Electron movement in a wire: some crude numbers

Let us have a 1 cm thick and 1 meter long wire made of a metal. Its weight is roughly 1 kg, and it contains some 10²⁵ atoms.

There are

       ~ 10²⁵ conducting electrons

in the wire. The mass of these electrons is

       ~ 10⁻⁵ kilograms.

The charge of these electrons is

       ~ 10⁶ coulombs.

We want to create a 1 ampere current in the wire.

https://en.wikipedia.org/wiki/Inductance

The inductance of the wire is

       L ~ 1 microhenry.

The voltage to increase the current I in the wire is

       V = L dI / dt.

We put a voltage V of 1 microvolt over the wire for 1 second. After that time, the current is 1 ampere.

The Coulomb force of 1 microvolt / meter on the charge 10⁶ coulombs is

        F ~ 1 newton.

The energy of the magnetic field after the operation is

       1/2 L I² ~ 1 microjoule.

The "impulse" which the voltage exerted on the conducting electrons was

       p = 1 newton second.

The collective velocity of the electrons is only

       v = 1 micrometer / second.

It is as if the "inertial mass" of the electrons were a whopping

       10⁶ kilograms.

The kinetic energy and the momentum calculated from the rest mass of the electrons is very small. The momentum is only

       10⁻¹¹ newton seconds,

and the kinetic energy is

       10⁻¹⁷ joules.

Radiation pressure at 4 K is negligible on an object moving 1 micrometer per second

Suppose that we have an object carrying those one million coulombs of charge and moving at v = 1 micrometer per second. That corresponds to a current of one ampere.

We want to calculate the frictional force that the reflection of black body radiation at 4 kelvins imposes on the object.

Let us assume that the area A of the object is one square meter. The power of black body radiation is

       P = A σ T⁴,

where σ = 5.67 * 10⁻⁸ W/(m² K⁴) is the Stefan-Boltzmann constant and T is the temperature. We have

       P = 1.4 * 10⁻⁵ W

at 4 K. The radiation pressure force by the power P is

       P / c = 5 * 10⁻¹⁴ N.

The pressure is almost the same on the each side of the object, except for the Doppler shift caused by the tiny velocity v = 1 μm/s.

The effect of the Doppler shift is

       4 v / c,

because the reflection adds a factor of two, and the effect is on both sides of the object.

We conclude that the frictional force on the object by black body radiation is

       F ~ 4 v / c * P / c

           = 7 * 10⁻²⁸ N.

We calculated above that, to create the magnetic field for a 1 ampere current in a one meter wire, we have to expend 1 newton second of impulse. The force F is negligible relative to that.

The friction from black body radiation is negligible at 4 K. The resistivity at a low temperature has to come from collisions with the lattice. Those collisions move a large amount of impulse from the lattice to the electrons.

Problems with time crystals and superconductivity: a perpetuum mobile is created?

In an ordinary crystal, the lowest energy state is static, except of zero-point vibrations of the atoms. In a time crystal, the lowest energy state involves movement of particles. A moving lowest energy state is hard to grasp from the classical point of view. In classical mechanics we are used to configurations where one can reduce the energy of the system by slowing down the motion.

The simplest time crystal: a particle in a box in the lowest energy state

Let us consider the simplest textbook example of quantum mechanics at work: a particle in a box.

                                                   |

                                                   |   spoon

                                                   0

                               particle

               |               <-- • -->               |

                           box, length L

Classically, the "crystal" would be a state where the particle has been stopped at a certain location. In quantum mechanics, the particle bounces back and forth, so that its de Broglie wavelength is double the length of the box.

Suppose that we insert a spoon in the box and try to slow down the particle. Why this cannot succeed?

                               ________

                             /                 \ 

                      jumping rope rotates

The wave function of the particle is like the jumping rope of children rotating around its ends fixed at the walls of the box.

Any disturbance to the wave moves the system to a higher energy state. When we try to insert the spoon into the box, we feel "pressure" resisting the insertion procedure. We must do work against the pressure.

If the particle is at the lowest energy level, then the time crystal of its movement is the slowest possible movement. The time crystal is stable simply because the particle cannot move any slower.

A time crystal involves "frame-dragging": any change in the motion requires additional energy

From general relativity we know frame-dragging around a rotating neutron star or a black hole. The lowest energy state of an approaching test mass is one where the test mass moves along with the rotating body. If we want to keep the approaching mass static relative to the global frame, we have to supply "kinetic energy" to it.

The lowest energy state of a time crystal drags the frame of electrons.

Let us compare this to an ordinary crystal. There the lowest energy state is static (except for zero-point oscillation). A time crystal probably behaves just like an ordinary crystal, if we take into account that the frame is dragged here and there.

If we push a single atom in an ordinary crystal, we have to do work. Furthermore, the momentum which we gave to the atom quickly disperses to the entire crystal.

An ordinary crystal may act as an easy-to-grasp analogue of a time crystal.

The time crystal of conducting electrons plus lattice vibrations in a superconductor

At a high temperature, conducting electrons behave much like a gas. A gas is not a time crystal since it is not ordered.

             lattice of ions

                    +        +

           o     o o     o o     o    <----

           o     o o     o o     o    <---- vibration

           o     o o     o o     o    <----

           o     o o     o o     o    <----

                    ^        ^

                    |        |

                    e-       e-

         conducting electrons

We conjecture that in a superconductor many, or all, conducting electrons form a time crystal together with lattice vibrations. Electrons like to move in zones where a vibration has concentrated positive charge.

The electrons move some 1,000 km/s. It cannot be an ordinary crystal where electrons would be confined each to a small space.

The orbitals of a molecule are an analogue for the time crystal in the lattice. Electrons must keep on moving. Some electrons may have orbits which ship them around for the whole length of the molecule.

We conjecture that a superconductor is like a giant molecule where denser zones of the lattice ions are like nuclei in a molecule, except that these denser zones move at the speed of 5 km/s. Electrons move 200 times faster and will order and synchronize their own movement according to these zones.

