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

Question

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 1

The entropy increase comes from the assumption that you can close the system on the kinetic level, thereby (i) making the dynamics tractable and getting a transport equation, and (ii) disregarding extremely high frequency contributions and paying for this with an entropy increase.

Any interaction leads to collision terms; the details only matter for the particular form of the collision integral but not for its existence.

There are different ways to obtain the Boltzmann equation, but all share the above features. The molecular chaos assumption works only for classical ideal gases. For a modern derivation of kinetic equations and in particular the Boltzmann equation from fundamental principles (i.e., quantum field theory), see

Yu. B. Ivanov, J. Knoll, and D. N. Voskresensky, Self-Consistent Approximations to Non-Equilibrium Many-Body Theory, Nucl. Phys. A 657 (1999), 413--445. hep-ph/9807351

and related papers. See also Good reading on the Keldysh formalism

Edit: In an operator-based formalism, the kinetic approximation forces the density matrix to take the form $e^{-S/k_B}$, where $S$ is a 1-particle operator. This eliminates lots of (not all) high frequency contributions, as the exact dynamics destroys this form, so the approximation must project it back to it instantaneously. For understanding how the projection works see the book by Grabert on Projection operator techniques.

Calzetta did some work on kinetic theory in curved spaces (search the arXiv: http://lanl.arxiv.org); maybe this is more directly related to your question.

Answer 2

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.