"Phonons -- are some sort of vibratinal lattice quarks."
Not true, phonons are not quarks but quasiparticles, more here. Phonons are also bosons.
Wikipedia misleadingly shows that phonon is not an elementary particle or a boson but my lecturer and Wikipedia article about bosons support the bosonness.
Please, note that the term
"quasiparticle" differ between contextes. According to Wikipedia,
"These fictitious particles are typically called "quasiparticles" if they are fermions (like electrons and holes), and called "collective excitations" if they are bosons". This is about elementary particles so do not mix them up. I understand this so that phonon is not an elementary particle so the above definitions do not hold for it. Anyway, notice that the same word
"quasiparticle" is confusingly used in similar context for totally different meaning!
Phonon is a quantitative piece of energy in vibrational lattice (
"fononi=kiteen värähtelyenergian kvantti", ~p.163). This quantification is eventually a result of $E=hf$. I don't use the word
"particle" because phonon is not traditional elementary particle like electron, proton or a quark. It is some sort of fictious particle that appears during interaction with other materials. For example, light travelling in photons do not contain phonons but when the high-energy light hits wall, the phonon lattice emerges.
Example: Silica and Diamond
Silica and diamond have the same structure but Silica does not conduct heat. Please, note that electricity conductance do not mean heat conductance, both Silica and Diamond do not conduct electricity because of no moving electrons. In order to explain the heat conductance of diamond, one needs to consider frequencies and hence energies. In layman terms, diamond lattice is able to sustain frequencies of higher energy while Silica not. This may be different in extreme temperatures such as close to absolute zero.
The heat conductivity is a feature due to the phonon lattice.
The lattice vibrates according to partial derivatives aka Hamiltonian equations here. The Hamiltonian consists of kinetic energy and rotational energy for each thing
where we don't have potential energy between the particles because it is very close to zero (gravitational field effect very small). For me, the Hamiltonian equations look similar to the wave equation here but the earlier Hamiltonian looks different.
ERR I am probably messing something up here, cannot yet see how the two different Hamiltonians are the same...checking, the picture source here.
Bullet points from Wikipedia
I. "The thermodynamic properties of a solid are directly related to its phonon structure."
II. "At absolute zero temperature, a crystal lattice lies in its ground state, and contains no phonons"
II.I. "If more than one ground state exists, they are said to be degenerate."
I. Is phonon a particle?
II. What does it look like?
III. Can you see the phonon with microscope?
Phonon is a boson, more here and a picture here.
'These fictitious particles are typically called "quasiparticles" if they are fermions (like electrons and holes), and called "collective excitations" if they are bosons (like phonons
and plasmons),1 although the precise distinction is not
universally agreed.' Wikipedia