How does the superconductor time crystal deal with impurities, or borders of crystals of the metal?

An ordinary crystal grows in a way where it tries to avoid impurity atoms or molecules. A time crystal probably behaves in the same way. It avoids impurities in the lattice.

How about borders of metal crystals? In a non-superconducting metal, borders of crystals cause scattering of conducting electrons. For the time crystal there are two options at a border:

1. The structure of the time crystal is so strong that it prevents scattering. Any vibration caused in the structure by crossing the border is a part of the time crystal itself. There cannot be any dissipation because it is the lowest energy state. The vibration cannot escape from the time crystal to the outside world: it remains as a part of the time crystal itself.

2. Some of the electrons are scattered at the border, but they remain as a part of the time crystal. There cannot be any dissipation of phonons or electromagnetic waves because it is the lowest energy state.

How does a thermal phonon behave in the time crystal of a superconductor?

Let us have a time crystal, attached to the center of the laboratory floor, in a thermodynamic equilibrium with its environment. The environment is a cavity filled with black body radiation.

Photons in surrounding space are turned into phonons as they enter the time crystal.

Since it is an equilibrium, the movements in the time crystal cannot systematically change, e.g., the angular momentum relative to the center of the laboratory. The time crystal should behave just like an ordinary crystal in a thermodynamic equilibrium.

An ordinary crystal can refract or reflect photons. It can absorb energy from photons, as long as the total flux of energy in and out of the crystal is zero.

Let us try to model how a time crystal reacts to a phonon which enters the crystal at a random location.

       "diffraction grating" by the phonon

                    e-  electron flow

                    |

                    v

            +         +         +         +    zones of positive charge 

                                        ^

                                        |

                                        e-   electron flow

The phonon concentrates positive charge in the lattice into zones marked with symbols +. It is like a diffraction grating for the approaching electrons.

The phonon contains a small amount of energy. The energy, or a part of it, is absorbed as vibrations in the time crystal.

The vibrations may be longitudinal in the electron flows, or they may be transverse, making the flows move sideways.

If the time crystal would not be in the lowest energy state, then there would be free kinetic energy in the electron flows, and the vibrations would steal kinetic energy: there would be resistance.

But what exactly is the reason why we cannot disturb much the movement in the electron flows? A hydrogen atom may appear completely neutral to the outside world, even though the electron is moving rapidly around the proton. Could it be that the electron flow plus associated lattice vibrations are not coupled to the thermal phonon at all?

If the negative charge in the electron flows is completely canceled by the positive charge concentration in the lattice vibrations, then the system might appear neutral.

Theorem. Any system of charges in its lowest energy state must have a static electromagnetic field if observed from a distance. Otherwise, the system would radiate.

The theorem suggests that the diffraction grating meets a continuous flow of electrons. It must not see individual electrons, because if it would, there would be an electromagnetic wave radiating from the system.

   

                              + charge

                   e- --->

               o o o o o o o o o o o o 

                     long molecule          

               |                                  |

            ======================  frame

An analogous setup: we have a long molecule where an electron orbits from end to end. The molecule is attached to a frame. We put a positive charge close to the molecule. There probably will be some dipole force between the molecule and the electron. The force has to be constant. If it were periodic, then the molecule would radiate electromagnetic waves.

We conclude that the electron flows appear to the diffraction grating as essentially continuous flows of charge. The grating disturbs the flows only a little.

Electron flows in a time crystal are like "continuous" flows of charge

Hypothesis. Electron flows in a time crystal look like continuous flows of charge to a disturbance like a thermal phonon. There is no scattering of individual electrons.

                                                 \   scattered wave

                                            \

                                       \

               ------------------      ^

               ------------------      |

               +    +    +    +              periodic potential

               ------------------      ^

               ------------------      | e- free electron

The scattering of a free electron looks like the diagram above. The periodic potential perturbs the electron wave and creates a weak scattered wave.

If there can be no scattering, the diagram above cannot be the right depiction of the process. What kind of a wave might describe a continuous flow of charge?

The perpetuum mobile problem if we assume a potential wall between different states of the current in a superconductor

Suppose that we have a superconducting wire in the lowest energy state. It is a time crystal.

          e- ---->                                               e- ---->

       ---------------- ================== -------------

      ordinary       superconductor

      wire

Then we add an electron to the left end of the superconductor, or alternatively, remove an electron from the right end.

Let us assume that the added electron "joins" a flow of electrons in the time crystal.

There are problems, though: how does the electron know to join a flow which goes exactly from end to end? Also, the claim that an extra electron can "join" a flow without spoiling the properties of the time crystal, is very ad hoc. Why should it be true?

                            resistor

                      ------- ### -------

                    |                           |  

                     =============

                             e-  ----> 

                      superconductor

                              <-------

                                I = ε

Let us consider the following setup. There is a superconductor in a loop with an ordinary conductor whose resistance is extremely small. We put a small initial current 

       I = ε

into the loop.

Let us assume that the electron flow in the superconductor now is in some kind of a local lowest energy state with the current ε flowing. There is a potential wall which prevents the system from decaying to an I = 0 state.

Then we would have a perpetuum mobile which generates heat in the resistor for ever.

How to prevent the existence of a perpetuum mobile? Since there is now a small voltage over the superconductor, maybe that voltage creates the opposite current -ε within the superconductor, so that the total current is zero?

But that does not work. If there is a potential wall which prevents the decay ε -> 0, then there probably is a potential wall which prevents the transition 0 -> -ε.

Question. If we try to explain superconductivity by a (local) energy minimum argument, does that always lead to the existence of a perpetuum mobile?

Conclusions

Energy minimum arguments may explain the behavior of a time crystal in the lowest energy state. The time crystal in this case is a closed system.

