statistical mechanics - What physical processes may underly the collisional term in the Boltzmann equation, and how do they increase entropy?

Consider particles interacting only by long-range (inverse square law) forces, either attractive or repulsive. I am comfortable with the idea that their behavior may be described by the collsionless Boltzmann equation, and that in that case the entropy, defined by the phase space integral $-\int f \log f \, d^3x \, d^3v$, will not increase with time. All the information about the initial configuration of the particles is retained as the system evolves with time, even though it becomes increasingly harder for an observer to make measurements to probe that information (Landau damping).

But after a long enough time most physical systems relax to a Maxwellian velocity distribution. The entropy of the system will increase for this relaxation to occur. Textbooks tend to explain this relaxation through a collisional term in the Boltzmann equation ('collisions increase the entropy'). A comment is made in passing that an assumption of 'molecular chaos' is being made, or sometimes 'one-sided molecular chaos.' My question is, how do the collisions that underlie the added term in the Boltzmann equation differ from any collision under an inverse square law, and why do these collisions increase entropy when it is clear that interactions with an inverse square law force do not generally increase entropy (at least on the time scale of Landau damping?) And finally, how valid is the so-called molecular chaos assumption?

EDIT: I should clarify that, if entropy is to increase, then it is probably necessary to invoke additional short-range forces in addition to the long range inverse square law forces. I suppose I could rephrase my question as 'what sort of short-range forces are necessary to explain the collisional term in the Boltzmann equation, and how do they increase entropy when inverse-square law collisions do not?' If the question is too abstract as written, then feel free to pick a concrete physical system such as a plasma or a galaxy and answer the question in terms of what happens there.

Answer

The statement that the entropy increases because of collisions is incorrect. The conservation of phase space volume is a theorem of Hamiltonian mechanics, and therefore applies to all known physical systems, regardless of whether they contain nonlinear forces, collisions or anything else.

What actually happens is that although the phase space volume doesn't change as you integrate the trajectories forward, it does get distorted and squished and folded in on itself until the system becomes experimentally indistinguishable from one with a bigger phase space volume. The information that was originally in the particles' velocity distribution ends up in subtle correlations between the particles' motions, and if you ignore those correlations, that's when you get the Maxwell distribution. The increase in entropy is not something that happens on the level of the system's microscopic dynamics; instead it occurs because some of the information we have about the system's initial conditions becomes irrelevant for making future predictions, so we choose to ignore it.

There is an excellent passage about this (in a slightly different context) in this paper by Edwin Jaynes, which gives a thorough criticism of the kind of textbook explanation that you mention. (See sections 4, 5 and 6.) It explains the issues involved in this much more eloquently than I can, so I highly recommend you give it a look.