The Kac ring model is used to test the validity of some conjectures about irreversibility. If the whole system is regarded as the universe, then it is demonstrated that all clocks (subsystems) run in the same direction during those times when the universe is not in equilibrium. In addition, mathematical techniques are introduced by means of which the appropriate order parameter for large, finite Kac ring models can be evaluated asymptotically. It is shown that the relaxation of this order parameter to its equilibrium value of zero is not exponential.