But a superconductor in a loop with an ordinary conductor is a not a closed system if there is a current ε flowing in the loop. A local energy minimum argument for the current ε seems to lead to the existence of a perpetuum mobile. That is, conservation of energy is broken.

We have to investigate this more. In earlier blog posts we tried to explain superconductivity with Bloch's theorem for amorphous matter. Maybe that is the way. Above we imagined that a single electron moves along an electron flow to the other end of the semiconductor. Maybe the current is a collective movement of the time crystal? The hypothetical "condensate" in BCS theory moves the charge with a collective motion.

How would we explain the magnetic flux quantum? If the fraction part of the flux is canceled by a superconducting current, is that compatible with our earlier claim that the time crystal of electrons circulating around the loop prefer to have a flux an integer times Φ₀ through the loop?

Since the fraction part of the flux is canceled by a superconducting current, that means that the magnetic field of a superconducting current is not quantized!

Resistivity and phonons

Let us look at how phonons cause resistivity in a metal. Then we can try to figure out how this resistance disappears in a superconductor.

https://en.wikipedia.org/wiki/Electrical_resistivity_and_conductivity#Temperature_dependence

An electron at the Fermi surface moves at a speed of some 1,000 km per second, while the speed of sound is only 5 km/s.

A typical thermal phonon at 1 kelvin has a wavelength of 50 nanometers and a frequency of 10¹¹ Hz.

A phonon has a very small momentum in its energy transport, but much larger in the movement of atoms in the lattice

A 0.4 meV phonon at 1 kelvin carries a tiny amount of energy at the speed of, say, 5 km/s. The momentum associated with the energy transport is very small.

An electron at the Fermi surface has a mass-energy 511 keV and moves 200 times faster. Its momentum is 250 billion times larger than that of the phonon.

But a phonon makes atoms in the lattice to vibrate. The momentum in the movement of the atoms is much larger than the momentum in the energy transport.

Electron scattering from a phonon

Let us assume that there is a macroscopic vibration of the lattice. The electron moves 200 times faster than the wave in the lattice. The electron feels a periodic potential which may scatter it.

           charge distribution by phonon

          +            +            +            +            +

          Λ = 50 nm

                                  ^

                                  |

                                  e-    electron

                                         de Broglie λ = 1 nm

The phonon concentrates the positive charge of the ions in the lattice at periodic locations.

The phonon makes a "diffraction grating" to the lattice, and the electron scatters from this grating. The spacing of the grating is typically Λ = 50 nm. The de Broglie wavelength of the electron is typically λ = 1 nm.

The first diffraction maximum is at the deflection angle

       λ / Λ radians.

The momentum of the electron is

       p = h / λ.

The change of the electron momentum in diffraction is

       Δp = p λ / Λ

             = h / Λ.

We may imagine that h / Λ is some kind of a "pseudomomentum" associated with the phonon. If the electron is scattered, it receives this pseudomomentum as real momentum from the lattice.

Question. The phonon momentum in energy transport is irrelevant. But is the momentum of atoms in the vibration relevant? Or can we treat the scattering like it would happen from a fixed diffraction grating in the lattice?

https://www.cond-mat.de/events/correl17/manuscripts/heid.pdf

Rolf Heid (2017) writes about electron-phonon coupling. He says that the cloud of conducting electrons will screen the charge distribution caused by a phonon in the lattice. Thus, the scattering process is quite complicated. We will not try to calculate the effect, and trust the literature that scattering by phonons is the reason for most of resistivity at temperatures higher than 10 K.

Conclusions

Our time crystal model on April 8, 2022 was based on a misunderstanding of how electrons are scattered by phonons. We thought that it is like electron scattering by photons in otherwise empty space. But phonons can give a much larger momentum to the electron because the momentum comes from the lattice.

We will next try to tune our time crystal model to accommodate our better understanding of scattering.

Magnetic flux quantization through a superconducting loop

In 1961 two independent research groups discovered that a small superconducting loop of a size ~ 1 micrometer only allows a magnetic flux

       Φ₀ = 2.068 * 10⁻¹⁵ T m²

            = h / (2 e),

or its multiple to pass through the loop.

https://en.wikipedia.org/wiki/Magnetic_flux_quantum

How can we explain this phenomenon?

A model where the charge carrier has the same charge e as the electron

Let us first assume that the charge carrier has the charge e and an arbitrary mass m. Initially the magnetic field is zero. We have a superconducting loop whose radius is r.

    

               ____

             /         \      superconducting

             \_____/      loop, radius r

                e -->

                charge carrier

                does two loops

We require that the wave function of the carrier returns to its original value after two rounds around the loop.

The angular momentum L of the carrier is a multiple of

       ħ / 2.

The momentum

       p = L / r

and the velocity

       v = L / (r m).

The cycle time is

       t = 2 π r / v

         = 2 π r² m / L.

The current running in the circle is

       I = e / t

         = e L / (2 π r² m).

The magnetic moment of the current loop is

       μ = I A

           = I * π r²

           = e L / (2 m).

Let us compare the energy difference of two successive states 0 and 1 of the charge carrier, where

       L₀ = N ħ / 2,

       L₁ = (N + 1) ħ / 2.

We assume that N is a large positive integer. The kinetic energy

        E = p² / (2 m)

           = L² / (2 m r²).

If we increase the value of L by ħ / 2, the square of L grows approximately by L ħ. The energy difference between successive states is approximately

        ΔE = L ħ / (2 m r²).

We want to add or reduce the energy of the carrier by one such step. We add a magnetic field B which is normal to the loop. The magnetic moment then contributes to the energy of the carrier, or reduces it.

       ΔE = μ B

  <=>

       B = ΔE / μ

          = L ħ / (2 m r²) * (2 m) / (L e)

          = ħ / (e r²).

The magnetic flux through the loop is then

       B π r² = ħ π / e

                  = h / (2 e)

                  = Φ₀.

Note that we did not need to use Berry's phase, the magnetic vector potential, or the minimal coupling in the calculation. It is enough to treat μ B as a potential under which the charge carrier moves.

