UPDATE March 30, 2022: The "phonon" and the "roton" of Lev Landau seem to be something like an atom, or a lighter or a heavier particle moving and carrying the energy p² / (2 m) and the momentum p. There, m is the mass of the particle. Landau's phonon is not a vibration of a lattice of atoms. A high-energy phonon may be a single helium atom. A roton is a group of ~ 5 helium atoms.
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UPDATE March 28, 2022: In this blog post we treat an "excitation" as a phenomenon of classical mechanics. Helium atoms are relatively heavy objects, and groups of them even heavier. Therefore, a classical treatment makes sense. Friction between solid bodies is usually modeled as a classical process.
How would we quantize a classical excitation? For example, in Newton's cradle, the shock wave carries very large momentum compared to its energy. The simplest example of a shock wave is a single atom flying and carrying momentum. The momentum and energy in such a setup is not quantized, or, the "quantum" is a single atom.
The problem of quantization of excitations in a lattice is a fundamental one. We will look at this problem.
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It is Chapter 5 in the above link.
Superfluid in a capillary hits the wall
capillary
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ooooooooooo ----> v superfluid M
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<---- p' momentum from the hit
A body of a superfluid is moving in a capillary at the velocity v.
Let us first work in the frame where the superfluid is static. The fluid may be hit by the wall of the capillary and receive the momentum p' to the left.
The momentum p' is the total momentum received by the whole body of the fluid. Let p be the momentum of the created excitation in the fluid body. There is no obvious reason why p and p' should be equal.
Landau assumes that the hit produces a single excitation of the fluid, such that the vibrational energy of the excitation can only take a single value determined by a function
ε(p).
Is the assumption of a single value for the energy function realistic?
We may imagine that we have harmonic oscillators attached to the body of the fluid. The oscillators may have various masses m. Let an oscillator absorb the momentum p. The vibrational energy after that is
p² / (2 m).
Besides p, the energy depends on m. The mass m can be at most the total mass M of the superfluid. There is no obvious lower limit for m.
An "oscillator attached to the fluid" may be just some volume of the fluid body, e.g., 1/10 of the total fluid. We may divide the fluid into 10 such oscillators. The oscillators are coupled to each other.
We conclude that ε(p) is not a function. It may have many values for the same p and the values may be very large.
Let us then work in the laboratory frame. We assume that P = M v >> | p |. If
E = P² / (2 M),
then
dE / dP = 2 P / (2 M) = P / M = v.
If v is small, then the body of the fluid, when it hits the wall, loses little energy compared to the momentum (p') that it loses.
On the other hand, an excitation typically contains a lot of energy compared to the momentum it holds.
If the excitation would be required to hold the entire momentum p' lost by the body of the fluid, then the excitation would not be able to happen. There would not be enough energy available to create the excitation.
This is the error of Landau: he assumed that the momentum p of the excitation has to be the same as the momentum p' lost by the body of the superfluid. When the body of the superfluid hits the wall, there is an interaction between the different parts of the fluid. The entire momentum p' is not required to go to the excitation.
Conclusions
Lev Landau considered a body of a superfluid that is hit by the wall of a capillary. His analysis produced a counter-intuitive result: the kinetic energy lost by the body cannot go to vibrations of the fluid body because there is not enough energy available.
The claim is strange. We know that if we have a moving rod of a material, and the rod hits an obstacle, then the lost kinetic energy does go to vibrations.
The paradox is solved through the realization that the vibrations do not hold the entire momentum p' lost by the rod. Landau did not consider this.
The other error made by Landau is that he assumed that the energy of an excitation ε(p) is uniquely determined by its momentum p.
The third error is to assume that the hit only produces a single excitation. If a rod hits an obstacle, many different vibrations are born at the same time.
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