Journal of Statistical Physics, Vol. 42, Nos. 5/6, 1986. p. 1115-1125.
Zermelo and Loschmidt pointed out that the equations of classical mechanics are recurrent and reversible, while those of macroscopic physics are nonrecurrent and irreversible. These observations cast doubt on the possibility of deriving the macroscopic equations from classical mechanics. Therefore an example is presented to show that nonrecurrent equations can be derived from recurrent ones, and another example to show that irreversible equations can be derived from reversible ones. The irreversible equation derived in the second example describes either decaying, growing, or undamped motions, depending upon the initial conditions. Thus the specification of initial conditions introduces the irreversibility. These demonstrations may help to clarify previous resolutions of the recurrence and reversibility paradoxes.