I have a pure math background and I am currently self-learning physics.
To mathematically justify and understand the Second Law of Thermodynamics, mathematicians and physicists have studied the motion of dilute gases governed by Boltzmann's equation.
Within this concrete mathematical model, one can prove that the gas converges to equilibrium (under some assumptions). To the best of my knowledge, this was first rigorously shown by Toscani and Villani (see Theorem $20$ from here).
Is there such a mathematical analysis for liquids too (even with simplified models)?
I am asking because the second law of thermodynamics applies to all kinds of systems, so it seems strange (and unsatisfactory) that I found rigorous justification for it, only for dilute gases.