  There are other approaches to understanding the approach to equilibrium at variance with the approaches that rely on mixing phenomena. O. Lanford, for example, has shown that for an idealized infinitely dilute gas one can show, for very small time intervals, an overwhelmingly likely behavior of the gas according to the Boltzmann equation. Here the interpretation of that equation by the Ehrenfests, the interpretation suitable to the mixing approach, is being dropped in favor of the older idea of the equation describing the overwhelmingly probable evolution of a system. This derivation has the virtue of rigorously generating the Boltzmann equation, but at the cost of applying only to one severely idealized system and then only for a very short time (although the result may be true, if unproven, for longer time scales). Once again an initial probability distribution is still necessary for time asymmetry.