What is the distribution of speeds in an ideal liquid?

Question

By ideal, I mean similar conditions to the ideal gas-in-a-box with perfect spherical perfectly elastic homogenous atoms, so no inter-atomic forces, evaporation, gravity, rotations or phase changes. Collisions with the container are perfectly elastic. I envisage a liquid fully filling an ideal sphere. I suppose there must be some small extra space to provide slack for movement but with the aforementioned conditions I hope to be able to neglect the possibility of large bubbles. Assume temperature and pressure conditions far from any state transitions.

I have seen plenty of assertions that Maxwell-Boltzmann applies but no proof of that.

My naive expectation is that the distribution is very narrow as atoms are already in contact with some others so bumps gets transmitted and dispersed locally very quickly. This related question has not been answered and it differs by allowing the complication of an adjacent empty space into which atoms can escape.

Answer 1

According to a comment pointing to Micheal Seifert's answer on Chemistry SE, the fast answer is that the Maxwell-Boltzmann distribution applies to any phase, provided the conditions of temperature and pressure allowing the use of classical statistical mechanics are satisfied.

Since we are on PSE, I think it would be helpful to elaborate slightly more.

Your definition of an "ideal liquid" corresponds to the model fluid of particles interacting through a hard sphere interaction. However, such a limitation is not necessary. Whatever the intermolecular interaction, the distribution of the molecular speeds remains the Maxwell-Boltzmann.

The proof is more or less trivial, depending on the equilibrium ensemble one uses. In the case of the canonical ensemble, the factorization of the kinetic energy Boltzmann factor (used in the previously cited answer) makes the result evident. In an ensemble like the microcanonical ensemble, the result is less obvious; nevertheless, it can be obtained as an asymptotic result valid at the thermodynamic limit.