The model allows the mass m of the charge carrier to be arbitrary. It only set the charge strictly to e.

How to explain magnetic flux quantization?

Let us start from a zero magnetic field. Charge carriers in the loop might be in states L = N ħ / 2, where N is an integer. This would be analogous to the Bohr atomic model, but we allow also a half integer orbital angular momentum.

Does the superconductor prefer to order the charge carriers in the way in which they would be if B = 0? That might be true. If B ≠ 0, then symmetric orbits to the opposite directions, angular momenta L and -L, along the loop will have different energies, because their magnetic moments are opposite. It might be that the superconductor prefers symmetry.

What if we add a magnetic field

       B = n Φ₀ / A,

where n is a "smallish" integer? Then the orbits L and -L no longer have the same energy. Their energy difference is

        2 n ΔE,

where ΔE is the energy difference of successive states close to L or -L. It may be that the superconductor tolerates this asymmetry. That is, the asymmetric state has essentially the same energy as the original state.

What if the magnetic field B is not an integer multiple of Φ₀ / A?

Then the energy levels of symmetric orbits will have an energy difference which may be an arbitrary fraction of the energy difference ΔE between successive states. It may be that this asymmetry is too ugly for the superconductor. The state would have a much higher energy, and the superconductor rather generates a current which sets the magnetic flux to an integer times Φ₀.

Conclusions

Flux quantization offers us a clue about how large is the electric charge of the carrier in superconductivity.

We introduced a model where the carrier has the charge e, and its wave function must return to the original value after two loops. Usually, people assume a model where the carrier has the charge 2 e, and its wave function returns after one loop.

We explained magnetic flux quantization from a hypothesis that the superconductor prefers a symmetry in the energy levels of symmetric orbits L and -L.

Flux quantization may help us to design a time crystal model for superconductivity.

Orbital angular momentum can be any fraction of reduced Planck constant in a coil

We use the particle-in-a-box model. We make a "coil" from a tube so that the ends of the tube are joined. The particle is inside the tube.

If the coil has just one loop, then it is the torus. The de Broglie wavelength of the particle is

      λ = h / p,

where p is the momentum of the particle.

Let the radius of the torus be r. A stationary state must have an integer number n of wavelengths along the tube:

       2 π r = n λ

                 = n h / p.

The angular momentum L of the particle around the center of the torus is

       L = p r

          = n h / (2 π)

          = n ħ.

We have the familiar quantization of angular momentum.

Let us make more loops in the coil, say N loops. Now the radius of the coil is

       r' = r / N,

and the angular momentum is

       L = p r'

           = n / N * ħ.

We have a simple example to support our claim in the previous blog post: the angular momentum of a complex system can be any fraction of ħ.

Conclusions

A superconducting current (= supercurrent) inside the lattice of a superconductor is a complex system. Many electrons, as well as lattice vibrations are involved. There is no a priori reason why the angular momentum of the system should be an integer multiple of ħ.

A macroscopic uncertainty principle in quantum mechanics: we cannot measure angular momentum precisely?

In our previous blog post we analyzed Anthony J. Leggett's claim that a circulating superconducting current in a loop cannot decay to a lower angular momentum state because there is a huge potential wall which prevents this transition.

The angular momentum of a complex system does not need to be an integer multiple of ħ

For a single microscopic particle, quantum mechanics dictates that its "orbital" angular momentum to the z direction in a stationary state has to be a multiple of ħ. Does the same hold for a macroscopic system?

That does not hold for a complex microscopic system, if we allow the system to change its state. The system does not need to return to the original state after one rotation, because the system may have a different state during the 2nd rotation, yet another state during the 3rd rotation, and so on. The angular momentum can be any fraction of ħ.

What if we demand that the state of the system is the same after one rotation? It is not clear what "the same state" exactly means for a complex system.

A rotating donut: can we change its angular momentum by a fraction of ħ?

We assume that we have a macroscopic rotating donut in otherwise empty space. The temperature of the donut is exactly 0 kelvin.

                                                         ~~~~~

        radius 1 meter

                _____

              /           \_____ mirror       ~~~~~

              \______/  1 meter

                 ---->  rotation

                                                         ~~~~~

                                                    1,000-meter-wide

                                                    laser beam

                                                    λ = 1 meter

The donut has an attached mirror whose size is one meter.

We have a laser beam which is 1,000 meters wide and whose wavelength is 1 meter. The beam is defocused by rougly 0.001 radians because the beam is 1,000 meters wide.

Can the laser reduce or increase the angular momentum of the donut by a fraction of ħ?

We have

       λ = h / p,

where p is the momentum of the photon. Since λ = 1 meter, the numerical value of p is the same as the Planck constant h has.

We assume that the donut rotates very slowly. When the laser is switched on, the mirror is at a small angle relative to the laser. The mirror will scatter incoming photons to various directions, but most of the photons get deflected very little. We can put a photographic plate far away from the mirror and measure where each photon hits the plate.

If a photon is deflected by, say 0.01 radians, then it probably was reflected or scattered by the mirror. It gave some 0.01 h = 0.06 ħ of angular momentum to the donut.

We can give a fraction of ħ as new angular momentum to the donut.

Adding angular momentum to a system in fractions of ħ

Assume that we have a (macroscopic) system whose angular momentum is quantized to be an integer times ħ. Using the method sketched in the previous section we add angular momentum to it in fractions of ħ. What happens?

If we just send a single photon at a time, then the system can only scatter the photon in the way where the angular momentum of the system stays as an integer times ħ.

If the system is macroscopic and the laser beam is macroscopic, then the system must behave classically, because of the principle of the classical limit. There cannot be any problem in absorbing angular momentum from the laser beam, even though in terms of individual photons it comes as fractions of ħ.

An uncertainty principle for a macroscopic system

Maybe there cannot exist a macroscopic system whose angular momentum is quantized as an integer times ħ?

