In Lectures on Gas Theory, the Boltzmann equation is derived in Sections 3 and 4 of Chapter I. In Section 5, he replaces the term (f ΄ F1΄ – fF1)dρ/dω in eq. (25) with lf (f ΄ F1΄ – fF1)dρ in eq. (29), etc., where lf is his notation for ln f and dρ=σ2g cosϑdωdω1dλ. This reduces to lf = 1/dω. Having substituted lf for 1/dω, he gets the mathematically inevitable result that he seeks, which is that dH/dt yields an expression that is always less than or equal to zero. This result has nothing to do with physics.
However, there must surely be a dynamical solution to the problem of the second law, since deterministic ideal gas models generate the energy distribution. The third sentence in Chapter I gives an example of the dispersal of energy in an ideal gas, where a hypothetical collision of two molecules of one unit of energy each results in one molecule with zero energy and another with energy = 2. Extrapolating this example illustrates that the maximum energy of a molecule in an ideal gas increases with time. For instance, a subsequent collision of two molecules, each with energy = 2, can result in one molecule with zero energy and another with energy = 4, etc. This process will continue until the maximum molecular energy equals the total system energy E, after which time we will be at or near equilibrium.
Starting from the same example, Boltzmann focuses instead on velocity, from which he seeks to derive a transport equation. One would expect that his velocity model would display the same asymmetry implicit in his energy example. However, on page 41 he says “Each molecule flies from one collision to another one so far away that one can consider the occurrence of another molecule, at the place where it collides the second time, with a definite state of motion, as being an event completely independent (for statistical calculations) of the place from which the first molecule came…” This removes the conditionality implicit in his example. Since conditionality is a necessary property of a transport equation, evolution toward equilibrium cannot occur and only equilibrium can be described. This is apparently why he needed to switch to the H-theorem.
Revised 8/11/22