  If a macroscopic (random) classical system is put into a random state in
phase space, it will of course the most likely have an almost maximal entropy
according to second law of thermodynamics. We will show, however, the following
theorem: If it is enforced to be periodic with a given period $T$ in advance,
the distribution of the entropy for the otherwise random state will be much
more smoothed out, and the entropy could be very likely much smaller than the
maximal one. Even quantum mechanically we can understand that such a lower than
maximal entropy is likely. A corollary turns out to be that the entropy in such
closed time-like loop worlds remain constant.