Uncertainty principle for a macroscopic system. We cannot measure the angular momentum of a macroscopic system at the precision of ħ.

What chance we might have for measuring a macroscopic system at the precision of ħ? We would need to measure the state of a huge number of particles. The measurement would add a lot of energy and angular momentum into the system.

In quantum mechanics we do not like hidden variables. If we cannot measure something, even in principle, then it does not possess a sharp value at all.

General uncertainty principle for a macroscopic system. It is not possible to measure any property of a macroscopic system at the precision allowed for a microscopic system.

Can we find a counter-example for the general principle?

Conclusions

Our macroscopic uncertainty principle may solve some other paradoxes, in addition to the quantization of angular momentum.

Let us have a macroscopic pulse of laser light. How many photons are in it? We have earlier written that since the pulse can excite oscillators of various frequencies, it does not make sense to claim that the pulse contains some precise integer number N of photons of a precise wavelength λ. The set of photons in the pulse is inherently fuzzy: we can decide at which frequencies we harvest energy from the pulse. The pulse, in this sense, is strictly a classical entity. It cannot be quantized. "Quantization" only happens at the moment when we use quantum harmonic oscillators to harvest energy from the pulse.

If the donut in our example would have its angular momentum quantized in units of ħ or in some fractions of ħ, then that would severely restrict how individual photons can scatter from the mirror. But in the classical limit, a macroscopic laser beam must scatter from the mirror in the classical way. Does this show that the angular momentum of the macroscopic donut cannot be quantized?

We will investigate this more. The fuzzy line between the quantum world and the classical world may be clarified.

BCS theory: why there is superconductivity? Leggett's proof seems to be wrong

UPDATE April 19, 2022: If we have a closed electric circuit where a part of the circuit is superconducting, and there is a potential wall which prevents changes in the current in the superconductor, then we can build a perpetuum mobile. Just put a little resistor into the loop, and it will keep producing energy for ever.

Superconductivity cannot be based on such a potential wall.

----

UPDATE April 13, 2022: Leggett's argument below is based on a superconducting current in a closed superconducting loop. Superconductivity happens also in a linear wire which is just a part of of an ordinary, non-superconducting, current loop. Why would an argument for a closed loop apply to the linear case?

Also, suppose that we have a superconducting wire and add a few electrons to one end. If there would be a large potential wall which blocks changes in the current in the wire, then the electrons would not be able to move. The superconductor would appear as an insulator for small amounts of charge.

----

Let us analyze the Bardeen, Cooper, and Scrieffer model from 1957 (BCS theory).

https://journals.aps.org/pr/abstract/10.1103/PhysRev.108.1175

Why the time crystal of the electron cloud in a single atom cannot vibrate?

In an earlier blog post we remarked about the analogy between a many electron atom and the Fermi sea in a metal.

In an atom, only the electrons at the top energy level scatter from a photon. The result is an excited atom or an ion plus an electron.

The electron cloud of an atom is a time crystal, too. Why a photon cannot make the time crystal of the electron cloud to vibrate? The orbitals of electrons would be in a collective oscillating motion.

The reason has to be that the resonant frequencies of the cloud are very high. The cloud rather breaks apart than starts to vibrate.

                 ● /\/\/\ ●

Let us have two particles bound by a potential. The potential pit may be so shallow that the first excited energy level  breaks the system apart.

                ● /\/\/\ ● /\/\/\ ● /\/\/\ ●

However, if we make a chain of many particles, the chain will have low-frequency resonant frequencies. The vibration for a single pair is so gentle that it does not break their binding.

A crystal made of many atoms does have excited states where the orbitals of the electrons vibrate.

The surface of the Fermi sea and scattering from phonons: BCS theory

An electron at the top of the Fermi sea can scatter from a phonon and move to a higher energy state. The analogue is a photon which hits an atom and moves a single electron to a higher energy state.

In BCS theory electrons can form bound Cooper pairs through an (obscure) interaction which is mediated by lattice vibrations.

How does BCS theory prevent electrons from scattering from phonons?

On pages 1177 - 1178 the authors write:

"Our theory also accounts in a qualitative way for those aspects of superconductivity associated with infinite conductivity and a persistent current flowing in a ring. When there is a net current, the paired states (k₁↑, k₂ ↓) have a net momentum k₁ + k₂ = q, where q is the same for all virtual pairs. For each value of q, there is a metastable state with a minimum in free energy and a unique current density."

"Nearly all fluctuations will increase the free energy; only those which involve a majority of the electrons so as to change the common q can decrease the free energy. These latter are presumably extremely rare, so that the metastable current carrying state can persist indefinitely."

The claims above remind us of the Landau model for superfluidity which we covered in our March 26, 2022 blog post.

Bardeen, Cooper, and Schrieffer describe a model where conducting electrons form an ordered structure. But they do not give any reason why this structure would be a superconductor. The sentence "These latter are presumably extremely rare" is their only comment on the subject.

Question. What kind of moving structures can survive a hit from a random phonon without losing kinetic energy of the structure?

Superconductivity lectures by Anthony J. Leggett: the proof of superconductivity breaks conservation laws

https://physics.stackexchange.com/questions/339747/what-is-the-link-between-the-bcs-ground-state-and-superconductivity

At the Physics Stackexchange, two users recommend the following lectures:

https://courses.physics.illinois.edu/phys598sc1/fa2015/

Anthony J. Leggett (2015) has written a course on superconductivity. Lecture 14 concerns the stability of the superconducting current. Let us look at it.

Leggett considers a superconducting current in a circular loop of a wire. He assumes that quantum mechanics dictates that the angular momentum of the rotating electron-phonon system must be quantized as an integer multiple of ħ.

"We are particularly interested in the probability of a fluctuation that takes us to the lowest saddle-point in the free energy barrier that separates states of different winding number."

On page 9 Leggett calculates that there is a huge potential barrier which prevents the rotating system from decaying to a lower angular momentum state.

