set of obstacles
2. Obstacles whose density is constant but they are not evenly spaced: the amorphous case. If the obstacles are atoms of a glass, then they are not evenly spaced, but their density varies very little when we look at a large set of obstacles.
| zone 1 | zone 2 | zone 1 | zone 2 |
● ● ● ●
obstacles
Let us simplify the problem. We divide space into two zones of an equal volume: if a wave is born in the first zone, then its phase is π / 2. If a wave is born in the second zone, its phase is -π / 2. That is, we sum amplitudes which can be either 1 or -1.
If the obstacles would be placed in space totally randomly, then the sum of N waves would have an amplitude whose expected value is
~ sqrt(N).
This is because the variance of the sum is the sum of the variances of individual random variables. There would be no destructive interference.
Presumably, in most real world cases obstacles are packed in a way where the number of them in the zone 1 is almost exactly equal to the number in the zone 2. There is almost complete destructive interference of waves.
We conclude that if a chain of spheres slides past an ordered lattice of obstacles, or an "amorphous" set of obstacles, the created sound wave or electromagnetic wave may be very small. Superfluidity is possible.
o o o o o o o o o --->
<--- o o o o o o o o o
What about two chains of spheres sliding past each other? In this case there could be strong sound waves in the vertical direction as the spheres jump up and down. Transverse electromagnetic waves would be very weak. It is not clear if chains can slide past each other without friction.
o o o o o o o o o o --->
o oo o o o oo
<--- o o o o o o o o o o
On the other hand, sliding past an ordinary "fluid", or an amorphous structure, might happen without friction. The argument is the same as in the case of the chain of spheres and an amorphous set of obstacles. Maybe ordinary fluid helium-4 is required as a "lubricant" to let lattices of superfluid helium-4 to slide past each other? This brings into mind the plate model of superfluidity which we presented on March 24, 2022.
The time crystal
Let us assume that the superfluid is an ordered lattice at the atomary scale. If it can move with no friction or viscosity, then it is, by definition, a time crystal when it moves.
An open problem: why thermal vibrations do not spoil the frictionless flow of the chain of spheres?
random hit
o \ radiation or sound
| \
v \
o o o o o o o o o o ----->
● ● ● ● ● ●
Suppose that a sphere in a "normal fluid" component hits the sliding chain. The sphere is moving because of thermal energy. It pushes the chain down and causes some sphere to hit an obstacle strongly. We get a burst of radiation or sound waves. At the same time the sphere in the chain loses some of its momentum to the right as it bumps into the obstacle. The loss of momentum means friction or viscosity.
Since the process is random, destructive interference does not wipe out the outgoing radiation or sound.
How can we explain the frictionless flow? Is the chain so strong that the hit does not make much of a dent into the chain? Or is there a thick lattice of superfluid atoms, say, 100 atoms thick, so that the hit is spread evenly over a large area and does not make much of a difference?
Conclusions
Randomness seems to create dissipation of energy or momentum if we have spheres, or atoms, moving.
Dissipation can be reduced by ordering the spheres into a chain or a lattice. Destructive interference can wipe out dissipation waves almost entirely.
Landau's critical velocity for superfluidity may follow from our model.
Are atoms in superfluid helium-4 ordered into chains or lattices? Let us look at the research. Does X-ray diffraction tell us anything?
No. Richard Feynman in his 1955 paper states that in X-ray and neutron diffraction studies, superfluid helium-4 looks like an ordinary fluid.
We conclude that the mechanism of superfluidity is an open problem of physics. Our own model might explain it if we find a reason why the impacts from thermal movement do not spoil our model.
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