I know, for gases, the distribution on the speed of particles is the Maxwell-Boltzmann distribution. What is the corresponding distribution for liquids (e.g. water)? You can assume that the liquid is still, with no big external forces e.g. swirling it around or making it flow. Just like how the Maxwell-Boltzmann distribution doesn't account for if a wind is blowing.
My thought is that it would also be a chi-distribution with 3 degrees of freedom, just with different constants.
Equilibrium distribution of molecule velocities is given by the Maxwell-Boltzmann distribution in classical statistical physics, that is, Newtonian, non-quantum theory. In this theory, it does not matter whether the macrostate is classified as gas or liquid or solid (and I think even a neutral plasma, although showing that poses some difficulties). Certainly in any phase with no ionization, the assumptions implying the probability of a microstate is $e^{-E/k_B T}/Z$ - the Boltzmann distribution law - are the same and are assumed valid. These assumptions are : all phase space points are allowed distinguishable states that can carry some probability, inter-molecular forces decay fast enough with distance, so system energy can be linearly proportional to its volume (thus, no gravity forces allowed), and phase space is finite (the system has to be in a box, and the inter-molecular forces must not be too singular function of inter-molecular distance). Thus given these assumptions, the result, the Boltzmann distribution law, is valid irrespective of the phase the molecules are in. Expressing the Boltzmann law in velocity 3D space, instead of on a 1D energy space, we get the Maxwell-Boltzmann distribution. Thus in this classical theory, speeds of neutral water molecules in liquid water are predicted to obey the Maxwell-Boltzmann distribution.
This classical theory ignores two things: special relativity, and quantum theory.
There is a generalization of the Maxwell-Boltzmann distribution taking into account special relativity, called the Juttner distribution. For usual temperatures, it is indistinguishable from the Maxwell-Boltzmann distribution, except for total lack of molecules with speeds higher than $c$, so we can expect a small shift of the average speed towards lower speeds.
In quantum statistical physics, the assumptions about what the system is are somewhat different: we have a different underlying set of allowed microstates that can carry probability - we can have either a fermion gas obeying the Fermi-Dirac probability distribution law, or a boson gas, obeying the Bose-Einstein probability distribution law, instead of the Boltzmann law.
Which probability law is appropriate depends on whether the particles moving around are all bosons, or fermions (I don't know what happens when we have a mixture of these two types).
An even number of neutrons means the molecule is a boson, thus pure helium 4 with two neutrons is a boson system, but pure helium 3 with one neutron is a fermion system. Water molecule has either 8 or 10 neutrons, and so it is a boson.
For low enough temperatures or large enough densities, these quantum probability distributions deviate significantly from Boltzmann's distribution. In case of a boson gas, instead of a probability of state proportional to $\frac{1}{e^{\beta (\epsilon-\mu)}}$ like in non-quantum statistical physics, we have probability proportional to $\frac{1}{e^{\beta (\epsilon-\mu)} - 1}$. This quantum theory of boson gas was applied to ideal gas by Einstein in his paper on Quantum theory of the monoatomic ideal gas, see here:
https://www.thphys.uni-heidelberg.de/~amendola/otherstuff/einstein-paper-v2.pdf
This theory predicts severe "condensation" of the gas, if the temperature is low enough and density is high enough. The condensation happens in "velocity space", thus many molecules drop out of translation motion completely, but their position is undetermined. The distribution of velocities will be affected by this - many molecules will concentrate around zero velocity, something that does not happen in the Maxwell-Boltzmann distribution.
It is hard to get a real boson gas to condense in this way, because other things usually happen due to interaction of the particles - e.g. water freezes into a crystal, and the boson gas theory does not seem to be applicable then.
But pure Helium 4 does not freeze - it remains liquid even at very low temperatures, and although it is not exactly a non-interacting boson gas, and does not perfectly obey its theory, its behaviour (superfluidity, extremely high heat conductivity) is "kind of" similar to boson condensation predicted.