He concludes that small quanta like phonons cannot move the system to a lower angular momentum state. The superconducting current keeps flowing almost indefinitely.

But Leggett's model, or calculation, cannot be correct if the rotating system can scatter quanta to the direction of the rotation. We can extract significant energy and angular momentum from the rotating system by letting it scatter a large number of quanta. If the system would keep rotating at the original speed, we would have a breach of conservation of energy and angular momentum.

The rotating system consists of moving electrons and lattice vibrations. It will certainly interact with, say, moving charges inside the superconductor. It would be very surprising if it cannot scatter any quanta.

Leggett's model or analysis is probably incorrect.

To prove that a superconducting current can flow for ever, we probably need to prove the claim in classical or semiclassical physics. Quantum mechanics does not help us because the rotating system is a macroscopic object, and the phonon gas, as a whole, is macroscopic, too.

UPDATE April 12, 2022: Our new blog post today shows that one really cannot say that the angular momentum of a macroscopic system is quantized in units of ħ.

Conclusions

The original paper by Bardeen, Cooper, and Schrieffer describes a model which might explain the phase change of the conducting electron system close to the critical temperature. That is, the model may explain the specific heat and some other properties of the low temperature phase.

Their paper does not try to prove that the model implies superconductivity. They simply assume that the low energy phase is superconducting.

At the Physics Stackexchange, various users give different explanations for superconductivity. None of the explanations is detailed enough to be convincing.

Anthony J. Leggett's lectures aim to prove superconductivity of the BCS model, but the proof breaks conservation of energy and angular momentum if the rotating system is able to scatter something. The model or the proof is probably incorrect.

We conclude that the proof of superconductivity is an open problem of theoretical physics.

For our own time crystal model we argued in an earlier blog post that it might let phonons go through without scattering. We need to elaborate our model and our analysis of it.

We need to check if the BCS model can superconduct.

Second order phase transition and a time crystal

A phase transition which happens at a constant temperature is called a first order phase transition. Some examples are melting of ice or boiling of water.

https://en.wikipedia.org/wiki/Phase_transition

A second order phase transition happens gradually when we lower or raise the temperature. Freezing of saline water in a puddle happens gradually as the temperature drops.

Specific heat measurements suggest that the transition to superconductivity or superfluidity happens gradually when the temperature is lowered. The heat from the phase transition is released over an interval of from one kelvin to several kelvins.

Melting of a time crystal versus an ordinary crystal

An ordinary crystal is an almost static lattice where atoms or molecules sit. There is vibration of the atoms around their mean position, as well as collective vibrations of the lattice, but in the big picture the system is simple and static. There is just one, or a few, compact (not fiber-like) building blocks from which we build the lattice in 3D space.

Melting of an ordinary crystal is a simple monotonic process and can happen completely at a sharp temperature.

On the other hand, a time crystal may be extremely complex because it is built from atoms and their movements. Movements can take complicated forms. An atom and its movement in spacetime constitutes a thread. Building a time crystal requires weaving these threads together in a 4D space.

A time crystal might be analogous to a 3D crystal built from polymers. Such a crystal presumably does not have a sharp melting point.

Conjecture. Most phase transitions of time crystals are second order. There is no sharp temperature which destroys all the order in a time crystal.

An example: the conducting electron gas in a metal crystal undergoing zero-point vibration

Let us have a crystal of metal. Our initial configuration is a uniform gas of conducting electrons.

       1.      +  +  +           +  +  +

       2.      +    +    +    +    +    +

                          crystal

The lattice of the crystal is undergoing vibrations which move positive charge back and forth between the center and the ends of the crystal. The stages of the process are marked 1 and 2 in the diagram.

We can presumably lower the energy of the system by letting an electron adjust its path so that it masks some of the extra positive charge at the stage 1.

We can mask all of the extra positive charge by adjusting the path of several electrons. We created a time crystal in the electron gas and the lattice.

How does the time crystal "melt" if we raise the temperature?

If we remove one electron from the time crystal and release to it to the random gas of electrons, then the rest of the electrons are more tighly bound because there is now more excess positive charge at the stage 1.

The time crystal melts gradually as we raise the temperature. It is a second order phase transition.

In this example it was not essential that the system is a time crystal. The essential point was that the excess positive charge at the stage 1 is large and that it requires many electrons to mask the excess charge.

Our example is similar to the ionization of a many electron atom. We have to raise the temperature to get more electrons out.

Conclusions

Melting of an ordinary crystal at a precise temperature is a special case where the simplicity of the process allows it to happen at a sharp temperature.

Since a time crystal in a lattice of a metal is a more complicated system, it probably cannot melt at a sharp temperature.

A time crystal model of superconductivity

UPDATE April 16, 2022: The model of electron scattering from a phonon is erroneous in this blog post. The phonon makes a "diffraction grating" to the lattice and the electron scatters from the grating.

----

In the blog post on April 5, 2022 we listed the obstacles to superconductivity in a metal:

1. phonons from thermal vibrations;

2. impurities in crystals;

3. borders of crystals.

Let us investigate how a time crystal model of superconductivity can overcome the obstacles 1, 2, and 3.

Phonons in a metal

https://en.wikipedia.org/wiki/Phonon

Wien's displacement law states that the black body radiation at 1 K has its peak at the wavelength 3 mm, which corresponds to the frequency 10¹¹ Hz. The energy of a typical photon is h f = 6.6 * 10⁻²³ J, or 400 microelectronvolts.

Let us place a block of the metal in a cavity which contains the 1 K black body radiation. The metal interacts with the radiation primarily through oscillations of its electron content, just like in a radio antenna.

The oscillation of electrons presumably creates oscillation in the lattice of atoms. These oscillations are phonons.

The speed of sound in a typical metal is 5 km/s. Thus, the typical thermal phonon has a wavelength of 50 nm at 1 kelvin.

