Do gases have phonons?

The only mention of this subject I can recall seeing is an aside in Xiao-Gang Wen's book, Quantum Field Theory of Many-Body Systems. Footnote on page 86:

A sound wave in air does not correspond to any discrete quasiparticle. This is because the sound wave is not a fluctuation of any quantum ground state. Thus, it does not correspond to any excitation above the ground state.

I'm not completely sure that I buy this, but it does certainly identify a crucial point. Plasmons or phonons in a condensed matter setting both have a restoring force, which lets one identify a minimum energy state to excite. In your typical view of an ideal gas, in which atoms mostly travel freely but occasionally collide with one another in some short-ranged way, this is not really true. You can make all sorts of density patterns in which the atoms are still not actually touching and thus the energy is not increased.

One might be tempted to get around this by taking a continuum limit somehow and considering a smooth quantum fluid, but then you are by definition trying to quantize a macroscopic field, which does not seem to make sense in even a formal way. In particular, since the field is a coarse-graining of the true system, one has necessarily thrown away some degrees of freedom, which means that the state of the field is never a pure quantum state and is more likely very close to a fully decohered statistical mixture.

In contrast, in a system with long-range interactions, and some boundary conditions, I would assume that phonon-like excitations are possible because the restoring force from mutual repulsion provides a well-defined ground state. This is a Coulomb crystal (1). But clearly this is very far from an ideal gas.

Edit: I should emphasize, as @Xcheckr has, that the above answer is interpreting the OP's question to refer to a Maxwell-Boltzmann ideal gas in a high-temperature state. There is of course no obstacle to defining the ground state of a BEC of an alkali gas, and such a ground state does indeed have phonon excitations (assuming a weak interaction). Similar remarks apply to a degenerate Fermi gas.

This is what I thought at first:

You need a restoring force in order to have a phonon: after all, phonons result from the quantization of the lattice energy written as a sum of harmonic oscillators. The hamiltonian of a gas cannot be written in this way: there are no restoring forces (you cannot "pull" a gas), so there can be no phonons.

As Rococo wrote, quoting Xiao-Gang Wen, a sound wave (SW for short) in a gas is different from a SW in a solid. If you think about it, a SW in a gas cannot be thought of as an "excitation" above some "fundamental vibrational state": it is just energy transiting through the medium. This is why you cannot quantize a sound wave in a gas, and hence you cannot introduce phonons in its description.

Then I talked with my professor of physics of liquids, and it seems like the subject is a bit more complicated than what I thought. He said that you can have collective excitations, hence phonons, in liquids, but there are some things to consider.

First, while the structure of a solid remains unchanged when a SW passes through it, this is not true for a liquid: the liquid can "restructure" itself locally with a certain speed. So, if the perturbation's frequency is greater than the inverse of the time it takes for the system to restructure itself, we will have a coherent wave. But if the frequency is smaller, the liquid will restructure faster than the propagation of the wave and the wave will undergo decoherence. The time it takes for the liquid to restructure can be estimated from the time it takes for the dynamic structure factor $S(\vec k, t)$ to go to $0$ (see figure below)

Another thing is that the wave vector $\vec k$ could be a bad quantum number to describe this kind of collective motion, since the liquid has no periodic long-range order and it is difficult to define for example a Brillouin zone. We can talk of normal modes of vibration, but we cannot apply the same formalism we use on crystals. This is also true for glasses and other amorphous solids.

I think many of these insights can be applied to gases too. The situation there is of course even more different from a crystal because short-range order (which is present in liquids) is lacking, too. So we can have collective motion only at very low $\vec k$, i.e. on large scales where the medium can be seen as a continuum.

As you mentioned, a phonon is a quantized sound wave. Let us ask then what is the difference between a quantum and classical sound wave in a crystal? At room temperature, for most solids, the specific heat is pretty close to $3Nk$. This is the classical Dulong-Petit answer.

However, at low temperature, the specific heat deviates away from the Dulong-Petit answer and exhibits a $T^3$ relation. By this measure, one could then say that phonons are really observable at low temperature. The problem then is that most gases at room temperature become solids at these low temperatures. However, there are important exceptions. Helium remains a liquid at low temperatures and indeed quantized sound waves are observed in this case. See for instance:

http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.24.646

Probably the most well-known kind of low temperature gas is the Bose-Einstein condensate, and here phonons have been observed as well:

http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.83.2876

I borrowed a lot of this response from the link below where the answer is discussed more thoroughly:

https://thiscondensedlife.wordpress.com/2015/09/20/why-is-sound-in-a-solid-different-from-sound-in-a-liquid/