Systems of vibrations occurring within atoms that make up solids or liquids.

Example: I strike the end of a metal bar with a hammer. That hammer causes atoms at the stuck end to compress slightly, which, in turn, compresses the adjacent, non-compressed atoms. A compression wave (phonon) then travels along the bar, causing the opposite end to vibrate.

Would it be somewhat accurate to say it's a mathematical trick that makes it easier to analyze the propagation of the compression wave by pretending its a moving particle instead, simplifying the math?

are those compression waves similar to that of peristalisis in biology?

Phonons are often used in situations where you're analysing things at a quantum level, where wave-particle duality kicks in - particles can be more accurately modeled as waves, and visa versa, depending on the scenario in question - and for reasons that I don't understand well enough to ELI5 this applies to sound waves, which are therefore also particles.

I dont mean to be rude, but that wasn't my question.

Sorry, I'll be more explicit: The answer to your question is "No" because of the explanation in my previous post.

I don’t deal with sound waves but in my field phonons are absolutely a mathematical trick. With some materials, you need more than just a photon to produce the photo electric effect, you also need a phonon which is in this case, a quantized amount of momentum. Phonons are a virtual Particle and therefore aren’t really existing.

No, because phonons aren’t strictly related to forces applied externally. Molecules are naturally experiencing vibrations—spring like oscillations—along all of the interatomic bonds. When you have a bunch of molecules linked together, all of those vibrations are now interconnected and affecting each other, which is what we call a phonon.

Think of a chunk of crystalline solid. The atoms inside this solid are very ordered, and as such, if there is any small movement or vibration of the atoms, it will propagate as waves. This vibration will have different frequencies/wavelengths and energies associated with it. In quantum mechanics, the energy of waves are discrete and a phonon is just a fancy way of describing one unit of energy of a vibration at a particular frequency/wavelength. At room temperature, you will have many of these waves or phonons in a crystalline solid due to the atoms vibrating a lot.