The typical spacing of metal atoms in a crystal lattice is 0.1 nm.

The lattice of the metal is distorted by a phonon, which in turn creates an electric field. A conducting electron can interact with this electric field and scatter.

In umklapp scattering, the phonon exchanges momentum also with the lattice. 

The time crystal of electrons must let phonons pass through in superconductivity

In electric resistance, a phonon scatters from an individual electron. We can remove that resistance if the phonon can pass through the time crystal of electrons without scattering.

Actually, it is enough that the energy of the phonon does not change if it is scattered. If the electron drift velocity would slow down, the electrons would lose energy to phonons.

1. The structure of the time crystal must be so strong that a 400 microelectronvolt phonon cannot kick an electron out of the structure.

2. The time crystal must be ordered or uniform enough at the scale of 50 / 2 = 25 nanometers, so that destructive interference removes almost all of the scattered wave.

3. Phonons must not be reflected by the moving parts in the time crystal. If the spatial form of the time crystal is static, then this probably is the case.

4. The time crystal should not have a resonant oscillation frequency at the phonon frequency, so that the phonon cannot lose energy to the time crystal. How can we accomplish that?

Item 4 is the hardest part. Let us assume that the potential which binds together individual parts of the time crystal is not steep. However, the energy required to detach a part from the crystal is large. In BCS theory it is assumed that some collective effect keeps the "condensate" together and makes the dissociation energy large for any individual part.

For an ordinary crystal, the energy required to detach an inner atom is huge. We would need to split the crystal into pieces.

Since the potential is not steep, vibration frequencies of the time crystal may be very low, much lower than the 10¹¹ Hz of a phonon at 1 K.

With these assumptions we obtain: the phonon is not able to break the time crystal, nor is the phonon able to be absorbed by the time crystal. The phonon passes through the time crystal without a change in energy. The time crystal can keep transporting electrons at the original velocity.

An analogous setup is a block of glass moving in a bath of black body radiation. If the block is very small, photons will scatter from it and the block loses its momentum. But if the block is large, photons will pass through the block without losing energy.

The time crystal in a superconductor must be large compared to the phonon wavelength 50 nm.

The time crystal and impurities

Impurity atoms scatter electrons and make them to lose their drift velocity. How can a time crystal avoid this?

                      crystal

                   o o o o o o ---->

                  --------/\--------

            surface with a spike

Let us have an ordinary crystal sliding over a solid surface with a one atom spike.

Since the crystal and the surface are macroscopic objects, we can measure very precisely the position of the crystal and the spike. Therefore, we can use classical physics, and in classical physics it is self-evident that the spike will create vibrations in the crystal.

We conclude that the time crystal must avoid impurity atoms. Otherwise, there will be power dissipation and resistance.

Can the time crystal avoid them? Crystallization means that a system falls to a lower energy state which is reached through creating order in the system. If there would be an impurity atom within the time crystal, and it would make electrons to scatter, it would presumably destroy order. That would require energy which is missing from the system.

Can impurities prevent the formation of a time crystal altogether? We have to check what empirical research says about this.

Quantum mechanics prevents the electron in the hydrogen atom from falling to a very low orbit. A short-wavelength orbit would possess so much kinetic energy p² / (2 m) that the energy cannot be recovered from the lower potential energy. There is a quantum force which stops the electron from falling.

Similarly, a quantum force may keep the time crystal away from impurities which cause scattering and break the time crystal.

Borders of crystals of the metal

This is a major obstacle to superconductivity. The time crystal can avoid impurity atoms, but it cannot avoid borders of the crystals in the metal.

The border means some kind of a potential hill, and it will scatter electrons back unless the electrons are bound in a time crystal.

               --------->  

          o o o o o o o              /

          o o o o o o o            /

          o o o o o o o          /

          o o o o o o o        /

               crystal         obstacle

Let us study an ordinary crystal sliding over a line-like obstacle. The atoms on different rows in the diagram will generate sound waves or electromagnetic waves whose phase is

       n α,

where n is the number or the row and 0 < α < 2 π.

We assume that

       α > s / v * 2 π f,

where s is the spacing of adjacent atoms of the crystal meeting the obstacle, v is the speed of sound or of electromagnetic waves, and f is the frequency of the wave.

Then the phase of the waves, when observed from a faraway location, is

       n β,

where 0 < β < 2 π. We can argue like in our April 1, 2022 blog post that destructive interference wipes out almost all of the waves.

If the analysis carries over to the time crystal, then borders of metal crystals do not prevent superconductivity, unless the border is normal to the flow of the time crystal. We may argue like in the previous section that the time crystal will avoid such normal borders.

Conclusions

We now have a time crystal model of superconductivity. The model addresses the problems of phonons, impurities, and crystal borders.

We will next compare our model to BCS theory.

The structure of the hypothetical time crystal is unclear. The crystal must include both lattice vibrations and conducting electrons. Electrons move much faster than lattice vibrations. How can the time crystal accommodate such different speeds of its components?

Cooper pairs in BCS theory might correspond to bindings between electrons in the time crystal model. The "condensate" of BCS theory corresponds to the time crystal itself. The concept of a Cooper pair as well as the condensate are fuzzy concepts in BCS theory.

We cannot calculate the orbitals of a many electron atom. It looks like we have the same problem in solid state physics. The details of the wave functions of electrons in the Fermi sea are unknown. To understand superconductivity well, we should know the wave functions.

Bloch's theorem for amorphous matter

Bloch's theorem states that scattering of a wave from an ordered lattice structure is insignificant:

https://en.wikipedia.org/wiki/Bloch%27s_theorem

We demonstrated in our April 1, 2022 blog post that destructive interference in such a case cancels almost all the scattered waves. We also argued that the scattering from an amorphous material is insignificant, provided that its density is very precisely constant for zones which produce scattered waves of a specific phase φ

When can scattering be significant? Destructive interference does not cancel scattering from random obstacles. Also, if the obstacles are ordered and the incoming wave "resonates" with them, there is very significant scattering.

