Abstract: The classical theory of the ergodic hypothesis (EH) asserts that to be
ergodic, trajectories may not start everywhere from ergodic surfaces. This
remarkable but abstract ergodic condition has never been understood in
physical terms. In the 2000s the speaker developed a physical theory of EH
for quantum and classical many-body systems known as the ergometric
theory. It does not have this ergodic condition. However when applied to
classical systems to test EH, the ergometric theory agrees with
Birrkhoff's theorem, perhaps the most celebrated math theory on EH.
To trace the ergodic condition, I have turned to trajectories in the
logistic map. According to the theorem of Sharkovskii, chaos exists where
3-cycle exists. To show chaos, one has to first solve for the fixed points
of 3-cycle, which turns out to be a sextic equation. The analytical
solutions allow one to realize chaos and chaotic trajectories via an aleph
cycle. An aleph cycle spans a field of all irrational points in an
interval but not of rational points. One can thus see that the ergodic
condition refers to a set of points of measure zero on ergodic surfaces.
When time averages are actually calculated for dynamical variables of
macroscopic systems, this set of points of measure zero does not
contribute. This is why this particular ergodic condition does not appear
in the ergometric theory.