Reversibility vs. Irreversibility

The seeming paradox of reversible underlying dynamics leading to irreversible macroscopic behavior has been wrestled with since the time of Boltzmann. For the case of gas dynamics, Boltzmann derived his famous equation on the basis of the questionable statistical assumption of “Stosszahlansatz”. From the Boltzmann equation one may derive the “H theorem” which says that the time derivative of the entropy is greater than or equal to zero. The “Stosszahlansatz” or “molecular chaos assumption” says that the probability distributions for colliding molecules should be uncorrelated. As has been pointed out many times (see for example: pp. 46-88 of [Chapman and Cowling, 1958] and pp. 28-32 of the Statistical Physics volume in [Landau and Lifshitz, 1960-1981]), because the underlying dynamics is reversible, for every state with its entropy increasing, there is a corresponding state with its entropy decreasing. In fact, one may show from this argument that the stosszahlansatz can hold only when the time derivative of entropy is zero. There is no intrinsically special direction in time: If one starts with a random state with low entropy, the entropy increases if one follows the evolution of the state either backwards or forwards in time. There are many more states with high entropy than low, and so a system is likely to be in a high entropy state regardless of the time…