Bloch's theorem for amorphous matter. Let us have an observer very far away from the amorphous material which is being tested. Let Z be the zone which produces a scattered wave whose phase is in the range [φ₁, φ₂). If the density of the material in each such zone is "very precisely" the same, then the scattered wave is insignificant.

The phrase "very precisely" means that the variance of the density must be much less than what would result from a totally random replacement of atoms, where we would also allow the overlapping of atoms.

The wave must not lose energy when it is scattered from the material. The quantum of the wave must not be able to kick an atom of the material from its place. The material must not have a resonant frequency at the frequency of the wave.

The theorem is essentially another way to say that destructive interference wipes out almost all of the scattered wave.

Examples

1. Why visible light is not scattered by glass?

Let us consider a 500 nm laser beam entering a block of glass. The atom spacing is only some 0.1 nm in glass. The wave interacts primarily with volumes of the material where the size of a volume is a half of the wavelength.

Since each volume contains a huge number of atoms, the volumes are very similar with respect to their scattering capability.

We can now apply the ordinary version of Bloch's theorem: the "lattice" of the volumes is ordered and does not cause scattering.

Even though glass is amorphous, that fact is not significant at the level of 250 nanometers. The material is very uniform at such a scale. We do not need Bloch's theorem for amorphous materials in this case.

2. Scattering of a conducting electron from atoms (ions) of liquid metal. The Fermi velocity of a conducting electron is something like 1,000 km/s. Its de Broglie wavelength is 0.7 nm. Since the spacing of atoms is 0.1 nm, we cannot claim that the material is uniform at the scale 0.35 nm.

We can resort to Bloch's theorem for amorphous matter. Let us have an observer very far away. We divide the metal in zones which produce each phase φ of the scattered wave. The zones are very much like each other and cause an almost total destructive interference of the scattered wave.

The resistivity of a liquid metal just above its melting point is roughly double the solid metal. The free path of a conducting electron is about a half of that in the solid. Bloch's theorem for amorphous matter explains in part why scattering of conducting electrons does not explode when a metal melts.

Conclusions

Bloch's theorem about scattering from a periodic potential is just a special case of a more general theorem: if the zones, which produce different phases of the scattered wave, are "very similar", then destructive interference wipes out almost all the scattering.

If we place scattering objects totally randomly in space, then the zones are not "very similar".

If we have a material which is essentially "uniform" at the scale of the wavelength, then scattering is insignificant.

What causes resistivity in metals?

To understand superconductivity, we need to understand what causes resistivity.

The free electron model

https://en.m.wikipedia.org/wiki/Free_electron_model#Mean_free_path

The free electron model of Arnold Sommerfeld describes the electron gas of the conducting electrons in a metal.

Electrons fly almost like non-interacting particles of a gas, but collide with phonons, defects and borders of crystals, and impurities once for every tens of nanometers of free flight.

https://en.wikipedia.org/wiki/Electrical_resistivity_and_conductivity

The resistivity of copper is almost linearly dependent on the temperature at 20 K - 1,200 K. At less than 20 K, copper retains some resistivity which seems to come from impurities and defects in the crystal structure. Copper does not become a superconductor at low temperatures.

Why electrons do not bump into atoms in the crystal lattice or into each other?

https://en.wikipedia.org/wiki/Bloch%27s_theorem

Bloch's theorem explains why a single electron moving in a lattice potential does not get scattered. Its wave function is a plane wave modulated by a periodic function. We showed in our previous blog post that in such a situation, there is an almost total destructive interference of scattered waves. Thus, there is essentially no scattering.

However, in a metal there is a huge number of free electrons which could bump into each other. Bloch's theorem is an inaccurate description of the system.

The analogy of a many electron atom and a crystal of a metal

It is an empirical fact that electrons seem to fly freely and do not bump into atoms of a perfect crystal of a metal, or into each other. As far as we know, there exists no adequate theoretical model presently.

A similar empirical fact is that the electrons in an arbitrary atom seem to fill orbitals which are similar to the orbitals of hydrogen. For an unknown reason, we can solve the system for just one electron (i.e., hydrogen), and we obtain a qualitatively correct solution for a many electron system.

The "Fermi sea" of electrons in an atom consists simply of the orbitals of the electrons. In a metal, electrons seem to settle to different energy levels in a manner similar to a single atom. The orbitals in a metal might look like the solutions in Bloch's theorem.

Conclusions

The free electron model of Arnold Sommerfeld is an empirical result.

Empirically, resistance is caused by:

1. thermal vibrations of the lattice (but not by zero-point vibrations?);

2. defects and borders of crystals;

3. impurities in a crystal.

We have no adequate theoretical model for resistivity.

To explain superconductivity, we need to explain why items 1, 2, and 3 above do not cause resistance in a superconductor.

Superfluidity: osmotic pressure of excitations and a time crystal

The excellent paper by Russell J. Donnelly (2009) helped us to understand the second sound in superfluid helium-4.

The osmotic pressure of excitations within a fluid

https://www.physics.umd.edu/courses/Phys404/Anlage_Spring10/Donnelly%2520Two%2520Fluid%2520LHe.pdf

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.

https://arxiv.org/abs/1001.0785

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

https://en.wikipedia.org/wiki/Roton

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.

http://www.w2agz.com/Library/Classic%2520Papers%2520in%2520Superconductivity/Feynman,%2520%25281957%2529%2520APPLICATION%2520OF%2520QUANTUM%2520MECHANICS%2520TO%2520LIQUID%2520HELIUM.pdf

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.

https://royalsocietypublishing.org/doi/10.1098/rspa.1947.0108

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.

https://proofwiki.org/wiki/Sum_of_Exponential_of_i_k_x

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.

https://en.wikipedia.org/wiki/Bloch%27s_theorem

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?

https://www.sciencedirect.com/science/article/abs/pii/S0079641708600773

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.